Adaptive Bayes exactly tracks information over intrinsic time

# Adaptive Bayes exactly tracks information over intrinsic time
Source: [https://arxiv.org/html/2607.08789](https://arxiv.org/html/2607.08789)
###### Abstract

Bayesian and multiplicative\-weights updates reweight experts, models, or actions from sequential feedback\. We show that the regret of any such update obeys an exact information\-accounting identity\. On each round, the learner’s excess loss to any chosen comparator is the sum of an immediate payment for the uncertainty exposed by the round and a reduction in the information distance from the learner’s current weights to the comparator\. The cumulative payment defines a pathwise uncertainty clock, the*intrinsic time*of the realized sequence\. Summing one\-step balances yields two exact adaptive decompositions of cumulative regret, one for each natural way of composing the update across rounds\. Because the decompositions are exact rather than upper bounds, favorable stochastic or low\-noise regimes appear as self\-bounding properties of the realized intrinsic time, not as slack in worst\-case analyses\. The same calculus covers Hedge, optimistic and side\-information variants, continuous priors, boosting, online convex optimization, contextual bandits, and repeated games: the pathwise account is the same in every case\.

## 1Introduction

Bayesian and multiplicative updates are the main mechanisms for learning from sequential feedback\. In the classical experts problem they reweight experts after each loss vector; in Bayesian model averaging they reweight models after each observation; and in many modern pipelines they reweight candidate actions, hypotheses, or responses after partial evidence\. The common primitive is simple: keep a distribution over candidates, observe feedback, and move mass toward candidates that explain that feedback better\.

The main point of this paper is that the performance of these updates can be written as an exact ledger of information\. On each round, the learner’s excess loss to a chosen benchmark splits into an immediate payment for the uncertainty exposed by the current feedback and a reduction in the information gap to the benchmark\. Summing those one\-step balances produces two exact adaptive decompositions: one for a prior\-retempered update that recomputes the current posterior from the original prior using cumulative scores, and one for a local update that moves only from the current weights using the current score vector\. In both cases, information is the common accounting currency\. It plays many simultaneous roles: measuring comparator complexity, governing learning\-rate effects, and recording how difficult the realized sequence actually was\.

This update template applies not only to prediction with expert advice, but also much more broadly\. It encompasses Bayesian model averaging, optimistic online learning, bandit algorithms, boosting, online convex optimization over continuous domains, repeated\-game play, contextual decision\-making, multiscale aggregation, and softmax preference learning\. All are instances of a single Bayes\-rule update with different choices of scores, side information, and learning rate\.

On roundttthe learner chooses weightspt∈Δ​\(\[K\]\)p\_\{t\}\\in\\Delta\(\[K\]\)overKKexperts, observes expert lossesℓt∈ℝK\\ell\_\{t\}\\in\\mathbb\{R\}^\{K\}, and incurs mixture loss⟨pt,ℓt⟩\\langle p\_\{t\},\\ell\_\{t\}\\rangle\. For a comparator distributionρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\), the cumulative regret up to timeTTis

RTℓ​\(ρ\):=∑t=1T⟨pt,ℓt⟩−⟨ρ,LT⟩LT:=∑t=1TℓtR\_\{T\}^\{\\ell\}\(\\rho\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}\\rangle\\qquad\\qquad L\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}Point\-mass comparators recover the usual regret to a single expert, while diffuseρ\\rhocompare to arbitrary mixtures\. A comparator is simply the benchmark distribution we want to match in hindsight, and regret is the learner’s extra cumulative loss relative to that benchmark\. Throughout the paper the loss sequence may be arbitrary, so every statement is pathwise, over the realized trajectory of the learner\.

The classic Hedge/exponential\-weights algorithm\[[27](https://arxiv.org/html/2607.08789#bib.bib27)\]is the special casect=ℓtc\_\{t\}=\\ell\_\{t\}\. More generally, one may first incorporate side information through positive factorsst​\(i\)s\_\{t\}\(i\)by settingut​\(i\):=−ηt−1​log⁡st​\(i\)u\_\{t\}\(i\):=\-\\eta\_\{t\}^\{\-1\}\\log s\_\{t\}\(i\)andct​\(i\):=ℓt​\(i\)\+ut​\(i\)c\_\{t\}\(i\):=\\ell\_\{t\}\(i\)\+u\_\{t\}\(i\)\. This simple composite\-loss reduction already covers optimistic methods, compensators, side priors, and several partial\-information reductions\[[39](https://arxiv.org/html/2607.08789#bib.bib39),[15](https://arxiv.org/html/2607.08789#bib.bib15),[70](https://arxiv.org/html/2607.08789#bib.bib70)\]\. Once that reduction is made, the rest of the analysis depends only on the induced sequencectc\_\{t\}and the chosen prior\.

This viewpoint is especially natural in settings where one repeatedly reweights proposals, actions, or hypotheses from sequential evidence\. It underlies optimistic repeated\-game play\[[39](https://arxiv.org/html/2607.08789#bib.bib39),[15](https://arxiv.org/html/2607.08789#bib.bib15)\], adaptive step\-size selection in online and stochastic optimization\[[22](https://arxiv.org/html/2607.08789#bib.bib22)\], and, more recently, KL\-regularized policy updates in reinforcement learning from human feedback\[[59](https://arxiv.org/html/2607.08789#bib.bib59)\]and related post\-training pipelines for language models, where the same softmax reweighting step governs how candidate responses are promoted or suppressed by sequential evidence\.

##### Relation to game\-theoretic concentration\.

The present paper is self\-contained, but the same objects also have a complementary game\-theoretic interpretation\. A complementary reading views a Bayes/exponential\-weights round as a one\-step game in which the learner commits to a distributionptp\_\{t\}and a temperatureηt\\eta\_\{t\}, nature reveals a centered score vector with a constrained one\-scale cumulant generating function \(CGF\), and the terminal payoff is the support function of a KL ball around the prior\. Under that interpretation, the per\-round CGF increment is the intrinsic\-time incrementQt​\(c\)Q\_\{t\}\(c\)analyzed here, the log\-partition value function is the terminal potentialAt​\(η\)A\_\{t\}\(\\eta\)of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2), and the exact retempering driftDTD\_\{T\}is the price of switching between fixed\-scale games as the temperature changes\. Thus the present paper can be read simply as the pathwise information\-accounting calculus; the game formulation is an optional dual viewpoint\.

Once written in this form, the prior plays two roles\. It initializes the update, and it also serves as a decoder for the comparator class\. For a point comparator concentrated on expertii, the description cost is−log⁡π​\(i\)\-\\log\\pi\(i\); for a mixed comparatorρ\\rhoit isKL​\(ρ∥π\)\\mathrm\{KL\}\(\\rho\\\|\\pi\)\. The paper therefore measures difficulty relative to the realized sequence and the chosen decoder, not relative to any assumed data\-generating law\[[36](https://arxiv.org/html/2607.08789#bib.bib36)\]\. This is close in spirit to the predictive\-complexity and game\-theoretic traditions of online learning, where one fixes the sequential prediction rule and asks how hard the realized path is for that rule\[[19](https://arxiv.org/html/2607.08789#bib.bib19),[73](https://arxiv.org/html/2607.08789#bib.bib73),[52](https://arxiv.org/html/2607.08789#bib.bib52),[43](https://arxiv.org/html/2607.08789#bib.bib43),[74](https://arxiv.org/html/2607.08789#bib.bib74),[67](https://arxiv.org/html/2607.08789#bib.bib67)\]\.

### 1\.1The common mechanism: one\-step information balance

The heart of the paper is a single one\-step identity\. Ifqtq\_\{t\}is the current distribution,qt\+1q\_\{t\+1\}is the next distribution produced by exponential reweighting on composite lossesctc\_\{t\}, andρ\\rhois any comparator, then

⟨qt,ct⟩−⟨ρ,ct⟩=δt​\(c\)\+KL​\(ρ∥qt\)−KL​\(ρ∥qt\+1\)ηt\\langle q\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle=\\delta\_\{t\}\(c\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|q\_\{t\}\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{t\+1\}\)\}\{\\eta\_\{t\}\}\(1\)whereδt​\(c\)\\delta\_\{t\}\(c\)is the round’s immediate payment: the gap between the learner’s mean composite loss and the centered logarithmic mix loss for that round\. Read from left to right, the identity says that excess loss is exactly split into what is paid now and what is converted into a reduction of the information gap to the comparator\. No approximation has yet been made\.

This one\-step balance is the paper’s basic accounting statement\. It holds for every Bayesian update over experts, regardless of how the temperature is chosen\. At fixed temperature the KL term telescopes, giving the familiar fixed\-rate exact identities\. When the temperature varies, the same one\-step balance can be composed in two natural ways, depending on how the new temperature enters the update\.

### 1\.2Two ways to compose the same update

Once the temperature varies, two natural adaptive updates appear\. The first is theprior\-retempered update, which recomputes the current posterior from the original prior at the current temperature:

pt​\(i\)∝π​\(i\)​e−ηt​Ct−1​\(i\)Ct​\(i\):=∑s=1tcs​\(i\)p\_\{t\}\(i\)\\propto\\pi\(i\)e^\{\-\\eta\_\{t\}C\_\{t\-1\}\(i\)\}\\qquad\\qquad C\_\{t\}\(i\):=\\sum\_\{s=1\}^\{t\}c\_\{s\}\(i\)Its exact decomposition \(Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)\) says that for every comparatorρ\\rhoand timeTT,

RTc​\(ρ\)=∑t=1Tηt​Qt​\(c\)⏟intrinsic\-time payment\+DT⏟drift from changing temperature\+BT​\(ρ\)⏟terminal comparator informationR\_\{T\}^\{c\}\(\\rho\)=\\underbrace\{\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\}\_\{\\textup\{intrinsic\-time payment\}\}\+\\underbrace\{D\_\{T\}\}\_\{\\textup\{drift from changing temperature\}\}\+\\underbrace\{B\_\{T\}\(\\rho\)\}\_\{\\textup\{terminal comparator information\}\}\(2\)HereQt​\(c\)≥0Q\_\{t\}\(c\)\\geq 0is the exact round\-difficulty increment revealed by the update itself\. The cumulative quantityVT​\(c\):=∑t=1TQt​\(c\)V\_\{T\}\(c\):=\\sum\_\{t=1\}^\{T\}Q\_\{t\}\(c\)is the*intrinsic time*: the uncertainty clock exposed by the realized path rather than imposed from outside\.

The second is thelocal update,

pt\+1​\(i\)∝pt​\(i\)​e−ηt​ct​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}which updates only from the current weights and the current score vector\. Its exact cumulative identity \(Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13)\) is built from the same one\-step paymentQt​\(c\)Q\_\{t\}\(c\), but the cumulative bookkeeping is different: instead of a temperature\-drift term, it produces an exact cumulative normalization identity, and for comparator classes it is naturally expressed through the terminal masspT\+1​\(E\)p\_\{T\+1\}\(E\)assigned to an event or classEE\.

The practical distinction mirrors the algebra, with different learning rate schedules being appropriate for the two updates\. The prior\-retempered update pairs naturally with a monotone second\-order schedule driven by cumulative intrinsic time\. The local update pairs naturally with a “pressure\-target" line search that chooses the current temperature so that the one\-step normalized loss hits a prescribed target\. Algorithm[1](https://arxiv.org/html/2607.08789#alg1)makes the two update families explicit\. At fixed temperature the two updates agree; under adaptive rates they genuinely diverge, even though the same one\-step information balance drives both of them\.

Algorithm[1](https://arxiv.org/html/2607.08789#alg1)displays the full2×22\\times 2design space\. The update geometry \(prior\-retempered versus local\) and the rate controller \(square\-root clock versus pressure target\) are distinct choices\. We refer to the four cells asRet\-Sqrt,Ret\-Press,Loc\-Sqrt, andLoc\-Press\. The paper’s main online pairings areRet\-SqrtandLoc\-Press; the two crossed pairings are recorded explicitly in Section[3\.1](https://arxiv.org/html/2607.08789#S3.SS1)\.

Algorithm 1The2×22\\times 2adaptive Bayes design space\. Rows choose the update geometry; columns choose the learning\-rate controller\.qt,η​\(i\):=π​\(i\)​e−η​Ct​\(i\)∑jπ​\(j\)​e−η​Ct​\(j\),At​\(η\):=−η−1​log​∑iπ​\(i\)​e−η​Ct​\(i\)\.q\_\{t,\\eta\}\(i\):=\\frac\{\\pi\(i\)e^\{\-\\eta C\_\{t\}\(i\)\}\}\{\\sum\_\{j\}\\pi\(j\)e^\{\-\\eta C\_\{t\}\(j\)\}\},\\qquad A\_\{t\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\sum\_\{i\}\\pi\(i\)e^\{\-\\eta C\_\{t\}\(i\)\}\.

HereΓ\>0\\Gamma\>0is a comparator\-complexity budget and, unless a different constant is stated, the square\-root controller uses the optimized valueC=1/2C=1/\\sqrt\{2\}from Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)\. The exact intrinsic incrementQt​\(c\)Q\_\{t\}\(c\)is defined in Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2);Vt​\(c\)V\_\{t\}\(c\)is the exact cumulant clock, whileWt​\(c\)W\_\{t\}\(c\)denotes a quadratic relaxation used only as an optional controller or concentration scale\. For the pressure cells, the targetata\_\{t\}is a one\-round free\-energy level\. In the local row the line search is causal becauseptp\_\{t\}is already fixed beforectc\_\{t\}is observed\. In the prior\-retempered row the same line search naturally sets the next temperatureηt\+1\\eta\_\{t\+1\}, since the current action distribution itself depends on the temperature\.

### 1\.3The meaning of intrinsic time

The meaning of \([2](https://arxiv.org/html/2607.08789#S1.E2)\) is straightforward\. One part of regret,∑tηt​Qt​\(c\)\\sum\_\{t\}\\eta\_\{t\}Q\_\{t\}\(c\), is the cumulative price of online uncertainty over the measured candidates\. Another part is the distortion introduced by changing the strength of the update, namelyDTD\_\{T\}\. The last part is the information needed to specify the comparator relative to the prior, namelyBT​\(ρ\)B\_\{T\}\(\\rho\)\. Thus the central question is not only how large regret can be, but how the realized sequence splits into online uncertainty, schedule\-induced distortion, and comparator description cost\. The paper keeps this exact variational form intact for as long as possible: up through specific schedule\-dependent calculations, the main statements are identities or exact two\-sided envelopes, and one\-sided inequalities enter only when the exact terms are deliberately relaxed to simpler online surrogates\.

It is important to keep two clocks separate\. The exact intrinsic clock isVt​\(c\):=∑s=1tQs​\(c\)V\_\{t\}\(c\):=\\sum\_\{s=1\}^\{t\}Q\_\{s\}\(c\), where each incrementQs​\(c\)Q\_\{s\}\(c\)is the finite\-temperature cumulant actually paid by the update\. A useful quadratic relaxation isWt​\(c\):=∑s=1t12​Vari∼ps​\(cs​\(i\)\)W\_\{t\}\(c\):=\\sum\_\{s=1\}^\{t\}\\tfrac\{1\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{s\}\}\(c\_\{s\}\(i\)\), which keeps only the ordinary second\-order statistics of the played distributions; the factor12\\tfrac\{1\}\{2\}is dictated by the small\-temperature expansionQs​\(c\)=12​Vari∼ps​\(cs​\(i\)\)\+o​\(1\)Q\_\{s\}\(c\)=\\tfrac\{1\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{s\}\}\(c\_\{s\}\(i\)\)\+o\(1\), so thatWt​\(c\)W\_\{t\}\(c\)is the leading\-order Taylor relaxation ofVt​\(c\)V\_\{t\}\(c\)\. The two coincide only in the small\-temperature limit\. One may drive a schedule by either clock, but onlyVt​\(c\)V\_\{t\}\(c\)appears in the exact identities;Wt​\(c\)W\_\{t\}\(c\)is a practical surrogate, not the quantity being paid\. A third, distinct object appears only in the concentration statements of Section[3\.2](https://arxiv.org/html/2607.08789#S3.SS2): the*predictable quadratic variation*Vt⊔:=∑s=1tVari∼ps​\(cs​\(i\)\)=2​Wt​\(c\)V^\{\\\!\\sqcup\}\_\{t\}:=\\sum\_\{s=1\}^\{t\}\\mathrm\{Var\}\_\{i\\sim p\_\{s\}\}\(c\_\{s\}\(i\)\)=2\\,W\_\{t\}\(c\)of the sampling martingale, which carries no factor12\\tfrac\{1\}\{2\}because it is the Bernstein variance of the increments rather than a relaxation of the paid clock\.

The intrinsic\-time incrementQt​\(c\)Q\_\{t\}\(c\)is therefore not a proxy for a variance term\. Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)shows that it is an exact average of tilted variances generated by the actual Bayes\-rule update on that round, verified to machine precision across a synthetic family of score configurations in Appendix[C\.3\.3](https://arxiv.org/html/2607.08789#A3.SS3.SSS3)\. In that sense, the algorithm does not try to approximate some hidden difficulty parameter; it*reveals*the structure of the realized path\. Changing the proposal distribution changes the very variance geometry being measured, rather than merely improving a loose upper bound\.

The same per\-round information functional—the intrinsic\-time incrementQt​\(c\)Q\_\{t\}\(c\), read as a centered finite\-temperature cumulant of the current played distribution—appears in both the retempered and local decompositions, though with different prefactors and different cumulative bookkeeping\. This is the precise sense in which the two update families share one mechanism: the round\-by\-round information cost is computed by the*same functional of*\(pt,ct\)\(p\_\{t\},c\_\{t\}\)in both, and only the way it accumulates differs\. The realized values ofQt​\(c\)Q\_\{t\}\(c\)are nonetheless algorithm\-dependent, since under a variable schedule the two recursions generally play differentptp\_\{t\}on the same loss path; the two coincide round\-by\-round only when the temperature schedule is fixed, so that the played distributions agree\.

### 1\.4Side information and a unified perspective

The exact decomposition also dissolves several distinctions that are usually presented as separate tradeoffs\. TableLABEL:tab:tradeoffssummarizes some of these\. While many individual pieces of the picture have appeared before, a contribution of this paper is to place them inside one exact accounting framework\. That perspective will be revisited in the related\-work section, where the rows of the table are used to organize the surrounding literature\.

Table 1:Supposed dichotomies of online learning, and their explanations through the lens of information\.Apparent tradeoffConventional manifestationExact informational formHard vs\. easy sequencesWorst\-case and stochastic analyses require separate theorems\.Hardness is relative to a comparator: the same identity yields worst\-caseO​\(VT\)O\(\\sqrt\{V\_\{T\}\}\)or fast rates depending on how much the intrinsic time self\-bounds\. Pure noise remains hard across comparator choices\.Second\-order vs\. first\-order boundsSecond\-order \(variance\-sensitive\) bounds need extra assumptions or different algorithms\.First\-, second\-, and higher\-order bounds are successive Taylor relaxations of the same exact cumulant term; the order is chosen by the analyst, not the algorithm\.Different comparator classesPoint experts, mixtures, and/or sleeping/shifting comparators need separate analyses\.Every comparatorρ\\rhoincurs regret according to its information\-theoretic costKL​\(ρ∥π\)\\mathrm\{KL\}\(\\rho\\\|\\pi\); richer classes simply have larger terminal terms in the same identity\.Parameter\-free vs\. adaptive“Parameter\-free” methods are presented as removing tuning, while adaptive methods still expose learning\-rate or regularization choices\.Budgets, temperatures, and targets are observable constraints on comparator complexity or revealed difficulty\. Outer controllers may learn them online, but the same exact identity underlies both the tuned and simultaneous versions\.Algorithmic upper vs\. lower envelopesAlgorithmic penalty upper bounds and actual incurred penalties are usually analyzed separately, and the gap between them is hidden inside worst\-case inequalities\.For any realized path, the algorithm’s internal payment decomposes exactly\. There is no slack at the identity level, and any later gap comes only from deliberately relaxing exact terms such asQt​\(c\)Q\_\{t\}\(c\)orDTD\_\{T\}\.Variance definitionsMultiple variance notions \(empirical, conditional, quadratic\) appear in different results\.All approximate the exact intrinsic\-time density: the algorithm’s own play\-conditional cumulantQt​\(c\)Q\_\{t\}\(c\)\.Time horizonBounds depend on the horizonTT, which must be known or guessed\.The intrinsic\-time processVT​\(c\)V\_\{T\}\(c\)replaces the horizon; adaptive schedules track it online without knowingTT\.Bounded vs\. unbounded lossesBounded losses \(e\.g\.ct​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\]\) are needed for low regret; boundedness and Taylor expansions go hand in hand to control the cumulant generating function\.The exact identity requires only that the one\-step log\-normalizer is finite at the chosen temperature\. Boundedness enters only when relaxing the exact cumulantQt​\(c\)Q\_\{t\}\(c\)via a Taylor or range\-based bound; without such relaxations, the pathwise identity itself remains meaningful whenever the exponential moment exists\.Full vs\. partial feedbackBandit and partial\-feedback settings require fundamentally different algorithms and analyses\.The same identity applies to estimated losses; the only new terms are an explicit predictable bias correction and martingale terms from the estimation procedure\. The intrinsic time is computed on the estimates, and graph or sparsity structure enters only through the observation ratios that govern those corrections\.The composite\-loss viewpoint is especially useful and general when some part of the next loss vector is predictable\. Ifmt∈ℝKm\_\{t\}\\in\\mathbb\{R\}^\{K\}is a forecast available before roundtt, takingut=−mtu\_\{t\}=\-m\_\{t\}yields the optimistic update\. The composite loss becomesct=ℓt−mtc\_\{t\}=\\ell\_\{t\}\-m\_\{t\}, so only the forecast residual enters the intrinsic\-time cost\. The original\-loss regret then separates into an online term for the unpredictable residual and an explicit predictable mismatch term\. This is the form used by optimistic Hedge in adaptive game playing, model\-based repeated games, and other settings where one predicts the opponent or the environment before acting\[[39](https://arxiv.org/html/2607.08789#bib.bib39),[15](https://arxiv.org/html/2607.08789#bib.bib15)\]\. Good side information does not change the exact theorem; it changes the sequence to which the theorem is applied\.

The main decomposition is distribution\-free\. A*stochastic\-luckiness*result adds a second, optional interpretation\. It says that if the realized sequence happens to satisfy a self\-bounding relation — roughly, if the same intrinsic\-time term that drives the adversarial bound is controlled by the comparator’s own excess loss — then the typicalO​\(VT\)O\(\\sqrt\{V\_\{T\}\}\)scaling of regret collapses to a constant or other fast rate\. This handles adversarial robustness and stochastic easiness in one stroke, on the observed sequence\. The algorithm remains the same, but the realized sequence turns out to be easy for the chosen decoder\. Section[4\.5](https://arxiv.org/html/2607.08789#S4.SS5)develops this comparator\-centered fast\-rate interpretation in PAC\-Bayesian form\.

A useful historical reference point is the scaling\-time question in\[[25](https://arxiv.org/html/2607.08789#bib.bib25)\], which asked for regret guarantees controlled by internal variance\-like quantities rather than only horizon and number of experts\. Our starting point is to derive the exact identity first and only then upper bound it\. That is what makes the result simultaneous over distributional comparators, directly compatible with side information, and naturally interpretable as an individual\-sequence complexity statement rather than only as a worst\-case upper bound\.

Adaptive online optimization methods choose step sizes from realized geometry rather than from horizon alone\[[76](https://arxiv.org/html/2607.08789#bib.bib76),[22](https://arxiv.org/html/2607.08789#bib.bib22)\]\. The same philosophy appears here, but with entropic geometry rather than Euclidean geometry\. Two contrasting learning\-rate families are studied in Section[3](https://arxiv.org/html/2607.08789#S3): one chooses the temperature from the intrinsic\-time process, and the other uses the current score geometry to hit a prescribed pressure target\. In that sense this paper gives an entropic analogue of adaptive step\-size selection: the update strength is chosen from the difficulty already revealed by the sequence\.

This viewpoint also clarifies the usual distinction between*parameter\-free*and*adaptive*\. In our notation, quantities such as a comparator budgetΓ\\Gammaor a pressure targetata\_\{t\}are not hidden nuisance parameters; they encode observable constraints or goals against which the information flow is being measured\. One may learn them online by an outer controller—for instance, Section[4\.4](https://arxiv.org/html/2607.08789#S4.SS4)aggregates a grid of budgets—but the underlying Bayes update and exact regret identity are the same either way\.

Viewed this way, optimistic updates, control\-variate corrections, compensators, and side priors are not separate tricks layered on top of Hedge\. They change which part of the round’s information is treated as predictable and which part remains to be learned online\. In the composite\-loss notation, all of these variants feed the same Bayes update a different residual sequence\. When the look\-ahead signal is accurate, that residual is genuinely easier, the intrinsic\-time clock slows down, and the exact regret identity records the improvement directly\. Lower regret in these variants is therefore a statement about reduced unexplained information, not merely about a more convenient proof or a clever analytic trick\. This is reflected in the many information\-theoretic identities and equalities in this paper, which persist over any trajectory\.

### 1\.5Contributions

The paper contributes five main pieces\.

1. 1\.A generic Bayes\-rule reduction\.Section[2\.1](https://arxiv.org/html/2607.08789#S2.SS1)converts multiplicative evidence, side priors, and predictable corrections into additive composite losses\. This is not only an analysis device: it identifies the residual information that still has to be learned online\. After the reduction, the rest of the analysis depends only on the induced sequencectc\_\{t\}and the prior complexity term\.
2. 2\.One shared one\-step balance, two parallel three\-piece cumulative decompositions\.Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)proves the exact prefix and terminal identities for the prior\-retempered updatept​\(i\)∝π​\(i\)​e−ηt​Ct−1​\(i\)p\_\{t\}\(i\)\\propto\\pi\(i\)e^\{\-\\eta\_\{t\}C\_\{t\-1\}\(i\)\}, splitting regret into intrinsic\-time payment, temperature\-change drift, and terminal comparator\-information terms\. Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)proves a structurally parallel three\-piece identity for the local updatept\+1​\(i\)∝pt​\(i\)​e−ηt​ct​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}under pressure\-target schedules: the same intrinsic\-time payment, an Abel\-form transport drift, and a terminal\-mass partition\. Both updates use the same composite\-loss reduction and the same per\-round information functional \(a centered cumulant of the current played distribution\); only the cumulative bookkeeping differs, and at fixed temperature, where the two recursions play the sameptp\_\{t\}, they collapse to one identity\.
3. 3\.Schedule design from the same information clock\.Section[3](https://arxiv.org/html/2607.08789#S3)studies two natural controller families\. The prior\-retempered update supports an intrinsic\-time square\-root schedule, driven either by the exact clockVt​\(c\)V\_\{t\}\(c\)or by the quadratic relaxationWt​\(c\)W\_\{t\}\(c\), yielding pathwise sampled\-expert identities, high\-probability and anytime consequences, upper law\-of\-the\-iterated\-logarithm behavior, and optimistic corollaries through predictable side information\. The local update supports a pressure\-target schedule∑ipt​\(i\)​e−ηt​\(ct​\(i\)−at\)=1\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}\(c\_\{t\}\(i\)\-a\_\{t\}\)\}=1, yielding exact weighted and class\-conditioned identities that are especially natural in boosting\-style applications\.
4. 4\.Comparator\-rich and side\-information extensions\.Later sections return to the original losses, derive shifting\-comparator and simultaneous\-quantile theorems, show how comparator\-centered low\-noise conditions turn the same pathwise identity into fast PAC\-Bayes rates, and transfer the calculus to repeated games, bandits, feedback graphs, contextual bandits, continuous\-action online convex optimization \(OCO\), boosting, and logarithmic pooling\.
5. 5\.An empirical diagnostic viewpoint, with numerical evidence\.Section[7](https://arxiv.org/html/2607.08789#S7)defines the regret\-decomposition plot derived from the prefix split and uses it to distinguish stochastic, adversarial, and mixed individual\-sequence regimes\. It then carries out the protocol numerically: the predicted decomposition signatures appear cleanly on each of the four canonical sequence families; the realized intrinsic\-time paymentPTP\_\{T\}is shown to track the lower side of the two\-sided envelope of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)on cycling\-adversarial paths \(the empirical ratioPT/Γ​VT​\(c\)P\_\{T\}/\\sqrt\{\\Gamma V\_\{T\}\(c\)\}approaches the asymptotic envelope coefficient2​C≈1\.412C\\approx 1\.41asTTgrows\); the paper’s two schedules are compared against AdaHedge, NormalHedge, AdaNormalHedge, FTRL\-Tsallis, and Squint across twelve seeds with a broken\-axis presentation that keeps the paper algorithms legible; and the shifting\-comparator analysis of Section[4\.3](https://arxiv.org/html/2607.08789#S4.SS3)is exhibited on a planted\-change\-point sequence\.

Whenever a numerical default is needed for the square\-root schedule, we useC=1/2C=1/\\sqrt\{2\}, which minimizes the leading coefficient2​C\+C−12C\+C^\{\-1\}in \([28](https://arxiv.org/html/2607.08789#S3.E28)\); the identities and envelopes keepC\>0C\>0symbolic\.

This paper can therefore be read as a study of how sequential information is spent and gained by Bayes/exponential weights\. The composite\-loss reduction determines what is encoded into the composite losses\. The two update families determine how that evidence is composed over time\. The schedule determines how aggressively that evidence is trusted\. And the theorem records—exactly—what is paid immediately, what is shifted by changing the temperature, and what is ultimately charged to comparator complexity\.

##### Guide to the paper\.

Section[2](https://arxiv.org/html/2607.08789#S2)develops the core identities, beginning with the mixed\-coincidence reduction that converts side information into composite losses and continuing with the shared one\-step information balance and its two cumulative forms: the prior\-retempered update of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)and the local update of Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)\. Section[3](https://arxiv.org/html/2607.08789#S3)first records the full2×22\\times 2controller–update design space, then studies the two schedule families that naturally accompany the main pairings: the intrinsic\-time second\-order rule for the prior\-retempered update and the pressure\-target line search for the local update\. Section[4](https://arxiv.org/html/2607.08789#S4)returns to the original losses, compensators, richer comparators, and stochastic luckiness\. Section[5](https://arxiv.org/html/2607.08789#S5)collects further extensions to weighted\-entropy geometry, repeated games, continuum priors, continuous\-action OCO, and boosting\. Section[6](https://arxiv.org/html/2607.08789#S6)treats partial feedback with bandits\. Section[7](https://arxiv.org/html/2607.08789#S7)outlines an empirical protocol based on the prefix decomposition and reports numerical results that exhibit the predicted diagnostic signatures, validate the second\-order envelope of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2), and benchmark the paper’s schedules against current adaptive online learning baselines\. Section[8](https://arxiv.org/html/2607.08789#S8)closes with related work and discussion\. Appendix[A](https://arxiv.org/html/2607.08789#A1)records the connection to logarithmic pooling\.

##### Notation\.

The finite index set\[K\]:=\{1,…,K\}\[K\]:=\\\{1,\\dots,K\\\}may represent experts, actions, models, or hypotheses depending on the application\. Forq∈Δ​\(\[K\]\)q\\in\\Delta\(\[K\]\)andx∈ℝKx\\in\\mathbb\{R\}^\{K\}, we write⟨q,x⟩=∑i=1Kq​\(i\)​x​\(i\)\\langle q,x\\rangle=\\sum\_\{i=1\}^\{K\}q\(i\)x\(i\)\. The relative entropy isKL​\(q∥p\):=∑i=1Kq​\(i\)​log⁡q​\(i\)p​\(i\)\\mathrm\{KL\}\(q\\\|p\):=\\sum\_\{i=1\}^\{K\}q\(i\)\\log\\frac\{q\(i\)\}\{p\(i\)\}, with the usual convention thatKL​\(q∥p\)=\+∞\\mathrm\{KL\}\(q\\\|p\)=\+\\inftyunlessqqis absolutely continuous with respect topp\. We use the standard total variation distanceTV​\(q,p\):=12​∑i=1K\|q​\(i\)−p​\(i\)\|\\mathrm\{TV\}\(q,p\):=\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{K\}\|q\(i\)\-p\(i\)\|\. Ifμ\\muis a strictly positive measure on\[K\]\[K\], we writeH​\(q,μ\):=−∑i=1Kq​\(i\)​log⁡μ​\(i\)\\mathrm\{H\}\(q,\\mu\):=\-\\sum\_\{i=1\}^\{K\}q\(i\)\\log\\mu\(i\)for the generalized cross\-entropy; whenμ\\muis a probability distribution, this is the usual cross\-entropy\. We use PAC\-Bayes terminology when convenient, but throughout the main online\-learning statements the symbolρ\\rhodenotes an arbitrary comparison distribution\. The main algorithms use the notationptp\_\{t\}for the learner’s weights; generic one\-step identities useqtq\_\{t\}; and temperature\-indexed posteriors are written asqt,ηq\_\{t,\\eta\}\.

## 2Core identities

This section isolates the common one\-round mechanism and the first cumulative decomposition built from it\. Once side information has been absorbed into a composite loss, each round’s excess loss equals an immediate centered cumulant cost plus a change in relative entropy to the comparator\. What changes under adaptive temperatures is not the one\-step identity but the way those KL transport terms compose over time\. For the prior\-retempered update, they organize into an exact cumulative chain with an explicit drift term\. For the local recursion, the natural cumulative object is different, and we treat it separately in Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)together with the pressure\-target schedule\.

### 2\.1The composite\-loss reduction

The mixed\-coincidence identity is the paper’s gateway from side information to an ordinary additive loss sequence\. In static form it rewrites a geometric pool of measures as an entropy\-corrected variational problem\. In sequential form it shows that multiplicative side factors, predictable corrections, and ordinary losses can all be folded into a single composite loss sequence\. Once that reduction is made, both adaptive updates studied later operate on exactly the same objects\.

###### Theorem 2\.1\(Finite mixed coincidence identity\)\.

Letμ1,…,μW\\mu\_\{1\},\\dots,\\mu\_\{W\}be strictly positive measures on a finite set𝒳\\mathcal\{X\}, and letα=\(α1,…,αW\)∈ℝW\\alpha=\(\\alpha\_\{1\},\\dots,\\alpha\_\{W\}\)\\in\\mathbb\{R\}^\{W\}\. Define

Z​\(α\):=∑x∈𝒳∏w=1Wμw​\(x\)αw,pα⋆​\(x\):=∏w=1Wμw​\(x\)αwZ​\(α\)Z\(\\alpha\):=\\sum\_\{x\\in\\mathcal\{X\}\}\\prod\_\{w=1\}^\{W\}\\mu\_\{w\}\(x\)^\{\\alpha\_\{w\}\},\\qquad p^\{\\star\}\_\{\\alpha\}\(x\):=\\frac\{\\prod\_\{w=1\}^\{W\}\\mu\_\{w\}\(x\)^\{\\alpha\_\{w\}\}\}\{Z\(\\alpha\)\}Then for everyq∈Δ​\(𝒳\)q\\in\\Delta\(\\mathcal\{X\}\),

−log⁡Z​\(α\)\+KL​\(q∥pα⋆\)=∑w=1Wαw​H​\(q,μw\)−H​\(q\)\-\\log Z\(\\alpha\)\+\\mathrm\{KL\}\(q\\\|p^\{\\star\}\_\{\\alpha\}\)=\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{H\}\(q,\\mu\_\{w\}\)\-\\mathrm\{H\}\(q\)\(3\)whereH​\(q\):=−∑xq​\(x\)​log⁡q​\(x\)\\mathrm\{H\}\(q\):=\-\\sum\_\{x\}q\(x\)\\log q\(x\)is Shannon entropy\. If, in addition, eachμw\\mu\_\{w\}is a probability distribution andα∈Δ​\(\[W\]\)\\alpha\\in\\Delta\(\[W\]\), then

−log⁡Z​\(α\)=minq∈Δ​\(𝒳\)​∑w=1Wαw​KL​\(q∥μw\)\-\\log Z\(\\alpha\)=\\min\_\{q\\in\\Delta\(\\mathcal\{X\}\)\}\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{KL\}\(q\\\|\\mu\_\{w\}\)and the unique minimizer isq=pα⋆q=p^\{\\star\}\_\{\\alpha\}\.

To pass to the online setting, interpret one factor as the current posterior, one as the round\-loss update, and the rest as side information\. The resulting step is exactly Bayes/exponential\-weights updating with a predictable compensator\. From this one\-round identity we derive, in the remainder of this section, the shared one\-step information balance; Sections[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)and[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)then compose it into the two distinct cumulative identities — retempered and local — that give the paper’s two pathwise decompositions\.

###### Theorem 2\.2\(Sequential mixed coincidence identity\)\.

Fix a sequence of learner distributions\(qt\)t≥1\(q\_\{t\}\)\_\{t\\geq 1\}on\[K\]\[K\], lossesℓt∈ℝK\\ell\_\{t\}\\in\\mathbb\{R\}^\{K\}, predictable learning ratesηt\>0\\eta\_\{t\}\>0, and positive side factorsst∈\(0,∞\)Ks\_\{t\}\\in\(0,\\infty\)^\{K\}\. Consider the update

qt\+1​\(i\)=qt​\(i\)​e−ηt​ℓt​\(i\)​st​\(i\)Zt,Zt:=∑j=1Kqt​\(j\)​e−ηt​ℓt​\(j\)​st​\(j\)q\_\{t\+1\}\(i\)=\\frac\{q\_\{t\}\(i\)e^\{\-\\eta\_\{t\}\\ell\_\{t\}\(i\)\}s\_\{t\}\(i\)\}\{Z\_\{t\}\},\\qquad Z\_\{t\}:=\\sum\_\{j=1\}^\{K\}q\_\{t\}\(j\)e^\{\-\\eta\_\{t\}\\ell\_\{t\}\(j\)\}s\_\{t\}\(j\)\(4\)Define the side offset and the composite loss byut​\(i\):=−ηt−1​log⁡st​\(i\)u\_\{t\}\(i\):=\-\\eta\_\{t\}^\{\-1\}\\log s\_\{t\}\(i\)andct​\(i\):=ℓt​\(i\)\+ut​\(i\)c\_\{t\}\(i\):=\\ell\_\{t\}\(i\)\+u\_\{t\}\(i\)\. Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

−1ηt​log⁡Zt\+1ηt​KL​\(ρ∥qt\+1\)=1ηt​KL​\(ρ∥qt\)\+⟨ρ,ct⟩\-\\frac\{1\}\{\\eta\_\{t\}\}\\log Z\_\{t\}\+\\frac\{1\}\{\\eta\_\{t\}\}\\mathrm\{KL\}\(\\rho\\\|q\_\{t\+1\}\)=\\frac\{1\}\{\\eta\_\{t\}\}\\mathrm\{KL\}\(\\rho\\\|q\_\{t\}\)\+\\langle\\rho,c\_\{t\}\\rangle\(5\)If the side factor decomposes as

st​\(i\)=∏w=1Jtμt,w​\(i\)αt,ws\_\{t\}\(i\)=\\prod\_\{w=1\}^\{J\_\{t\}\}\\mu\_\{t,w\}\(i\)^\{\\alpha\_\{t,w\}\}\(6\)for strictly positive measuresμt,w\\mu\_\{t,w\}and coefficientsαt,w∈ℝ\\alpha\_\{t,w\}\\in\\mathbb\{R\}\(withw=1,…,Jtw=1,\\dots,J\_\{t\}\), then

−1ηt​log⁡Zt\+1ηt​KL​\(ρ∥qt\+1\)=1ηt​KL​\(ρ∥qt\)\+⟨ρ,ℓt⟩\+∑w=1Jtαt,wηt​H​\(ρ,μt,w\)\-\\frac\{1\}\{\\eta\_\{t\}\}\\log Z\_\{t\}\+\\frac\{1\}\{\\eta\_\{t\}\}\\mathrm\{KL\}\(\\rho\\\|q\_\{t\+1\}\)=\\frac\{1\}\{\\eta\_\{t\}\}\\mathrm\{KL\}\(\\rho\\\|q\_\{t\}\)\+\\langle\\rho,\\ell\_\{t\}\\rangle\+\\sum\_\{w=1\}^\{J\_\{t\}\}\\frac\{\\alpha\_\{t,w\}\}\{\\eta\_\{t\}\}\\mathrm\{H\}\(\\rho,\\mu\_\{t,w\}\)\(7\)

The next corollary rewrites the same identity in the mean\-centered form that is most convenient for composing into cumulative regret; it is the form to which both cumulative decompositions of the paper are applied\. The original form \([5](https://arxiv.org/html/2607.08789#S2.E5)\) records transport of comparator information; the centered form \([8](https://arxiv.org/html/2607.08789#S2.E8)\) records regret\.

###### Corollary 2\.3\(Centered one\-step mixed coincidence identity\)\.

In the setting of Theorem[2\.2](https://arxiv.org/html/2607.08789#S2.Thmtheorem2), let

mt:=−ηt−1​log⁡Zt,δt​\(c\):=⟨qt,ct⟩−mtm\_\{t\}:=\-\\eta\_\{t\}^\{\-1\}\\log Z\_\{t\},\\qquad\\delta\_\{t\}\(c\):=\\langle q\_\{t\},c\_\{t\}\\rangle\-m\_\{t\}Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

⟨qt,ct⟩−⟨ρ,ct⟩=δt​\(c\)\+KL​\(ρ∥qt\)−KL​\(ρ∥qt\+1\)ηt\\langle q\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle=\\delta\_\{t\}\(c\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|q\_\{t\}\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{t\+1\}\)\}\{\\eta\_\{t\}\}\(8\)Equivalently,⟨qt,ℓt⟩−⟨ρ,ℓt⟩=δt​\(c\)\+KL​\(ρ∥qt\)−KL​\(ρ∥qt\+1\)ηt\+⟨ρ,ut⟩−⟨qt,ut⟩\\langle q\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,\\ell\_\{t\}\\rangle=\\delta\_\{t\}\(c\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|q\_\{t\}\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{t\+1\}\)\}\{\\eta\_\{t\}\}\+\\langle\\rho,u\_\{t\}\\rangle\-\\langle q\_\{t\},u\_\{t\}\\rangle\. Ifηt≡η\\eta\_\{t\}\\equiv\\eta, then

RTc​\(ρ\)=∑t=1Tδt​\(c\)\+η−1​\(KL​\(ρ∥q1\)−KL​\(ρ∥qT\+1\)\)\\displaystyle R\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\delta\_\{t\}\(c\)\+\\eta^\{\-1\}\\bigl\(\\mathrm\{KL\}\(\\rho\\\|q\_\{1\}\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{T\+1\}\)\\bigr\)

Equation \([8](https://arxiv.org/html/2607.08789#S2.E8)\) is the one\-step information balance that drives the whole paper\. It is the shared mechanism behind both adaptive updates\. Following\[[20](https://arxiv.org/html/2607.08789#bib.bib20)\], we refer to the round’s immediate costδt​\(c\)\\delta\_\{t\}\(c\)as the centered mixability gap\. The KL difference is the amount of comparator information transported fromqtq\_\{t\}toqt\+1q\_\{t\+1\}\. When the learning rate is fixed, that transport telescopes exactly and one recovers the familiar fixed\-rate identity below\. When the rate varies, the coefficients change and the two adaptive updates diverge: the prior\-retempered update recomputes from the prior at each new temperature and tracks the resulting drift explicitly \(Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)\), while the local recursion compounds step\-by\-step and produces an exact cumulative normalization identity \(Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)\)\.

The idea of tuning the learning rate according to the mixability gap has an important history\. AdaHedge\[[20](https://arxiv.org/html/2607.08789#bib.bib20)\]and SafeBayes\[[37](https://arxiv.org/html/2607.08789#bib.bib37)\], for example, both use the gap to calibrate how aggressively the learner should update\. Our approach is close in spirit but different in aim\. Rather than designing one specific schedule that controls an upper bound on the gap, we keep the decomposition exact for any predictable schedule and study how information flows under that exact accounting\.

###### Corollary 2\.4\(Fixed\-rate exact comparator identity\)\.

Assumeηt≡η\>0\\eta\_\{t\}\\equiv\\eta\>0in \([4](https://arxiv.org/html/2607.08789#S2.E4)\), and define the one\-step mix lossmt:=−η−1​log⁡Ztm\_\{t\}:=\-\\eta^\{\-1\}\\log Z\_\{t\}\. Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

∑t=1Tmt\+η−1​KL​\(ρ∥qT\+1\)=⟨ρ,CT⟩\+η−1​KL​\(ρ∥q1\)\\sum\_\{t=1\}^\{T\}m\_\{t\}\+\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|q\_\{T\+1\}\)=\\langle\\rho,C\_\{T\}\\rangle\+\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|q\_\{1\}\)\(9\)

When the side factor has the decomposed form \([6](https://arxiv.org/html/2607.08789#S2.E6)\), the mismatch term admits a closed\-form expression in terms of cross\-entropies\.

###### Corollary 2\.5\(Cross\-entropy form of the side\-factor mismatch\)\.

Under the same assumptions as Corollary[2\.4](https://arxiv.org/html/2607.08789#S2.Thmtheorem4), if \([6](https://arxiv.org/html/2607.08789#S2.E6)\) holds, then the mismatch term is exactly

∑t=1T\(⟨ρ,ut⟩−⟨qt,ut⟩\)=∑t=1T∑w=1Jtαt,wη​\(H​\(ρ,μt,w\)−H​\(qt,μt,w\)\)\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle q\_\{t\},u\_\{t\}\\rangle\\right\)=\\sum\_\{t=1\}^\{T\}\\sum\_\{w=1\}^\{J\_\{t\}\}\\frac\{\\alpha\_\{t,w\}\}\{\\eta\}\\left\(\\mathrm\{H\}\(\\rho,\\mu\_\{t,w\}\)\-\\mathrm\{H\}\(q\_\{t\},\\mu\_\{t,w\}\)\\right\)\(10\)

### 2\.2The one\-step identity and the retempered chain

The one\-step identity from Section 2\.1 is shared by both update rules\. The adaptive issue is purely cumulative: when the coefficients1/ηt1/\\eta\_\{t\}vary with time, how do the KL transport terms compose? For the prior\-retempered update, the answer is obtained by recomputing the posterior from the original prior at the current temperature and keeping track of the resulting temperature\-change drift\. Section 3\.3 returns to the local recursion, whose cumulative bookkeeping is different and is most naturally presented together with the pressure\-target schedule\.

We therefore feed the composite losses into the standard variable\-rate exponential\-weights update

pt​\(i\)=π​\(i\)​e−ηt​Ct−1​\(i\)∑j=1Kπ​\(j\)​e−ηt​Ct−1​\(j\)Ct​\(i\):=∑s=1tcs​\(i\)p\_\{t\}\(i\)=\\frac\{\\pi\(i\)e^\{\-\\eta\_\{t\}C\_\{t\-1\}\(i\)\}\}\{\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{\-\\eta\_\{t\}C\_\{t\-1\}\(j\)\}\}\\qquad C\_\{t\}\(i\):=\\sum\_\{s=1\}^\{t\}c\_\{s\}\(i\)\(11\)whereπ∈Δ​\(\[K\]\)\\pi\\in\\Delta\(\[K\]\)is a fixed prior and the schedule\(ηt\)\(\\eta\_\{t\}\)is predictable\. The current distribution is always recomputed from the prior at the current temperature, so the cumulative mix\-loss analysis remains valid\. This is the prior\-retempered update of the framework; Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)develops the second update—the local recursionpt\+1​\(i\)∝pt​\(i\)​e−ηt​ct​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}—which agrees with \([11](https://arxiv.org/html/2607.08789#S2.E11)\) only at fixed temperature but produces its own parallel exact decomposition\. More importantly, the exact identities below hold for*every*predictable positive schedule: the second\-order rule is only one especially useful equalizer, not the premise of the theorem\.

Define

mt\\displaystyle m\_\{t\}:=−ηt−1​log⁡\(∑i=1Kpt​\(i\)​e−ηt​ct​\(i\)\),\\displaystyle:=\-\\eta\_\{t\}^\{\-1\}\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}\\right\),δt​\(c\)\\displaystyle\\delta\_\{t\}\(c\):=⟨pt,ct⟩−mt,\\displaystyle:=\\langle p\_\{t\},c\_\{t\}\\rangle\-m\_\{t\},ψt​\(λ\)\\displaystyle\\psi\_\{t\}\(\\lambda\):=log⁡𝔼i∼pt​\[exp⁡\(λ​\(ct​\(i\)−⟨pt,ct⟩\)\)\]\\displaystyle:=\\log\\mathbb\{E\}\_\{i\\sim p\_\{t\}\}\\left\[\\exp\\\!\\left\(\\lambda\(c\_\{t\}\(i\)\-\\langle p\_\{t\},c\_\{t\}\\rangle\)\\right\)\\right\]We also use the scaled cumulant

ϕt​\(η\):=ψt​\(−η\)η=η−1​log⁡𝔼i∼pt​\[exp⁡\(−η​\(ct​\(i\)−⟨pt,ct⟩\)\)\]\\phi\_\{t\}\(\\eta\):=\\frac\{\\psi\_\{t\}\(\-\\eta\)\}\{\\eta\}=\\eta^\{\-1\}\\log\\mathbb\{E\}\_\{i\\sim p\_\{t\}\}\\left\[\\exp\\\!\\left\(\-\\eta\(c\_\{t\}\(i\)\-\\langle p\_\{t\},c\_\{t\}\\rangle\)\\right\)\\right\]At the played temperature,ϕt​\(ηt\)=δt​\(c\)\\phi\_\{t\}\(\\eta\_\{t\}\)=\\delta\_\{t\}\(c\), soϕt​\(ηt\)\\phi\_\{t\}\(\\eta\_\{t\}\)is exactly the one\-round mixability gap andQt​\(c\)=ηt−1​ϕt​\(ηt\)=ηt−2​ψt​\(−ηt\)Q\_\{t\}\(c\)=\\eta\_\{t\}^\{\-1\}\\phi\_\{t\}\(\\eta\_\{t\}\)=\\eta\_\{t\}^\{\-2\}\\psi\_\{t\}\(\-\\eta\_\{t\}\)is its finite\-rate intrinsic\-time density\.

Equation \([11](https://arxiv.org/html/2607.08789#S2.E11)\) is the formal version of the generic Bayes\-rule template previewed in the introduction\. Ifgs​\(i\)\>0g\_\{s\}\(i\)\>0is any sequence of evidence factors andus​\(i\)u\_\{s\}\(i\)any predictable side offset, then the updated posterior

pt​\(i\)∝π​\(i\)​\(∏s=1t−1gs​\(i\)\)ηt​exp⁡\(−ηt​∑s=1t−1us​\(i\)\)p\_\{t\}\(i\)\\propto\\pi\(i\)\\left\(\\prod\_\{s=1\}^\{t\-1\}g\_\{s\}\(i\)\\right\)^\{\\eta\_\{t\}\}\\exp\\\!\\left\(\-\\eta\_\{t\}\\sum\_\{s=1\}^\{t\-1\}u\_\{s\}\(i\)\\right\)is exactly of the form \([11](https://arxiv.org/html/2607.08789#S2.E11)\) after settingcs​\(i\):=−log⁡gs​\(i\)\+us​\(i\)c\_\{s\}\(i\):=\-\\log g\_\{s\}\(i\)\+u\_\{s\}\(i\)\. Thus the analysis below applies to any online algorithm that accumulates expertwise evidence by Bayes’ rule and then chooses the strength of the update through a predictable learning rate\. Ordinary Hedge is the canonical special casegs​\(i\)=e−ℓs​\(i\)g\_\{s\}\(i\)=e^\{\-\\ell\_\{s\}\(i\)\}, but the algebra itself is agnostic about whether the evidence came from losses, likelihood ratios, side information, or some other Bayesian score\. Once an easy\-instance notion suggests a particular intrinsic\-time functional, the same calculus can be used both to analyze that Bayes\-rule update and to design bespoke learning\-rate schedules for it\.

The PAC\-Bayes variational reduction is most transparent in exact form\. The usual change\-of\-measure inequality is recovered only after dropping a nonnegative KL remainder\.

###### Lemma 2\.6\(Gibbs variational identity\)\.

Letπ∈Δ​\(\[K\]\)\\pi\\in\\Delta\(\[K\]\)andX∈ℝKX\\in\\mathbb\{R\}^\{K\}\. DefineqX⋆​\(i\):=π​\(i\)​eX​\(i\)∑j=1Kπ​\(j\)​eX​\(j\)q\_\{X\}^\{\\star\}\(i\):=\\frac\{\\pi\(i\)e^\{X\(i\)\}\}\{\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{X\(j\)\}\}\. Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\)withρ≪π\\rho\\ll\\pi,

log⁡\(∑i=1Kπ​\(i\)​eX​\(i\)\)=⟨ρ,X⟩−KL​\(ρ∥π\)\+KL​\(ρ∥qX⋆\)\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}\\pi\(i\)e^\{X\(i\)\}\\right\)=\\langle\\rho,X\\rangle\-\\mathrm\{KL\}\(\\rho\\\|\\pi\)\+\\mathrm\{KL\}\(\\rho\\\|q\_\{X\}^\{\\star\}\)\(12\)Equivalently, for everyη\>0\\eta\>0and everyx∈ℝKx\\in\\mathbb\{R\}^\{K\}, ifqη,x⋆​\(i\):=π​\(i\)​e−η​x​\(i\)∑j=1Kπ​\(j\)​e−η​x​\(j\)q\_\{\\eta,x\}^\{\\star\}\(i\):=\\frac\{\\pi\(i\)e^\{\-\\eta x\(i\)\}\}\{\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{\-\\eta x\(j\)\}\}, then

−η−1​log⁡\(∑i=1Kπ​\(i\)​e−η​x​\(i\)\)=⟨ρ,x⟩\+η−1​KL​\(ρ∥π\)−η−1​KL​\(ρ∥qη,x⋆\)\-\\eta^\{\-1\}\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}\\pi\(i\)e^\{\-\\eta x\(i\)\}\\right\)=\\langle\\rho,x\\rangle\+\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|q\_\{\\eta,x\}^\{\\star\}\)\(13\)In particular, the familiar PAC\-Bayes inequality follows by dropping the last term, and equality is attained atρ=qη,x⋆\\rho=q\_\{\\eta,x\}^\{\\star\}\.

The terminal potential is quantified by specializing the same variational identity to the cumulative loss\. It is worth recording separately because it closes the adaptive proof and records how much comparator information remains in the final posterior\.

###### Lemma 2\.7\(Terminal potential\)\.

Fort≥0t\\geq 0andη\>0\\eta\>0, defineAt​\(η\):=−η−1​log⁡\(∑i=1Kπ​\(i\)​e−η​Ct​\(i\)\)A\_\{t\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}\\pi\(i\)e^\{\-\\eta C\_\{t\}\(i\)\}\\right\)andqt,η​\(i\):=π​\(i\)​e−η​Ct​\(i\)/∑j=1Kπ​\(j\)​e−η​Ct​\(j\)q\_\{t,\\eta\}\(i\):=\\pi\(i\)e^\{\-\\eta C\_\{t\}\(i\)\}\\big/\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{\-\\eta C\_\{t\}\(j\)\}\. Then for everyq∈Δ​\(\[K\]\)q\\in\\Delta\(\[K\]\),

At​\(η\)\+η−1​KL​\(q∥qt,η\)=⟨q,Ct⟩\+η−1​KL​\(q∥π\)A\_\{t\}\(\\eta\)\+\\eta^\{\-1\}\\mathrm\{KL\}\(q\\\|q\_\{t,\\eta\}\)=\\langle q,C\_\{t\}\\rangle\+\\eta^\{\-1\}\\mathrm\{KL\}\(q\\\|\\pi\)\(14\)In particular,

At​\(η\)=minq∈Δ​\(\[K\]\)⁡\{⟨q,Ct⟩\+η−1​KL​\(q∥π\)\}A\_\{t\}\(\\eta\)=\\min\_\{q\\in\\Delta\(\[K\]\)\}\\left\\\{\\langle q,C\_\{t\}\\rangle\+\\eta^\{\-1\}\\mathrm\{KL\}\(q\\\|\\pi\)\\right\\\}\(15\)andAt​\(η\)A\_\{t\}\(\\eta\)is nonincreasing inη\\eta\.

The next theorem is the exact chain of identities that underlies the regret analysis\.

###### Theorem 2\.8\(Exact cumulant decomposition for variable\-temperature Bayes updating\)\.

Let\(ηt\)t=1T\(\\eta\_\{t\}\)\_\{t=1\}^\{T\}be any predictable positive schedule, and let\(pt\)\(p\_\{t\}\)be generated by \([11](https://arxiv.org/html/2607.08789#S2.E11)\)\. Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

RTc​\(ρ\)=∑t=1Tϕt​\(ηt\)\+∑t=1T−1\(At​\(ηt\)−At​\(ηt\+1\)\)\+KL​\(ρ∥π\)−KL​\(ρ∥qT,ηT\)ηTR\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\phi\_\{t\}\(\\eta\_\{t\}\)\+\\sum\_\{t=1\}^\{T\-1\}\\left\(A\_\{t\}\(\\eta\_\{t\}\)\-A\_\{t\}\(\\eta\_\{t\+1\}\)\\right\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{T,\\eta\_\{T\}\}\)\}\{\\eta\_\{T\}\}\(16\)whereRTc​\(ρ\):=∑t=1T⟨pt,ct⟩−⟨ρ,CT⟩R\_\{T\}^\{c\}\(\\rho\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,C\_\{T\}\\rangle\. If, moreover,\(ηt\)\(\\eta\_\{t\}\)is nonincreasing, then the temperature\-change drift satisfies

∑t=1T−1\(At​\(ηt\)−At​\(ηt\+1\)\)≤0\\displaystyle\\sum\_\{t=1\}^\{T\-1\}\\left\(A\_\{t\}\(\\eta\_\{t\}\)\-A\_\{t\}\(\\eta\_\{t\+1\}\)\\right\)\\leq 0

For fixed temperature, the exact adaptive chain collapses to a single terminal KL remainder\.

###### Corollary 2\.9\(Fixed\-rate exact PAC\-Bayes identity\)\.

Ifηt≡η\>0\\eta\_\{t\}\\equiv\\eta\>0, then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

RTc​\(ρ\)=∑t=1Tψt​\(−η\)η\+KL​\(ρ∥π\)−KL​\(ρ∥pT\+1\)ηR\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\frac\{\\psi\_\{t\}\(\-\\eta\)\}\{\\eta\}\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\}\{\\eta\}\(17\)wherepT\+1​\(i\):=π​\(i\)​e−η​CT​\(i\)/∑j=1Kπ​\(j\)​e−η​CT​\(j\)p\_\{T\+1\}\(i\):=\\pi\(i\)e^\{\-\\eta C\_\{T\}\(i\)\}\\big/\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{\-\\eta C\_\{T\}\(j\)\}\.

The exact chain becomes especially transparent once the cumulant itself is treated as the intrinsic quadratic increment\.

###### Theorem 2\.10\(Exact PAC\-Bayes second\-order identity\)\.

Let\(ηt\)t=1T\(\\eta\_\{t\}\)\_\{t=1\}^\{T\}be any predictable positive schedule, and let\(pt\)\(p\_\{t\}\)be generated by \([11](https://arxiv.org/html/2607.08789#S2.E11)\)\. Define

Qt​\(c\):=ϕt​\(ηt\)ηt,VT​\(c\):=∑t=1TQt​\(c\)Q\_\{t\}\(c\):=\\frac\{\\phi\_\{t\}\(\\eta\_\{t\}\)\}\{\\eta\_\{t\}\},\\qquad V\_\{T\}\(c\):=\\sum\_\{t=1\}^\{T\}Q\_\{t\}\(c\)\(18\)and setDT:=∑t=1T−1\(At​\(ηt\)−At​\(ηt\+1\)\)D\_\{T\}:=\\sum\_\{t=1\}^\{T\-1\}\\left\(A\_\{t\}\(\\eta\_\{t\}\)\-A\_\{t\}\(\\eta\_\{t\+1\}\)\\right\)andBT​\(ρ\):=\(KL​\(ρ∥π\)−KL​\(ρ∥qT,ηT\)\)/ηTB\_\{T\}\(\\rho\):=\\left\(\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{T,\\eta\_\{T\}\}\)\\right\)/\\eta\_\{T\}\. ThenQt​\(c\)≥0Q\_\{t\}\(c\)\\geq 0for everytt, and for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

RTc​\(ρ\)=DT\+BT​\(ρ\)\+∑t=1Tηt​Qt​\(c\)R\_\{T\}^\{c\}\(\\rho\)=D\_\{T\}\+B\_\{T\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\(19\)Ifηt≡η\\eta\_\{t\}\\equiv\\eta, thenDT=0D\_\{T\}=0and

RTc​\(ρ\)=BT​\(ρ\)\+η​VT​\(c\)R\_\{T\}^\{c\}\(\\rho\)=B\_\{T\}\(\\rho\)\+\\eta V\_\{T\}\(c\)\(20\)

##### Three structural pieces of the exact chain\.

Equation \([19](https://arxiv.org/html/2607.08789#S2.E19)\) separates adaptive regret into three orthogonal objects for the retempered update; Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)will show that the local update admits a parallel three\-piece structure with different cumulative bookkeeping\. First, the intrinsic timeVT​\(c\)=∑t≤TQt​\(c\)V\_\{T\}\(c\)=\\sum\_\{t\\leq T\}Q\_\{t\}\(c\)is computed under the learner’s own current distributionptp\_\{t\}, so it is the algorithm\-defined variance scale of the realized trajectory rather than an external clock\. This per\-round costQt​\(c\)Q\_\{t\}\(c\)is computed by the same functional in both updates—it is the common per\-round information functional in the one\-step information balance, evaluated at whateverptp\_\{t\}each recursion actually plays\. In that sense recent attempts to improve a second\-order algorithm by changing its proposal distribution should be read as changes to the intrinsic geometry that the algorithm reveals\. Second, the terminal posterior\-mismatch termBT​\(ρ\)B\_\{T\}\(\\rho\)is the unique place where structured comparator classes enter; ifπ\\piis read as a decoder, this term is also the remaining description cost of the comparator after the realized sequence has been processed\. Quantile and other uniform quantifiers are obtained precisely by choosingρ\\rhoto spread mass over a restricted set and thereby lowering the posterior complexity\. Third, the temperature\-change driftDTD\_\{T\}is the exact bookkeeping term for arbitrary learning\-rate changes\. Nothing is approximated there: for a nonmonotone schedule the drift is part of the identity, whereas for a nonincreasing schedule it becomes nonpositive and may be discarded\.

##### How to read the scaled cumulant and the intrinsic time\.

WriteXt:=ct​\(It\)−⟨pt,ct⟩X\_\{t\}:=c\_\{t\}\(I\_\{t\}\)\-\\langle p\_\{t\},c\_\{t\}\\rangleforIt∼ptI\_\{t\}\\sim p\_\{t\}, and for a hypothetical temperatureη\>0\\eta\>0writemt​\(η\):=−η−1​log​∑ipt​\(i\)​e−η​ct​\(i\)m\_\{t\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}\. Thenψt​\(λ\)=log⁡𝔼​\[eλ​Xt\]\\psi\_\{t\}\(\\lambda\)=\\log\\mathbb\{E\}\[e^\{\\lambda X\_\{t\}\}\]is the ordinary log\-mgf of the centered one\-round loss, whileϕt​\(η\)=η−1​log⁡𝔼​\[e−η​Xt\]=⟨pt,ct⟩−mt​\(η\)\\phi\_\{t\}\(\\eta\)=\\eta^\{\-1\}\\log\\mathbb\{E\}\[e^\{\-\\eta X\_\{t\}\}\]=\\langle p\_\{t\},c\_\{t\}\\rangle\-m\_\{t\}\(\\eta\)is exactly the mean\-to\-mix\-loss gap that would be paid at temperatureη\\etaif the current distribution were frozen\. ThusQt​\(c\)Q\_\{t\}\(c\)is not a proxy for a variance term; it is the exact finite\-rate curvature cost on roundtt\.

The connection with the familiar quadratic variation appears only as a small\-temperature interpretation\. Sinceψt​\(0\)=ψt′​\(0\)=0\\psi\_\{t\}\(0\)=\\psi\_\{t\}^\{\\prime\}\(0\)=0andψt′′​\(0\)=Vari∼pt​\(ct​\(i\)\)\\psi\_\{t\}^\{\\prime\\prime\}\(0\)=\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(c\_\{t\}\(i\)\), one hasϕt​\(η\)=η2​Vari∼pt​\(ct​\(i\)\)\+o​\(η\)\\phi\_\{t\}\(\\eta\)=\\frac\{\\eta\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(c\_\{t\}\(i\)\)\+o\(\\eta\)and thereforeQt​\(c\)=12​Vari∼pt​\(ct​\(i\)\)\+o​\(1\)Q\_\{t\}\(c\)=\\frac\{1\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(c\_\{t\}\(i\)\)\+o\(1\)asη↓0\\eta\\downarrow 0\. The analysis below keepsQt​\(c\)=ηt−2​ψt​\(−ηt\)Q\_\{t\}\(c\)=\\eta\_\{t\}^\{\-2\}\\psi\_\{t\}\(\-\\eta\_\{t\}\)itself throughout, so the variance picture is only a limit intuition and not a step in the proof\.

###### Proposition 2\.11\(Tilted\-variance representation of the intrinsic time\)\.

Fix a roundttand a temperatureη\>0\\eta\>0\. Letμt:=⟨pt,ct⟩\\mu\_\{t\}:=\\langle p\_\{t\},c\_\{t\}\\rangleand, fors∈\[0,1\]s\\in\[0,1\], define the exponentially tilted distribution

pt,s\(η\)​\(i\):=pt​\(i\)​exp⁡\(−s​η​\(ct​\(i\)−μt\)\)∑j=1Kpt​\(j\)​exp⁡\(−s​η​\(ct​\(j\)−μt\)\)p\_\{t,s\}^\{\(\\eta\)\}\(i\):=\\frac\{p\_\{t\}\(i\)\\exp\\\!\\left\(\-s\\eta\(c\_\{t\}\(i\)\-\\mu\_\{t\}\)\\right\)\}\{\\sum\_\{j=1\}^\{K\}p\_\{t\}\(j\)\\exp\\\!\\left\(\-s\\eta\(c\_\{t\}\(j\)\-\\mu\_\{t\}\)\\right\)\}Then

ϕt​\(η\)=η​∫01\(1−s\)​Vari∼pt,s\(η\)​\(ct​\(i\)\)​𝑑s,Qt​\(c\)=∫01\(1−s\)​Vari∼pt,s\(ηt\)​\(ct​\(i\)\)​𝑑s\\phi\_\{t\}\(\\eta\)=\\eta\\int\_\{0\}^\{1\}\(1\-s\)\\mathrm\{Var\}\_\{i\\sim p\_\{t,s\}^\{\(\\eta\)\}\}\(c\_\{t\}\(i\)\)\\,ds,\\qquad Q\_\{t\}\(c\)=\\int\_\{0\}^\{1\}\(1\-s\)\\mathrm\{Var\}\_\{i\\sim p\_\{t,s\}^\{\(\\eta\_\{t\}\)\}\}\(c\_\{t\}\(i\)\)\\,dsIn particular,Qt​\(c\)≥0Q\_\{t\}\(c\)\\geq 0, andQt​\(c\)=0Q\_\{t\}\(c\)=0if and only ifct​\(i\)c\_\{t\}\(i\)isptp\_\{t\}\-almost surely constant\.

##### Intrinsic time as revealed structure\.

Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)explains why both updates literally track the intrinsic time of the realized path\. For each round,Qt​\(c\)Q\_\{t\}\(c\)is an exact average of tilted variances generated by the actual Bayes\-rule update; no external variance proxy has to be guessed\. In that sense, the algorithm can be considered an unsupervised revealer of structure: once the update rule is fixed, the realized loss sequence itself determines the exact intrinsic\-time pathVT​\(c\)=∑t≤TQt​\(c\)V\_\{T\}\(c\)=\\sum\_\{t\\leq T\}Q\_\{t\}\(c\)\. This viewpoint also clarifies why changing the proposal distribution can matter so much in adaptive algorithms: it changes the very variance geometry being measured, rather than merely improving a loose upper bound\. Crucially, this revealing mechanism is the same in both updates: the retempered and local updates computeQt​\(c\)Q\_\{t\}\(c\)by the same functional of the played distribution and the composite\-loss vector\. The realized increments differ when the two recursions play differentptp\_\{t\}under a variable schedule; what is shared is the functional form, and how the cumulative cost∑tηt​Qt​\(c\)\\sum\_\{t\}\\eta\_\{t\}Q\_\{t\}\(c\)is organized and controlled is what distinguishes the two families\.

## 3Schedules and intrinsic time

This section turns the exact cumulant chain into concrete pathwise control\. We first record the complete2×22\\times 2controller–update design space \(Section[3\.1](https://arxiv.org/html/2607.08789#S3.SS1)\), including the two crossed combinations not used as the paper’s main pairings\. We then analyze the two pairings whose cumulative identities collapse most cleanly: the monotone square\-root schedule for the prior\-retempered update \(Section[3\.2](https://arxiv.org/html/2607.08789#S3.SS2)\), which interfaces directly with Theorem[2\.8](https://arxiv.org/html/2607.08789#S2.Thmtheorem8)and yields pathwise, high\-probability, and anytime second\-order envelopes; and the pressure\-target line search for the local update \(Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)\), whose reward is finer roundwise calibration rather than monotone drift control and whose native cumulative identity is an exact terminal\-mass partition\.

TableLABEL:tab:tradeoffsrows on “parameter\-free vs\. adaptive” and “horizon” should be read through this lens: both schedules replace the horizonTTby a quantity revealed by the realized path, but the retempered clock is a cumulative budget, while the pressure target is a one\-round calibration\.

### 3\.1The two crossed controller–update combinations

The two axes in Algorithm[1](https://arxiv.org/html/2607.08789#alg1)are independent\. The square\-root controller asks for a predictable global clock, while the pressure controller asks for a one\-round free\-energy target\. The prior\-retempered update composes evidence by recomputing a Gibbs posterior from the original prior at the current temperature, while the local update composes evidence by multiplying the current weights by the current likelihood factor\. The main pairings,Ret\-SqrtandLoc\-Press, are clean because the controller and the cumulative identity are matched\. The crossed pairings are still legitimate algorithms, and they are useful diagnostics because they show exactly which part of the analysis is due to the controller and which part is due to the update geometry\.

##### Pressure\-targeted cumulative retempering\.

For the cumulative, prior\-retempered geometry defineqt,η​\(i\):=π​\(i\)​e−η​Ct​\(i\)/∑jπ​\(j\)​e−η​Ct​\(j\)q\_\{t,\\eta\}\(i\):=\\pi\(i\)e^\{\-\\eta C\_\{t\}\(i\)\}/\\sum\_\{j\}\\pi\(j\)e^\{\-\\eta C\_\{t\}\(j\)\}andAt​\(η\):=−η−1​log​∑iπ​\(i\)​e−η​Ct​\(i\)A\_\{t\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\sum\_\{i\}\\pi\(i\)e^\{\-\\eta C\_\{t\}\(i\)\}\. After roundtthas revealedctc\_\{t\}, the cumulative analogue of the local pressure curve is

mtret​\(η\):=−η−1​log​∑iqt−1,η​\(i\)​e−η​ct​\(i\)=At​\(η\)−At−1​\(η\)\.m\_\{t\}^\{\\rm ret\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\sum\_\{i\}q\_\{t\-1,\\eta\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}=A\_\{t\}\(\\eta\)\-A\_\{t\-1\}\(\\eta\)\.A pressure\-retempered schedule chooses the next temperature by solving

mtret​\(ηt\+1\)=at,equivalently∑iqt−1,ηt\+1​\(i\)​e−ηt\+1​\(ct​\(i\)−at\)=1,m\_\{t\}^\{\\rm ret\}\(\\eta\_\{t\+1\}\)=a\_\{t\},\\qquad\\text\{equivalently\}\\qquad\\sum\_\{i\}q\_\{t\-1,\\eta\_\{t\+1\}\}\(i\)e^\{\-\\eta\_\{t\+1\}\(c\_\{t\}\(i\)\-a\_\{t\}\)\}=1,\(21\)and then sets the next posterior topt\+1=qt,ηt\+1p\_\{t\+1\}=q\_\{t,\\eta\_\{t\+1\}\}\. This is the precise cumulative\-update version of the pressure\-target rule\. The one\-step delay is important: using \([21](https://arxiv.org/html/2607.08789#S3.E21)\) to choose the already\-playedptp\_\{t\}would require knowingctc\_\{t\}before acting, while choosingηt\+1\\eta\_\{t\+1\}after observingctc\_\{t\}is causal\.

Unlike the local pressure curve,mtret​\(η\)m\_\{t\}^\{\\rm ret\}\(\\eta\)is not a fixed\-distribution mix\-loss curve, because the distributionqt−1,ηq\_\{t\-1,\\eta\}also changes withη\\eta\. Thus existence and uniqueness are not automatic from the interval conditionmini⁡ct​\(i\)<at<⟨qt−1,η,ct⟩\\min\_\{i\}c\_\{t\}\(i\)<a\_\{t\}<\\langle q\_\{t\-1,\\eta\},c\_\{t\}\\rangle\. The algorithmic rule should therefore specify a deterministic root\-selection convention on the admissible set, for example the smallest positive root in a preassigned interval\. Once the temperature path is selected, no new regret identity is needed: Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)applies verbatim, with the resulting retempering driftDTD\_\{T\}left in the ledger\.

The least arbitrary way to chooseata\_\{t\}is to choose a dimensionless information target first and let the pressure level be induced\. For a candidate temperature, define the current instance’s cumulative Bayesian surprise

Itret​\(η\):=KL​\(qt,η∥qt−1,η\)=η​\(mtret​\(η\)−⟨qt,η,ct⟩\)\.I\_\{t\}^\{\\rm ret\}\(\\eta\):=\\mathrm\{KL\}\(q\_\{t,\\eta\}\\\|q\_\{t\-1,\\eta\}\)=\\eta\\left\(m\_\{t\}^\{\\rm ret\}\(\\eta\)\-\\langle q\_\{t,\\eta\},c\_\{t\}\\rangle\\right\)\.\(22\)Choose an information quotaβt≥0\\beta\_\{t\}\\geq 0, solveItret​\(ηt\+1\)=βtI\_\{t\}^\{\\rm ret\}\(\\eta\_\{t\+1\}\)=\\beta\_\{t\}, and then set

at=mtret​\(ηt\+1\)=⟨qt,ηt\+1,ct⟩\+βtηt\+1\.a\_\{t\}=m\_\{t\}^\{\\rm ret\}\(\\eta\_\{t\+1\}\)=\\langle q\_\{t,\\eta\_\{t\+1\}\},c\_\{t\}\\rangle\+\\frac\{\\beta\_\{t\}\}\{\\eta\_\{t\+1\}\}\.\(23\)This turns the target from a raw loss\-scale number into an instance\-wise information calculation:βt\\beta\_\{t\}specifies how much posterior movement the current example is allowed to cause\. A practical default is an information\-fraction ruleβt=θ​Itret​\(η¯t\)\\beta\_\{t\}=\\theta I\_\{t\}^\{\\rm ret\}\(\\bar\{\\eta\}\_\{t\}\)with0<θ≤10<\\theta\\leq 1, whereη¯t\\bar\{\\eta\}\_\{t\}is a reference scale such as the current temperature or the unit scale, optionally clipped by a maximum per\-round information budget\. This is the clean way to formalize a gap\-retempered schedule: the observed mixability gap or Bayesian surprise determines the next cumulative temperature, and the free\-energy targetata\_\{t\}is then whatever pressure level realizes that information movement\.

##### Square\-root clock with the local update\.

The other crossed cell keeps the local recursionpt\+1​\(i\)∝pt​\(i\)​e−ηt​ct​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}but drives it by the square\-root clock

ηt=1if​Ut−1loc=0,ηt=min⁡\{1,C​Γ/Ut−1loc\}otherwise,\\eta\_\{t\}=1\\quad\\text\{if \}U\_\{t\-1\}^\{\\rm loc\}=0,\\qquad\\eta\_\{t\}=\\min\\\{1,C\\sqrt\{\\Gamma/U\_\{t\-1\}^\{\\rm loc\}\}\\\}\\quad\\text\{otherwise\},whereUtlocU\_\{t\}^\{\\rm loc\}may be the exact local cumulant clock∑s≤tQsloc​\(c\)\\sum\_\{s\\leq t\}Q\_\{s\}^\{\\rm loc\}\(c\)or its quadratic relaxation\. Withμt=⟨pt,ct⟩\\mu\_\{t\}=\\langle p\_\{t\},c\_\{t\}\\rangleandQtloc​\(c\):=ηt−2​log⁡𝔼i∼pt​e−ηt​\(ct​\(i\)−μt\)Q\_\{t\}^\{\\rm loc\}\(c\):=\\eta\_\{t\}^\{\-2\}\\log\\mathbb\{E\}\_\{i\\sim p\_\{t\}\}e^\{\-\\eta\_\{t\}\(c\_\{t\}\(i\)\-\\mu\_\{t\}\)\}, the ordinary one\-step identity gives, for every comparatorρ\\rho,

RTc​\(ρ\)=∑t=1Tηt​Qtloc​\(c\)\+KL​\(ρ∥p1\)η1−KL​\(ρ∥pT\+1\)ηT\+∑t=2TKL​\(ρ∥pt\)​\(1ηt−1ηt−1\)\.R\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}^\{\\rm loc\}\(c\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\}\{\\eta\_\{1\}\}\-\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\}\{\\eta\_\{T\}\}\+\\sum\_\{t=2\}^\{T\}\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\\left\(\\frac\{1\}\{\\eta\_\{t\}\}\-\\frac\{1\}\{\\eta\_\{t\-1\}\}\\right\)\.\(24\)The same square\-root envelope as Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)controls only the payment∑tηt​Qtloc​\(c\)\\sum\_\{t\}\\eta\_\{t\}Q\_\{t\}^\{\\rm loc\}\(c\)\. The extra Abel term is the cost of applying a cooling controller to the local geometry\. Since the square\-root clock is typically nonincreasing,1/ηt−1/ηt−1≥01/\\eta\_\{t\}\-1/\\eta\_\{t\-1\}\\geq 0, so this term is not a benign negative drift\. If one has an independent compression boundKL​\(ρ∥pt\)≤Λ\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\\leq\\Lambdaalong the path, the extra variation is at mostΛ​\(1/ηT−1/η1\)\\Lambda\(1/\\eta\_\{T\}\-1/\\eta\_\{1\}\)and remains on the same order as the square\-root term whenΛ\\Lambdais comparable toΓ\\Gamma; without such a bound it is an explicit additional price\. ThusLoc\-Sqrtis a sensible baseline when one wants a predictable local recursion, but it gives up both special simplifications: the nonpositive retempered drift ofRet\-Sqrtand the exact terminal\-mass normalization ofLoc\-Press\.

### 3\.2Second\-order schedules and intrinsic\-time consequences

We begin with the budgeted square\-root rule from Algorithm[1](https://arxiv.org/html/2607.08789#alg1)\. It is explicit, predictable, and nonincreasing, and it balances terminal comparator complexity against accumulated intrinsic time\.

##### Definition\.

FixΓ\>0\\Gamma\>0and a constantC\>0C\>0; the default optimized choice isC=1/2C=1/\\sqrt\{2\}\. The*second\-order schedule*with budgetΓ\\Gammaand tuningCCis the predictable positive sequence

ηt:=\{1,if​Vt−1​\(c\)=0min⁡\{1,C​ΓVt−1​\(c\)\}otherwise\\eta\_\{t\}:=\\begin\{cases\}1,&\\text\{if \}V\_\{t\-1\}\(c\)=0\\\\\[4\.30554pt\] \\min\\\!\\left\\\{1,\\,C\\sqrt\{\\frac\{\\Gamma\}\{V\_\{t\-1\}\(c\)\}\}\\right\\\}&\\text\{otherwise\}\\end\{cases\}\(25\)BecauseVt−1​\(c\)V\_\{t\-1\}\(c\)isℱt−1\\mathcal\{F\}\_\{t\-1\}\-measurable, eachηt\\eta\_\{t\}is predictable, and becauseVt−1​\(c\)V\_\{t\-1\}\(c\)is nondecreasing inttthe schedule is nonincreasing\.

###### Proposition 3\.1\(Range\-based bound for the intrinsic increment\)\.

Ifct​\(i\)∈\[at,bt\]c\_\{t\}\(i\)\\in\[a\_\{t\},b\_\{t\}\]for allii, then the standard bounded\-range exponential\-moment bound applied toct​\(i\)−⟨pt,ct⟩c\_\{t\}\(i\)\-\\langle p\_\{t\},c\_\{t\}\\ranglegivesQt​\(c\)≤\(bt−at\)2/8Q\_\{t\}\(c\)\\leq\(b\_\{t\}\-a\_\{t\}\)^\{2\}/8\. In particular, ifct​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\], thenQt​\(c\)≤1/8Q\_\{t\}\(c\)\\leq 1/8on every round\. Thus the exact intrinsic timeVT​\(c\)V\_\{T\}\(c\)remains a genuine quadratic\-variation object, now written directly in cumulant form\.

##### Exact and relaxed clocks\.

The exact controller analyzed below usesVt​\(c\)=∑s≤tQs​\(c\)V\_\{t\}\(c\)=\\sum\_\{s\\leq t\}Q\_\{s\}\(c\)\. A simpler quadratic summary is

Wt​\(c\):=∑s=1t12​Vari∼ps​\(cs​\(i\)\)W\_\{t\}\(c\):=\\sum\_\{s=1\}^\{t\}\\tfrac\{1\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{s\}\}\(c\_\{s\}\(i\)\)which keeps only the ordinary second moments of the played losses\. By the small\-temperature expansion of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2),Wt​\(c\)W\_\{t\}\(c\)is the leading\-order relaxation of the exact clockVt​\(c\)V\_\{t\}\(c\)\(the factor12\\tfrac\{1\}\{2\}matchesQs​\(c\)=12​Vari∼ps​\(cs​\(i\)\)\+o​\(1\)Q\_\{s\}\(c\)=\\tfrac\{1\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{s\}\}\(c\_\{s\}\(i\)\)\+o\(1\)\); the un\-halved sumVt⊔=2​Wt​\(c\)V^\{\\\!\\sqcup\}\_\{t\}=2W\_\{t\}\(c\)is reserved for the Bernstein concentration clock of the iterated\-logarithm statement\. One may therefore drive the square\-root controller by eitherVt−1​\(c\)V\_\{t\-1\}\(c\)orWt−1​\(c\)W\_\{t\-1\}\(c\)in practice\. The theory below is stated for the exact clock, because that is the quantity that appears in the identity itself\.

This is the default second\-order schedule because it balances comparator complexity against revealed intrinsic time in the right square\-root scale\. If one freezes a target intrinsic\-time levelV\>0V\>0and balances the proxyΓ/η\+η​V\\Gamma/\\eta\+\\eta V, the minimizer isη⋆​\(V\)=Γ/V\\eta^\{\\star\}\(V\)=\\sqrt\{\\Gamma/V\}\. Thus the scheduleηt∝Vt−1​\(c\)−1/2\\eta\_\{t\}\\propto V\_\{t\-1\}\(c\)^\{\-1/2\}is the predictable equalizer of terminal comparator complexity and cumulative centered cost\. The next theorem shows that, up to unavoidable discrete edge terms, this one\-step equalizer survives along the whole realized path\.

The point is not merely that the algorithm upper bounds its regret by a convenient surrogate\. Along every realized path it tracks the exact intrinsic\-time cost∑t=1Tηt​Qt​\(c\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\), and that cost itself is trapped between matching square\-root laws built from the same intrinsic timeVT​\(c\)V\_\{T\}\(c\)\.

###### Theorem 3\.2\(Two\-sided square\-root envelope for the second\-order schedule\)\.

Run the schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\), and define

Q∗T​\(c\):=max1≤t≤T⁡Qt​\(c\)Q\_\{\*\}^\{T\}\(c\):=\\max\_\{1\\leq t\\leq T\}Q\_\{t\}\(c\)Then

2​C​Γ​VT​\(c\)−C2​Γ≤∑t=1Tηt​Qt​\(c\)≤C2​Γ\+Q∗T​\(c\)\+2​C​Γ​VT​\(c\)2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\-C^\{2\}\\Gamma\\leq\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\\leq C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\(26\)Consequently, for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

2​C​Γ​VT​\(c\)−C2​Γ≤RTc​\(ρ\)−\(DT\+BT​\(ρ\)\)≤C2​Γ\+Q∗T​\(c\)\+2​C​Γ​VT​\(c\)2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\-C^\{2\}\\Gamma\\leq R\_\{T\}^\{c\}\(\\rho\)\-\(D\_\{T\}\+B\_\{T\}\(\\rho\)\)\\leq C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\(27\)If moreoverKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, thenBT​\(ρ\)≤Γ​ηT−1B\_\{T\}\(\\rho\)\\leq\\Gamma\\eta\_\{T\}^\{\-1\}and

RTc​\(ρ\)≤\(1\+C2\)​Γ\+Q∗T​\(c\)\+\(2​C\+C−1\)​Γ​VT​\(c\)R\_\{T\}^\{c\}\(\\rho\)\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\(28\)In particular, ifC=1/2C=1/\\sqrt\{2\}, then

RTc​\(ρ\)≤32​Γ\+Q∗T​\(c\)\+2​2​Γ​VT​\(c\)R\_\{T\}^\{c\}\(\\rho\)\\leq\\frac\{3\}\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2\\sqrt\{2\\Gamma V\_\{T\}\(c\)\}\(29\)

The two slack terms in \([26](https://arxiv.org/html/2607.08789#S3.E26)\) come from two different discrete effects, and both are unavoidable\.

- •For the upper side, the maximal jumpQ∗T​\(c\)Q\_\{\*\}^\{T\}\(c\)is necessary: withT=1T=1andQ1​\(c\)=qQ\_\{1\}\(c\)=q, we haveη1=1\\eta\_\{1\}=1, so∑tηt​Qt​\(c\)=q\\sum\_\{t\}\\eta\_\{t\}Q\_\{t\}\(c\)=qandVT​\(c\)=qV\_\{T\}\(c\)=q\. No bound of the form∑tηt​Qt​\(c\)≤A\+2​C​Γ​VT​\(c\)\\sum\_\{t\}\\eta\_\{t\}Q\_\{t\}\(c\)\\leq A\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}can hold for allqq, because the right\-hand side grows only likeq\\sqrt\{q\}\. A correction of order the largest single jump is therefore indispensable\.
- •For the lower side, the initialization taxC2​ΓC^\{2\}\\Gammais also forced by a one\-round example\. IfT=1T=1andQ1​\(c\)=C2​ΓQ\_\{1\}\(c\)=C^\{2\}\\Gamma, then stillη1=1\\eta\_\{1\}=1and∑tηt​Qt​\(c\)=C2​Γ\\sum\_\{t\}\\eta\_\{t\}Q\_\{t\}\(c\)=C^\{2\}\\Gamma, whereas2​C​Γ​VT​\(c\)=2​C2​Γ2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}=2C^\{2\}\\Gamma\. Therefore, any universal lower bound of the form2​C​Γ​VT​\(c\)−A≤∑tηt​Qt​\(c\)2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\-A\\leq\\sum\_\{t\}\\eta\_\{t\}Q\_\{t\}\(c\)must satisfyA≥C2​ΓA\\geq C^\{2\}\\Gamma\. The second\-order schedule therefore pays a real clipped\-initialization cost before the square\-root regime takes over\.

#### 3\.2\.1Pathwise sampled\-expert PAC\-Bayes regret

The exact chain also gives the promised sampled\-expert statement\. If the learner samples an expertIt∼ptI\_\{t\}\\sim p\_\{t\}on each round, then the sampled regret differs from the deterministic regret only by a martingale term\. The same randomized draw can also be implemented as a follow\-the\-perturbed\-leader \(FTPL\) step with i\.i\.d\. Gumbel perturbations, so the pathwise identities below equally apply to that realization of the action sample \(e\.g\.\[[3](https://arxiv.org/html/2607.08789#bib.bib3), Theorem 3\.9\]\)\.

###### Theorem 3\.3\(Pathwise sampled\-expert PAC\-Bayes identity\)\.

Assume that, conditionally on the past, the learner samplesIt∼ptI\_\{t\}\\sim p\_\{t\}independently on each round\. Define the sampled composite\-loss regretR^Tc​\(ρ\):=∑t=1Tct​\(It\)−⟨ρ,CT⟩\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\):=\\sum\_\{t=1\}^\{T\}c\_\{t\}\(I\_\{t\}\)\-\\langle\\rho,C\_\{T\}\\rangleand the martingale increment sumMTsam:=∑t=1T\(ct​\(It\)−⟨pt,ct⟩\)M\_\{T\}^\{\\mathrm\{sam\}\}:=\\sum\_\{t=1\}^\{T\}\\left\(c\_\{t\}\(I\_\{t\}\)\-\\langle p\_\{t\},c\_\{t\}\\rangle\\right\)\. Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

R^Tc​\(ρ\)=MTsam\+DT\+BT​\(ρ\)\+∑t=1Tηt​Qt​\(c\)\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\)=M\_\{T\}^\{\\mathrm\{sam\}\}\+D\_\{T\}\+B\_\{T\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\(30\)For the original losses,∑t=1Tℓt​\(It\)−⟨ρ,LT⟩=R^Tc​\(ρ\)\+∑t=1T\(⟨ρ,ut⟩−ut​\(It\)\)\\displaystyle\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}\(I\_\{t\}\)\-\\langle\\rho,L\_\{T\}\\rangle=\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-u\_\{t\}\(I\_\{t\}\)\\right\)\.

Applying the two\-sided envelope of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)to the identity above yields the following\.

###### Corollary 3\.4\(Second\-order bounds for the sampled\-expert regret\)\.

Under the schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\),

MTsam\+DT\+BT​\(ρ\)\+2​C​Γ​VT​\(c\)−C2​Γ≤R^Tc​\(ρ\)≤MTsam\+DT\+BT​\(ρ\)\+C2​Γ\+Q∗T​\(c\)\+2​C​Γ​VT​\(c\)M\_\{T\}^\{\\mathrm\{sam\}\}\+D\_\{T\}\+B\_\{T\}\(\\rho\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\-C^\{2\}\\Gamma\\leq\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\)\\leq M\_\{T\}^\{\\mathrm\{sam\}\}\+D\_\{T\}\+B\_\{T\}\(\\rho\)\+C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\(31\)IfKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, then

R^Tc​\(ρ\)≤MTsam\+\(1\+C2\)​Γ\+Q∗T​\(c\)\+\(2​C\+C−1\)​Γ​VT​\(c\)\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\)\\leq M\_\{T\}^\{\\mathrm\{sam\}\}\+\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\(32\)

#### 3\.2\.2High\-probability randomized prediction

A Hedge regret theorem\[[70](https://arxiv.org/html/2607.08789#bib.bib70)\]can be converted into a high\-probability guarantee for the randomized prediction rule that samples an action from the current weights\. In the present exact formalism the same conclusion is cleaner: Theorem[3\.3](https://arxiv.org/html/2607.08789#S3.Thmtheorem3)isolates the randomness in a single martingale, so the deterministic intrinsic\-time analysis survives unchanged and only one concentration step is needed\.

###### Theorem 3\.5\(High\-probability sampled\-expert PAC\-Bayes bound\)\.

Assumect​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\]for allttandii, and defineWT​\(c\):=∑t=1TVari∼pt​\(ct​\(i\)\)W\_\{T\}\(c\):=\\sum\_\{t=1\}^\{T\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(c\_\{t\}\(i\)\)\. Then for everyδ∈\(0,1\)\\delta\\in\(0,1\), with probability at least1−δ1\-\\delta,

R^Tc​\(ρ\)≤DT\+BT​\(ρ\)\+∑t=1Tηt​Qt​\(c\)\+2​WT​\(c\)​log⁡\(1/δ\)\+13​log⁡\(1/δ\)\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\)\\leq D\_\{T\}\+B\_\{T\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\+\\sqrt\{2W\_\{T\}\(c\)\\log\(1/\\delta\)\}\+\\tfrac\{1\}\{3\}\\log\(1/\\delta\)\(33\)simultaneously for all posteriorsρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\)\. Under the second\-order schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\), ifKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, then on the same event

R^Tc​\(ρ\)≤\(1\+C2\)​Γ\+Q∗T​\(c\)\+\(2​C\+C−1\)​Γ​VT​\(c\)\+2​WT​\(c\)​log⁡\(1/δ\)\+13​log⁡\(1/δ\)\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\)\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\+\\sqrt\{2W\_\{T\}\(c\)\\log\(1/\\delta\)\}\+\\tfrac\{1\}\{3\}\\log\(1/\\delta\)\(34\)SinceWT​\(c\)≤T/4W\_\{T\}\(c\)\\leq T/4, one also has

R^Tc​\(ρ\)≤\(1\+C2\)​Γ\+Q∗T​\(c\)\+\(2​C\+C−1\)​Γ​VT​\(c\)\+T2​log⁡\(1/δ\)\+13​log⁡\(1/δ\)\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\)\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\+\\sqrt\{\\tfrac\{T\}\{2\}\\log\(1/\\delta\)\}\+\\tfrac\{1\}\{3\}\\log\(1/\\delta\)

Thus the LS14\-style randomized\-prediction extension costs only the additional concentration of the sampling martingale\. The intrinsic\-time term itself is exactly the same one that already appears in the deterministic PAC\-Bayes theorem, and the bound is simultaneous over all posteriors because the martingale term does not depend onρ\\rho\.

The same martingale representation also yields an anytime form, with no union bound over times\.

###### Theorem 3\.6\(Anytime sampled\-expert confidence sequence\)\.

Fort≤Tt\\leq T, define the truncated sampled regretR^tc​\(ρ\):=∑s=1tcs​\(Is\)−⟨ρ,Ct⟩\\widehat\{R\}\_\{t\}^\{c\}\(\\rho\):=\\sum\_\{s=1\}^\{t\}c\_\{s\}\(I\_\{s\}\)\-\\langle\\rho,C\_\{t\}\\rangle, the truncated driftDt:=∑s=1t−1\(As​\(ηs\)−As​\(ηs\+1\)\)D\_\{t\}:=\\sum\_\{s=1\}^\{t\-1\}\(A\_\{s\}\(\\eta\_\{s\}\)\-A\_\{s\}\(\\eta\_\{s\+1\}\)\), the truncated terminal termBt​\(ρ\):=\(KL​\(ρ∥π\)−KL​\(ρ∥qt,ηt\)\)/ηtB\_\{t\}\(\\rho\):=\(\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{t,\\eta\_\{t\}\}\)\)/\\eta\_\{t\}, and the truncated intrinsic\-time objectsVt​\(c\):=∑s=1tQs​\(c\)V\_\{t\}\(c\):=\\sum\_\{s=1\}^\{t\}Q\_\{s\}\(c\)andQ∗,t​\(c\):=maxs≤t⁡Qs​\(c\)Q\_\{\*,t\}\(c\):=\\max\_\{s\\leq t\}Q\_\{s\}\(c\)\. For everyλ\>0\\lambda\>0, the processZt​\(λ\):=exp⁡\(λ​Mtsam−∑s=1tψs​\(λ\)\)Z\_\{t\}\(\\lambda\):=\\exp\\\!\\left\(\\lambda M\_\{t\}^\{\\mathrm\{sam\}\}\-\\sum\_\{s=1\}^\{t\}\\psi\_\{s\}\(\\lambda\)\\right\)is a nonnegative martingale\. Therefore, for everyδ∈\(0,1\)\\delta\\in\(0,1\), with probability at least1−δ1\-\\delta,

R^tc​\(ρ\)≤Dt\+Bt​\(ρ\)\+∑s=1tηs​Qs​\(c\)\+log⁡\(1/δ\)\+∑s=1tψs​\(λ\)λ\\widehat\{R\}\_\{t\}^\{c\}\(\\rho\)\\leq D\_\{t\}\+B\_\{t\}\(\\rho\)\+\\sum\_\{s=1\}^\{t\}\\eta\_\{s\}Q\_\{s\}\(c\)\+\\frac\{\\log\(1/\\delta\)\+\\sum\_\{s=1\}^\{t\}\\psi\_\{s\}\(\\lambda\)\}\{\\lambda\}\(35\)simultaneously for allt≤Tt\\leq Tand all posteriorsρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\)\. Under the second\-order schedule, ifKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, then on the same event

R^tc​\(ρ\)\\displaystyle\\widehat\{R\}\_\{t\}^\{c\}\(\\rho\)≤\(1\+C2\)​Γ\+Q∗,t​\(c\)\+\(2​C\+C−1\)​Γ​Vt​\(c\)\+log⁡\(1/δ\)\+∑s=1tψs​\(λ\)λ\\displaystyle\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*,t\}\(c\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{t\}\(c\)\}\+\\frac\{\\log\(1/\\delta\)\+\\sum\_\{s=1\}^\{t\}\\psi\_\{s\}\(\\lambda\)\}\{\\lambda\}\(36\)for allt≤Tt\\leq T\.

Mixing the martingalesZt​\(λ\)Z\_\{t\}\(\\lambda\)over a density onλ\\lambdayields curved anytime boundaries, which tend to give tighter instance\-specific bounds\[[42](https://arxiv.org/html/2607.08789#bib.bib42)\]\. Concretely, for an absolutely continuous mixing law with densityffon\(0,∞\)\(0,\\infty\)the mixtureZ¯t:=∫0∞Zt​\(λ\)​f​\(λ\)​𝑑λ\\overline\{Z\}\_\{t\}:=\\int\_\{0\}^\{\\infty\}Z\_\{t\}\(\\lambda\)\\,f\(\\lambda\)\\,d\\lambdais again a nonnegative supermartingale, and Ville’s inequality applied toZ¯t\\overline\{Z\}\_\{t\}gives a finite\-time, all\-tt\-simultaneous boundary; the boundary is recovered by a Laplace lower bound on the integrand near its peak, which contributes alog⁡f​\(λ⋆\)\\log f\(\\lambda^\{\\star\}\)density term rather than the mass of any single point\. The iterated\-logarithm*rate*for the sampled\-expert martingale is the asymptotic counterpart, in the spirit of\[[6](https://arxiv.org/html/2607.08789#bib.bib6)\]: along a geometric grid of rates one obtains the sharp2​Vt⊔​log⁡log⁡Vt⊔\\sqrt\{2V^\{\\\!\\sqcup\}\_\{t\}\\log\\log V^\{\\\!\\sqcup\}\_\{t\}\}growth constant, whereVt⊔=2​Wt​\(c\)V^\{\\\!\\sqcup\}\_\{t\}=2W\_\{t\}\(c\)is the predictable quadratic variation of the sampling martingale defined in the proposition below\.

###### Proposition 3\.7\(Iterated\-logarithm rate for the sampled\-expert martingale\)\.

Assumect​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\]for allttandii\. LetXt:=ct​\(It\)−⟨pt,ct⟩X\_\{t\}:=c\_\{t\}\(I\_\{t\}\)\-\\langle p\_\{t\},c\_\{t\}\\rangleandMtsam:=∑s=1tXsM\_\{t\}^\{\\mathrm\{sam\}\}:=\\sum\_\{s=1\}^\{t\}X\_\{s\}, and recall the predictable quadratic variationVt⊔:=∑s=1tVar​\(Xs∣ℱs−1\)=∑s=1tVari∼ps​\(cs​\(i\)\)V^\{\\\!\\sqcup\}\_\{t\}:=\\sum\_\{s=1\}^\{t\}\\mathrm\{Var\}\(X\_\{s\}\\mid\\mathcal\{F\}\_\{s\-1\}\)=\\sum\_\{s=1\}^\{t\}\\mathrm\{Var\}\_\{i\\sim p\_\{s\}\}\(c\_\{s\}\(i\)\)\. Then, almost surely on the event\{Vt⊔→∞\}\\\{V^\{\\\!\\sqcup\}\_\{t\}\\to\\infty\\\},

lim supt→∞\|Mtsam\|2​Vt⊔​log⁡log⁡Vt⊔≤1\.\\limsup\_\{t\\to\\infty\}\\;\\frac\{\|M\_\{t\}^\{\\mathrm\{sam\}\}\|\}\{\\sqrt\{2\\,V^\{\\\!\\sqcup\}\_\{t\}\\,\\log\\log V^\{\\\!\\sqcup\}\_\{t\}\}\}\\;\\leq\\;1\.\(37\)Consequently, for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\), the sampled regret satisfies the pathwise asymptotic bound

R^tc​\(ρ\)≤Dt\+Bt​\(ρ\)\+∑s=1tηs​Qs​\(c\)\+\(1\+o​\(1\)\)​2​Vt⊔​log⁡log⁡Vt⊔\\widehat\{R\}\_\{t\}^\{c\}\(\\rho\)\\leq D\_\{t\}\+B\_\{t\}\(\\rho\)\+\\sum\_\{s=1\}^\{t\}\\eta\_\{s\}Q\_\{s\}\(c\)\+\\bigl\(1\+o\(1\)\\bigr\)\\sqrt\{2\\,V^\{\\\!\\sqcup\}\_\{t\}\\,\\log\\log V^\{\\\!\\sqcup\}\_\{t\}\}\(38\)almost surely on\{Vt⊔→∞\}\\\{V^\{\\\!\\sqcup\}\_\{t\}\\to\\infty\\\}, simultaneously over all posteriorsρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\)\.

The key point, following\[[6](https://arxiv.org/html/2607.08789#bib.bib6)\], is that the growth constant in \([37](https://arxiv.org/html/2607.08789#S3.E37)\) is the sharp one: the iterated\-logarithm scalingVt⊔​log⁡log⁡Vt⊔\\sqrt\{V^\{\\\!\\sqcup\}\_\{t\}\\log\\log V^\{\\\!\\sqcup\}\_\{t\}\}is matched, not merely upper\-bounded by a worst\-caset\\sqrt\{t\}rate\. For the genuinely finite\-time, all\-tt\-simultaneous guarantee one uses the anytime confidence sequence of Theorem[3\.6](https://arxiv.org/html/2607.08789#S3.Thmtheorem6)\(a single mixing rateλ\\lambda, or its density mixture above\); the iterated\-logarithm constant is the price one pays asymptotically for the curvature of that boundary\.

##### Bandit analogue\.

The same argument applies to the martingale term in the generic bandit decomposition of Section[6](https://arxiv.org/html/2607.08789#S6): once the estimator increments admit the same bounded\-range or conditional sub\-Gaussian mgf bound, the centered bandit regret inherits the same asymptotic iterated\-logarithm rate after the explicit bias and intrinsic\-time terms are subtracted\.

##### Comparator\-centered cumulants\.

For a fixed posteriorρ\\rho, define

ψt,ρ​\(λ\):=log⁡𝔼i∼pt​exp⁡\(λ​\(ct​\(i\)−⟨ρ,ct⟩\)\),ϕt,ρ​\(η\):=η−1​ψt,ρ​\(−η\)\\psi\_\{t,\\rho\}\(\\lambda\):=\\log\\mathbb\{E\}\_\{i\\sim p\_\{t\}\}\\exp\\\!\\left\(\\lambda\(c\_\{t\}\(i\)\-\\langle\\rho,c\_\{t\}\\rangle\)\\right\),\\qquad\\phi\_\{t,\\rho\}\(\\eta\):=\\eta^\{\-1\}\\psi\_\{t,\\rho\}\(\-\\eta\)and setQtρ​\(c\):=\(ϕt​\(ηt\)−ϕt,ρ​\(ηt\)\)/ηtQ\_\{t\}^\{\\rho\}\(c\):=\\left\(\\phi\_\{t\}\(\\eta\_\{t\}\)\-\\phi\_\{t,\\rho\}\(\\eta\_\{t\}\)\\right\)/\\eta\_\{t\}\. Then the one\-round regret satisfies⟨pt,ct⟩−⟨ρ,ct⟩=ϕt​\(ηt\)−ϕt,ρ​\(ηt\)=ηt​Qtρ​\(c\)\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle=\\phi\_\{t\}\(\\eta\_\{t\}\)\-\\phi\_\{t,\\rho\}\(\\eta\_\{t\}\)=\\eta\_\{t\}Q\_\{t\}^\{\\rho\}\(c\), soRTc​\(ρ\)=∑t=1Tηt​Qtρ​\(c\)R\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}^\{\\rho\}\(c\)\. Section[4\.5](https://arxiv.org/html/2607.08789#S4.SS5)compares this exact comparator\-centered increment to the more familiar second\-moment quantityΨt​\(ρ\)\\Psi\_\{t\}\(\\rho\)\.

### 3\.3Local updates and pressure\-targeted schedules

The second\-order schedule of Section[3\.2](https://arxiv.org/html/2607.08789#S3.SS2)calibratesηt\\eta\_\{t\}against cumulative intrinsic time for the prior\-retempered update\. We now turn to the second update — the local recursionpt\+1​\(i\)∝pt​\(i\)​e−ηt​ct​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}— and its natural controller\.

##### Why nonmonotone schedules matter here\.

The exact chain of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)already allows arbitrary predictable schedules\. Monotone cooling was singled out there only because the retempered driftDT=∑t=1T−1\(At​\(ηt\)−At​\(ηt\+1\)\)D\_\{T\}=\\sum\_\{t=1\}^\{T\-1\}\\bigl\(A\_\{t\}\(\\eta\_\{t\}\)\-A\_\{t\}\(\\eta\_\{t\+1\}\)\\bigr\)is nonpositive and therefore easy to discard under cooling\. That monotonicity, however, is a simplifying choice rather than a structural requirement\. Under the local update, each round’s score is weighted directly by its own learning rate, and a schedule that can both decrease and increase is often the natural object: hard rounds may call for caution, later high\-edge rounds may justify a larger coefficient, and a purely monotone controller cannot express that\. This is the setting in which pressure targets are most natural: instead of asking for one cumulative clock to keep cooling forever, they calibrate the current round to a chosen target and let the exact cumulative identity keep track of the result\.

##### Definition of the pressure\-target rule\.

We first define the rule in its most general form:ata\_\{t\}is an arbitrary scalar target between the current minimum and mean score, andηt\\eta\_\{t\}is chosen so that the one\-step finite\-temperature cumulant hits that target\. The special interpretation ofata\_\{t\}as a*pressure*\(normalized free energy\) is natural but not necessary for the identities below; one may readata\_\{t\}purely as a scaled local CGF level, and all cumulative identities remain intact\.

The parallel with the retempered update is worth making explicit\. Both updates compose the shared one\-step information balance cumulatively\. The retempered chain produces a global cumulant decomposition with temperature\-change drift; the local chain produces an exact cumulative normalization identity where the final weights satisfypT\+1​\(i\)=p1​\(i\)​e−GT​\(i\)p\_\{T\+1\}\(i\)=p\_\{1\}\(i\)e^\{\-G\_\{T\}\(i\)\}with∑ip1​\(i\)​e−GT​\(i\)=1\\sum\_\{i\}p\_\{1\}\(i\)e^\{\-G\_\{T\}\(i\)\}=1\. The same per\-round information functionalQt​\(c\)Q\_\{t\}\(c\)appears in both cases \(its realized value tracking each recursion’s own playedptp\_\{t\}\), but the cumulative bookkeeping — and therefore the natural schedule family — is different\.

For a current distributionptp\_\{t\}and score vectorct∈ℝKc\_\{t\}\\in\\mathbb\{R\}^\{K\}, define the one\-round mix\-loss curve

mt​\(η\):=−η−1​log⁡\(∑i=1Kpt​\(i\)​e−η​ct​\(i\)\),η\>0m\_\{t\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}\\right\),\\qquad\\eta\>0Whenctc\_\{t\}is nonconstant,mtm\_\{t\}is continuous and strictly decreasing with

limη↓0mt​\(η\)=⟨pt,ct⟩,limη→∞mt​\(η\)=mini⁡ct​\(i\)\\lim\_\{\\eta\\downarrow 0\}m\_\{t\}\(\\eta\)=\\langle p\_\{t\},c\_\{t\}\\rangle,\\qquad\\lim\_\{\\eta\\to\\infty\}m\_\{t\}\(\\eta\)=\\min\_\{i\}c\_\{t\}\(i\)so every target between the current mean and the current minimum determines a unique positive temperature\.

###### Proposition 3\.8\(Pressure\-targeted normalization\)\.

Fixtt, assumectc\_\{t\}is nonconstant, and choose a scalarata\_\{t\}such thatmini⁡ct​\(i\)<at<⟨pt,ct⟩\\min\_\{i\}c\_\{t\}\(i\)<a\_\{t\}<\\langle p\_\{t\},c\_\{t\}\\rangle\. Then there is a uniqueηt\>0\\eta\_\{t\}\>0satisfying

−ηt−1​log⁡\(∑i=1Kpt​\(i\)​e−ηt​ct​\(i\)\)=at\-\\eta\_\{t\}^\{\-1\}\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}\\right\)=a\_\{t\}\(39\)Equivalently, for the shifted scoresc~t​\(i\):=ct​\(i\)−at\\tilde\{c\}\_\{t\}\(i\):=c\_\{t\}\(i\)\-a\_\{t\},

∑i=1Kpt​\(i\)​e−ηt​c~t​\(i\)=1\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}\\tilde\{c\}\_\{t\}\(i\)\}=1\(40\)With the exponential\-weights updatept\+1​\(i\)∝pt​\(i\)​e−ηt​ct​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}, one has, for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

KL​\(ρ∥pt\+1\)−KL​\(ρ∥pt\)=ηt​\(⟨ρ,ct⟩−at\)\\mathrm\{KL\}\(\\rho\\\|p\_\{t\+1\}\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)=\\eta\_\{t\}\\left\(\\langle\\rho,c\_\{t\}\\rangle\-a\_\{t\}\\right\)\(41\)Ifctc\_\{t\}andata\_\{t\}are simultaneously scaled by a constantb\>0b\>0, then the solution scales asηt/b\\eta\_\{t\}/b\.

###### Corollary 3\.9\(Unit\-potential normalization\)\.

The special choiceat=0a\_\{t\}=0is feasible whenever⟨pt,ct⟩\>0\>mini⁡ct​\(i\)\\langle p\_\{t\},c\_\{t\}\\rangle\>0\>\\min\_\{i\}c\_\{t\}\(i\)\. In that case the pressure\-target root condition \([40](https://arxiv.org/html/2607.08789#S3.E40)\) becomes∑i=1Kpt​\(i\)​e−ηt​ct​\(i\)=1\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}=1, which is the unit\-potential or pressure\-normalized schedule\.

##### Thermodynamic interpretation and scale\-freeness\.

The normalization viewpoint is not just linguistic\. Up to the usual sign convention,mt​\(η\)=−η−1​log​∑ipt​\(i\)​e−η​ct​\(i\)m\_\{t\}\(\\eta\)=\-\\eta^\{\-1\}\\log\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}is the one\-step free\-energy density associated with the current score vector, whileη−1​log​∑ipt​\(i\)​e−η​\(ct​\(i\)−at\)\\eta^\{\-1\}\\log\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta\(c\_\{t\}\(i\)\-a\_\{t\}\)\}is the corresponding pressure relative to the target levelata\_\{t\}\. Solving \([39](https://arxiv.org/html/2607.08789#S3.E39)\) therefore enforces a prescribed free\-energy level, or equivalently normalizes the shifted pressure to zero\. This is why we call the rule*pressure\-targeted normalization*\. It is also naturally scale\-free: simultaneous rescaling\(ct,at\)↦\(b​ct,b​at\)\(c\_\{t\},a\_\{t\}\)\\mapsto\(bc\_\{t\},ba\_\{t\}\)simply sends the solution toηt/b\\eta\_\{t\}/b, so only the relative geometry of the current round matters\.

###### Corollary 3\.10\(Pressure\-target regret identity\)\.

Under Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8), for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

⟨pt,ct⟩−⟨ρ,ct⟩=\(⟨pt,ct⟩−at\)\+KL​\(ρ∥pt\)−KL​\(ρ∥pt\+1\)ηt\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle=\\left\(\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\\right\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{t\+1\}\)\}\{\\eta\_\{t\}\}\(42\)Consequently,

RTc​\(ρ\)=∑t=1T\(⟨pt,ct⟩−at\)\+∑t=1TKL​\(ρ∥pt\)−KL​\(ρ∥pt\+1\)ηtR\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\left\(\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\\right\)\+\\sum\_\{t=1\}^\{T\}\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{t\+1\}\)\}\{\\eta\_\{t\}\}\(43\)Equivalently,

RTc​\(ρ\)=∑t=1T\(⟨pt,ct⟩−at\)\+KL​\(ρ∥p1\)η1−KL​\(ρ∥pT\+1\)ηT\+∑t=2TKL​\(ρ∥pt\)​\(1ηt−1ηt−1\)R\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\left\(\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\\right\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\}\{\\eta\_\{1\}\}\-\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\}\{\\eta\_\{T\}\}\+\\sum\_\{t=2\}^\{T\}\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\\\!\\left\(\\frac\{1\}\{\\eta\_\{t\}\}\-\\frac\{1\}\{\\eta\_\{t\-1\}\}\\right\)\(44\)If\(ηt\)\(\\eta\_\{t\}\)is nondecreasing, then

RTc​\(ρ\)≤∑t=1T\(⟨pt,ct⟩−at\)\+KL​\(ρ∥p1\)η1R\_\{T\}^\{c\}\(\\rho\)\\leq\\sum\_\{t=1\}^\{T\}\\left\(\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\\right\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\}\{\\eta\_\{1\}\}\(45\)

This bridges the fully analyzed second\-order schedule and the earlier root\-finding remark\. The second\-order theorem may be read as an especially explicit way of choosing target levelsat=mt​\(ηt\)a\_\{t\}=m\_\{t\}\(\\eta\_\{t\}\)and then controlling the resulting cumulative gap∑t\(⟨pt,ct⟩−at\)\\sum\_\{t\}\(\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\)on a square\-root scale\.

The key structural point is that Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8)defines a genuinely different adaptive algorithm from the prior\-retempered update \([11](https://arxiv.org/html/2607.08789#S2.E11)\): the temperature is chosen by line search against the current distributionptp\_\{t\}and the current score vectorctc\_\{t\}, and the recursion then proceeds locally viapt\+1​\(i\)∝pt​\(i\)​e−ηt​ct​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}\. Thus the schedule does not retemper the entire cumulative history at each step\. When the temperature is held fixed the local and retempered views coincide, but for variable rates the distinction is real\. In particular, the pressure\-target rule is genuinely local: onceptp\_\{t\},ctc\_\{t\}, and the targetata\_\{t\}are given, no separate cumulative budget such asVt−1​\(c\)V\_\{t\-1\}\(c\)needs to be tracked\. All historical information is compressed into the current weights themselves\.

This distinction changes the natural proof object\. For the prior\-retempered update, the exact chain in Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)is organized by the cumulative free energy

At​\(η\):=−η−1​log⁡\(∑iπ​\(i\)​e−η​Ct​\(i\)\)A\_\{t\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\\!\\left\(\\sum\_\{i\}\\pi\(i\)e^\{\-\\eta C\_\{t\}\(i\)\}\\right\)evaluated at the played temperature\. For the variable\-rate local recursion, that retempered free energy is generally not the state variable of the algorithm\. The native cumulative object is instead the exact normalization of the realized weighted scores from Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13)\. Consequently the primitive exact statement for the local schedule is a weighted\-regret identity; ordinary regret is then recovered from Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)by the Abel transform when one wants a scalar unweighted bound\.

#### 3\.3\.1Fixed\-temperature and direct\-adaptive variational forms

The fixed\-temperature profile is the common\-temperature specialization of the same update, while the direct\-adaptive statements below are the native identities for the realized pressure\-target path itself\. No bounded\-interval restriction on the temperatures is built into these formulas; one only needs the relevant one\-step log\-partition to be finite at the chosen value\.

###### Proposition 3\.11\(Fixed\-temperature variational identity for the local recursion\)\.

Fixη\>0\\eta\>0and run the local exponential\-weights recursion

p1\(η\):=πpt\+1\(η\)​\(i\)∝pt\(η\)​\(i\)​e−η​ct​\(i\)p\_\{1\}^\{\(\\eta\)\}:=\\pi\\qquad p\_\{t\+1\}^\{\(\\eta\)\}\(i\)\\propto p\_\{t\}^\{\(\\eta\)\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}For each round define

ψt\(η\)​\(λ\):=log⁡𝔼i∼pt\(η\)​\[exp⁡\(λ​\(ct​\(i\)−⟨pt\(η\),ct⟩\)\)\]\\psi\_\{t\}^\{\(\\eta\)\}\(\\lambda\):=\\log\\mathbb\{E\}\_\{i\\sim p\_\{t\}^\{\(\\eta\)\}\}\\left\[\\exp\\\!\\left\(\\lambda\\bigl\(c\_\{t\}\(i\)\-\\langle p\_\{t\}^\{\(\\eta\)\},c\_\{t\}\\rangle\\bigr\)\\right\)\\right\]Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

RTc,\(η\)​\(ρ\)=∑t=1Tψt\(η\)​\(−η\)η\+KL​\(ρ∥π\)−KL​\(ρ∥pT\+1\(η\)\)ηR\_\{T\}^\{c,\(\\eta\)\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\frac\{\\psi\_\{t\}^\{\(\\eta\)\}\(\-\\eta\)\}\{\\eta\}\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{\(\\eta\)\}\)\}\{\\eta\}\(46\)where

RTc,\(η\)​\(ρ\):=∑t=1T⟨pt\(η\),ct⟩−⟨ρ,CT⟩R\_\{T\}^\{c,\(\\eta\)\}\(\\rho\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}^\{\(\\eta\)\},c\_\{t\}\\rangle\-\\langle\\rho,C\_\{T\}\\rangle

###### Corollary 3\.12\(Self\-correcting fixed\-temperature envelopes\)\.

Under Proposition[3\.11](https://arxiv.org/html/2607.08789#S3.Thmtheorem11), letvt​\(η\)v\_\{t\}\(\\eta\)be any quantities satisfying

vt​\(η\)≥ψt\(η\)​\(−η\)ηfor all​tv\_\{t\}\(\\eta\)\\geq\\frac\{\\psi\_\{t\}^\{\(\\eta\)\}\(\-\\eta\)\}\{\\eta\}\\qquad\\text\{for all \}tThen

RTc,\(η\)​\(ρ\)=KL​\(ρ∥π\)η\+∑t=1Tvt​\(η\)−ΔTloc​\(ρ,η\)R\_\{T\}^\{c,\(\\eta\)\}\(\\rho\)=\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\}\+\\sum\_\{t=1\}^\{T\}v\_\{t\}\(\\eta\)\-\\Delta\_\{T\}^\{\\mathrm\{loc\}\}\(\\rho,\\eta\)\(47\)with the exact nonnegative gap

ΔTloc​\(ρ,η\):=KL​\(ρ∥pT\+1\(η\)\)η\+∑t=1T\(vt​\(η\)−ψt\(η\)​\(−η\)η\)≥0\\Delta\_\{T\}^\{\\mathrm\{loc\}\}\(\\rho,\\eta\):=\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{\(\\eta\)\}\)\}\{\\eta\}\+\\sum\_\{t=1\}^\{T\}\\left\(v\_\{t\}\(\\eta\)\-\\frac\{\\psi\_\{t\}^\{\(\\eta\)\}\(\-\\eta\)\}\{\\eta\}\\right\)\\geq 0

This is the clean variational form of the fixed\-temperature profile\. Any chosen surrogate for the centered cumulant, whether quadratic, self\-normalized, or otherwise, differs from the truth by an explicit slack variable\. One may of course insert a bounded\-range Hoeffding proxy when available, but no such assumption is part of the identity itself\.

###### Proposition 3\.13\(Cumulative normalization and direct weighted identity for the realized pressure\-target path\)\.

Under Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8), define

GT​\(i\):=∑t=1Tηt​\(ct​\(i\)−at\)G\_\{T\}\(i\):=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(c\_\{t\}\(i\)\-a\_\{t\}\\bigr\)Then

pT\+1​\(i\)=p1​\(i\)​e−GT​\(i\)and∑i=1Kp1​\(i\)​e−GT​\(i\)=1p\_\{T\+1\}\(i\)=p\_\{1\}\(i\)e^\{\-G\_\{T\}\(i\)\}\\qquad\\text\{and\}\\qquad\\sum\_\{i=1\}^\{K\}p\_\{1\}\(i\)e^\{\-G\_\{T\}\(i\)\}=1\(48\)Consequently, for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

∑t=1Tηt​\(⟨ρ,ct⟩−at\)=KL​\(ρ∥pT\+1\)−KL​\(ρ∥p1\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\langle\\rho,c\_\{t\}\\rangle\-a\_\{t\}\\bigr\)=\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\(49\)Equivalently, writing

μt:=⟨pt,ct⟩ψt​\(λ\):=log⁡𝔼i∼pt​\[exp⁡\(λ​\(ct​\(i\)−μt\)\)\]Qtloc​\(c\):=ηt−2​ψt​\(−ηt\)=ηt−1​\(μt−at\)\\mu\_\{t\}:=\\langle p\_\{t\},c\_\{t\}\\rangle\\qquad\\psi\_\{t\}\(\\lambda\):=\\log\\mathbb\{E\}\_\{i\\sim p\_\{t\}\}\\left\[\\exp\\\!\\left\(\\lambda\\bigl\(c\_\{t\}\(i\)\-\\mu\_\{t\}\\bigr\)\\right\)\\right\]\\qquad Q\_\{t\}^\{\\mathrm\{loc\}\}\(c\):=\\eta\_\{t\}^\{\-2\}\\psi\_\{t\}\(\-\\eta\_\{t\}\)=\\eta\_\{t\}^\{\-1\}\\bigl\(\\mu\_\{t\}\-a\_\{t\}\\bigr\)one has

∑t=1Tηt​\(μt−⟨ρ,ct⟩\)=∑t=1Tηt2​Qtloc​\(c\)\+KL​\(ρ∥p1\)−KL​\(ρ∥pT\+1\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\mu\_\{t\}\-\\langle\\rho,c\_\{t\}\\rangle\\bigr\)=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}^\{2\}Q\_\{t\}^\{\\mathrm\{loc\}\}\(c\)\+\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\(50\)

Equation \([48](https://arxiv.org/html/2607.08789#S3.E48)\) is the global effect of the local line search: the schedule exactly renormalizes the final exponentially weighted score, not merely up to a bound\. This is the local update’s counterpart of the exact intrinsic\-time payment in Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10): both are exact information\-tracking statements, but they track different cumulative objects\. The second\-order schedule for the retempered update controls its cumulative payment by a square\-root law in a monotone budget, whereas the pressure\-target schedule for the local update enforces a chosen one\-step target and therefore accumulates an exact cumulative normalization identity instead\. In both cases, the same exact\-cumulant functionalQt​\(c\)Q\_\{t\}\(c\)supplies the per\-round information cost—evaluated at each recursion’s own played distribution—and the terminal object measures how much comparator information remains in the final posterior\. The two updates therefore share the same composite\-loss reduction and the same round\-by\-round information functional, and differ both in the realized played distributions under a variable schedule and in how the cumulative bookkeeping is organized\. This coincidence at a shared rate and the controlled divergence under adaptive rates are confirmed numerically in Appendix[C\.3\.4](https://arxiv.org/html/2607.08789#A3.SS3.SSS4): the two updates’ realized increments agree to machine precision at a common temperature and separate monotonically with the total\-variation gap between their played distributions\.

#### 3\.3\.2Terminal\-mass identities and event\-conditioned weighted regret

The local normalization also has a finite\-scale coincidence interpretation\. Fix a roundttand set

Zt​\(η\):=∑i=1Kpt​\(i\)​e−η​ct​\(i\),μt:=⟨pt,ct⟩,rt​\(i\):=pt​\(i\)​e−ct​\(i\)Zt​\(1\)Z\_\{t\}\(\\eta\):=\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\},\\qquad\\mu\_\{t\}:=\\langle p\_\{t\},c\_\{t\}\\rangle,\\qquad r\_\{t\}\(i\):=\\frac\{p\_\{t\}\(i\)e^\{\-c\_\{t\}\(i\)\}\}\{Z\_\{t\}\(1\)\}Thusrtr\_\{t\}is the unit\-temperature posterior formed from the current weights and the current score vector, whether or not the algorithm actually plays temperature11\. The next proposition factors the one\-round normalizerZt​\(η\)Z\_\{t\}\(\\eta\)through the overlap between the present distributionptp\_\{t\}and that unit\-temperature posterior\.

###### Proposition 3\.14\(One\-round coincidence factorization of the local normalizer\)\.

For everyη\>0\\eta\>0,

Zt​\(η\)=Zt​\(1\)η​𝒞t​\(η\),𝒞t​\(η\):=∑i=1Kpt​\(i\)1−η​rt​\(i\)ηZ\_\{t\}\(\\eta\)=Z\_\{t\}\(1\)^\{\\eta\}\\mathcal\{C\}\_\{t\}\(\\eta\),\\qquad\\mathcal\{C\}\_\{t\}\(\\eta\):=\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)^\{1\-\\eta\}r\_\{t\}\(i\)^\{\\eta\}\(51\)Consequently, with

δtloc​\(η\):=μt\+η−1​log⁡Zt​\(η\)\\delta\_\{t\}^\{\\mathrm\{loc\}\}\(\\eta\):=\\mu\_\{t\}\+\\eta^\{\-1\}\\log Z\_\{t\}\(\\eta\)one has

δtloc​\(η\)=δtloc​\(1\)\+η−1​log⁡𝒞t​\(η\)=δtloc​\(1\)−1−ηη​Dη​\(rt∥pt\)\\delta\_\{t\}^\{\\mathrm\{loc\}\}\(\\eta\)=\\delta\_\{t\}^\{\\mathrm\{loc\}\}\(1\)\+\\eta^\{\-1\}\\log\\mathcal\{C\}\_\{t\}\(\\eta\)=\\delta\_\{t\}^\{\\mathrm\{loc\}\}\(1\)\-\\frac\{1\-\\eta\}\{\\eta\}D\_\{\\eta\}\(r\_\{t\}\\\|p\_\{t\}\)\(52\)where

Dη​\(r∥p\):=1η−1​log⁡\(∑i=1Kr​\(i\)η​p​\(i\)1−η\)D\_\{\\eta\}\(r\\\|p\):=\\frac\{1\}\{\\eta\-1\}\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}r\(i\)^\{\\eta\}p\(i\)^\{1\-\\eta\}\\right\)is the order\-η\\etaRényi divergence\. For0<η<10<\\eta<1,𝒞t​\(η\)∈\[0,1\]\\mathcal\{C\}\_\{t\}\(\\eta\)\\in\[0,1\]is the Hellinger/Rényi overlap betweenptp\_\{t\}andrtr\_\{t\}\.

Thus the local finite\-temperature payment is governed by a coincidence factor in the one\-step normalizer, not by the retempered free energy from Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)\. At any fixed round, changing the temperature changes the one\-step payment exactly through the overlap term𝒞t​\(η\)\\mathcal\{C\}\_\{t\}\(\\eta\), or equivalently through the Rényi divergence between the current weights and the unit\-temperature posterior\.

###### Proposition 3\.15\(Terminal mass and event\-conditioned weighted regret\)\.

Under the notation of Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13), letE⊆\[K\]E\\subseteq\[K\]be nonempty and define

Δ​\(E\):=\{ρ∈Δ​\(\[K\]\):ρ​\(E\)=1\},𝒵T​\(E\):=∑i∈Ep1​\(i\)​e−GT​\(i\)=pT\+1​\(E\)\\Delta\(E\):=\\\{\\rho\\in\\Delta\(\[K\]\):\\rho\(E\)=1\\\},\\qquad\\mathcal\{Z\}\_\{T\}\(E\):=\\sum\_\{i\\in E\}p\_\{1\}\(i\)e^\{\-G\_\{T\}\(i\)\}=p\_\{T\+1\}\(E\)If𝒵T​\(E\)\>0\\mathcal\{Z\}\_\{T\}\(E\)\>0, letpT\+1Ep\_\{T\+1\}^\{E\}denote the restriction ofpT\+1p\_\{T\+1\}toEE, namelypT\+1E​\(i\):=pT\+1​\(i\)/pT\+1​\(E\)p\_\{T\+1\}^\{E\}\(i\):=p\_\{T\+1\}\(i\)/p\_\{T\+1\}\(E\)fori∈Ei\\in E\. Then for every posteriorρ∈Δ​\(E\)\\rho\\in\\Delta\(E\),

−log⁡𝒵T​\(E\)\+KL​\(ρ∥pT\+1E\)=∑t=1Tηt​\(⟨ρ,ct⟩−at\)\+KL​\(ρ∥p1\)\-\\log\\mathcal\{Z\}\_\{T\}\(E\)\+\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{E\}\)=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\langle\\rho,c\_\{t\}\\rangle\-a\_\{t\}\\bigr\)\+\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\(53\)Equivalently,

∑t=1Tηt​\(μt−⟨ρ,ct⟩\)−KL​\(ρ∥p1\)=∑t=1Tηt2​Qtloc​\(c\)\+log⁡pT\+1​\(E\)−KL​\(ρ∥pT\+1E\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\mu\_\{t\}\-\\langle\\rho,c\_\{t\}\\rangle\\bigr\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}^\{2\}Q\_\{t\}^\{\\mathrm\{loc\}\}\(c\)\+\\log p\_\{T\+1\}\(E\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{E\}\)\(54\)Consequently,

−log⁡𝒵T​\(E\)=minρ∈Δ​\(E\)⁡\{∑t=1Tηt​\(⟨ρ,ct⟩−at\)\+KL​\(ρ∥p1\)\}\-\\log\\mathcal\{Z\}\_\{T\}\(E\)=\\min\_\{\\rho\\in\\Delta\(E\)\}\\left\\\{\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\langle\\rho,c\_\{t\}\\rangle\-a\_\{t\}\\bigr\)\+\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\\right\\\}\(55\)and

supρ∈Δ​\(E\)\{∑t=1Tηt​\(μt−⟨ρ,ct⟩\)−KL​\(ρ∥p1\)\}=∑t=1Tηt2​Qtloc​\(c\)\+log⁡pT\+1​\(E\)\\sup\_\{\\rho\\in\\Delta\(E\)\}\\left\\\{\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\mu\_\{t\}\-\\langle\\rho,c\_\{t\}\\rangle\\bigr\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\\right\\\}=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}^\{2\}Q\_\{t\}^\{\\mathrm\{loc\}\}\(c\)\+\\log p\_\{T\+1\}\(E\)\(56\)

WhenE=\[K\]E=\[K\], one haslog⁡pT\+1​\(E\)=0\\log p\_\{T\+1\}\(E\)=0, so \([56](https://arxiv.org/html/2607.08789#S3.E56)\) reduces to Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13)\. For smaller classes, the extra termlog⁡pT\+1​\(E\)≤0\\log p\_\{T\+1\}\(E\)\\leq 0is an exact class correction: any class that ends with very small terminal mass under the final posterior pays for that scarcity\. Typical choices ofEEinclude anϵ\\epsilon\-quantile set of experts, a sparsity pattern, a model family, or an agreement set singled out by side information\. If one also wants an ordinary unweighted regret bound, the same correction survives the Abel transform: for everyρ∈Δ​\(E\)\\rho\\in\\Delta\(E\),

RTc​\(ρ\)≤∑t=1T\(μt−at\)\+KL​\(ρ∥p1\)η1\+log⁡pT\+1​\(E\)ηT\+∑t=2TKL​\(ρ∥pt\)​\(1ηt−1ηt−1\)R\_\{T\}^\{c\}\(\\rho\)\\leq\\sum\_\{t=1\}^\{T\}\\bigl\(\\mu\_\{t\}\-a\_\{t\}\\bigr\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\}\{\\eta\_\{1\}\}\+\\frac\{\\log p\_\{T\+1\}\(E\)\}\{\\eta\_\{T\}\}\+\\sum\_\{t=2\}^\{T\}\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\\\!\\left\(\\frac\{1\}\{\\eta\_\{t\}\}\-\\frac\{1\}\{\\eta\_\{t\-1\}\}\\right\)by combining Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)withKL​\(ρ∥pT\+1\)=KL​\(ρ∥pT\+1E\)−log⁡pT\+1​\(E\)\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)=\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{E\}\)\-\\log p\_\{T\+1\}\(E\)\. Thus the terminal\-mass term is genuinely part of the regret accounting; the only extra ingredient needed to pass from weighted to ordinary regret is the reciprocal\-temperature variation term\.

##### Remark\.

If the composite scores come from the mixed\-coincidence reduction of Section[2\.1](https://arxiv.org/html/2607.08789#S2.SS1), namelyct​\(i\)=ℓt​\(i\)−ηt−1​log⁡st​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\-\\eta\_\{t\}^\{\-1\}\\log s\_\{t\}\(i\)withsts\_\{t\}of the form \([6](https://arxiv.org/html/2607.08789#S2.E6)\), then

pT\+1​\(E\)=exp⁡\(∑t=1Tηt​at\)​∑i∈Ep1​\(i\)​exp⁡\(−∑t=1Tηt​ℓt​\(i\)\)​∏t=1T∏w=1Jtμt,w​\(i\)αt,wp\_\{T\+1\}\(E\)=\\exp\\\!\\Bigl\(\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}a\_\{t\}\\Bigr\)\\sum\_\{i\\in E\}p\_\{1\}\(i\)\\exp\\\!\\Bigl\(\-\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\ell\_\{t\}\(i\)\\Bigr\)\\prod\_\{t=1\}^\{T\}\\prod\_\{w=1\}^\{J\_\{t\}\}\\mu\_\{t,w\}\(i\)^\{\\alpha\_\{t,w\}\}Thus the terminal mass ofEEis, up to the explicit scalar factorexp⁡\(∑t=1Tηt​at\)\\exp\(\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}a\_\{t\}\), a restricted mixed\-coincidence quantity\. The correctionlog⁡pT\+1​\(E\)\\log p\_\{T\+1\}\(E\)in \([54](https://arxiv.org/html/2607.08789#S3.E54)\) therefore measures how much cumulative geometric agreement remains inside the classEE\.

#### 3\.3\.3Centered scores and quadratic envelopes

###### Corollary 3\.16\(Centered\-score specialization and exact adaptive regret theorem\)\.

Assume that on each round one observes a score vectorgt∈ℝKg\_\{t\}\\in\\mathbb\{R\}^\{K\}satisfying⟨pt,gt⟩=0\\langle p\_\{t\},g\_\{t\}\\rangle=0\. Ifgtg\_\{t\}is nonconstant, choose a targetbtb\_\{t\}with0<bt<maxi⁡gt​\(i\)0<b\_\{t\}<\\max\_\{i\}g\_\{t\}\(i\)and letηt\>0\\eta\_\{t\}\>0be the unique solution of

log⁡\(∑i=1Kpt​\(i\)​eηt​gt​\(i\)\)=ηt​bt\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\\eta\_\{t\}g\_\{t\}\(i\)\}\\right\)=\\eta\_\{t\}b\_\{t\}\(57\)Ifgt≡0g\_\{t\}\\equiv 0, setbt:=0b\_\{t\}:=0and choose anyηt\>0\\eta\_\{t\}\>0\. Update

pt\+1​\(i\)∝pt​\(i\)​eηt​gt​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\\eta\_\{t\}g\_\{t\}\(i\)\}Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

∑i=1Kp1​\(i\)​exp⁡\(∑t=1Tηt​\(gt​\(i\)−bt\)\)=1\\sum\_\{i=1\}^\{K\}p\_\{1\}\(i\)\\exp\\\!\\left\(\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(g\_\{t\}\(i\)\-b\_\{t\}\\bigr\)\\right\)=1\(58\)and

∑t=1Tηt​\(⟨ρ,gt⟩−bt\)=KL​\(ρ∥p1\)−KL​\(ρ∥pT\+1\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\langle\\rho,g\_\{t\}\\rangle\-b\_\{t\}\\bigr\)=\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\(59\)Hence

∑t=1Tηt​⟨ρ,gt⟩≤KL​\(ρ∥p1\)\+∑t=1Tηt​bt\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\,\\langle\\rho,g\_\{t\}\\rangle\\leq\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}b\_\{t\}\(60\)If moreovergt​\(i\)=⟨pt,ℓt⟩−ℓt​\(i\)g\_\{t\}\(i\)=\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\ell\_\{t\}\(i\), then⟨ρ,gt⟩=⟨pt,ℓt⟩−⟨ρ,ℓt⟩\\langle\\rho,g\_\{t\}\\rangle=\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,\\ell\_\{t\}\\rangle, so \([59](https://arxiv.org/html/2607.08789#S3.E59)\) is an exact identity for theη\\eta\-weighted regret∑tηt​\(⟨pt,ℓt⟩−⟨ρ,ℓt⟩\)\\sum\_\{t\}\\eta\_\{t\}\(\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,\\ell\_\{t\}\\rangle\)of the realized line\-search schedule; Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)recovers the ordinary unweighted regret by an Abel transform\.

Combining Corollary[3\.16](https://arxiv.org/html/2607.08789#S3.Thmtheorem16)with Proposition[3\.15](https://arxiv.org/html/2607.08789#S3.Thmtheorem15)and takingct=−gtc\_\{t\}=\-g\_\{t\},at=−bta\_\{t\}=\-b\_\{t\}gives, for every nonemptyE⊆\[K\]E\\subseteq\[K\]withpT\+1​\(E\)\>0p\_\{T\+1\}\(E\)\>0,

supρ∈Δ​\(E\)\{∑t=1Tηt​⟨ρ,gt⟩−KL​\(ρ∥p1\)\}=∑t=1Tηt​bt\+log⁡pT\+1​\(E\)\\sup\_\{\\rho\\in\\Delta\(E\)\}\\left\\\{\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\langle\\rho,g\_\{t\}\\rangle\-\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\\right\\\}=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}b\_\{t\}\+\\log p\_\{T\+1\}\(E\)Thus the exact price of forcing the comparator to lie in a gain\-bearing classEEis again the negative logarithm of that class’s final posterior mass\.

###### Corollary 3\.17\(Exact quadratic\-penalty representation of the local line search\)\.

Under Corollary[3\.16](https://arxiv.org/html/2607.08789#S3.Thmtheorem16), define

κt:=btηt=1ηt2​log⁡\(∑i=1Kpt​\(i\)​eηt​gt​\(i\)\)\\kappa\_\{t\}:=\\frac\{b\_\{t\}\}\{\\eta\_\{t\}\}=\\frac\{1\}\{\\eta\_\{t\}^\{2\}\}\\log\\\!\\left\(\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\\eta\_\{t\}g\_\{t\}\(i\)\}\\right\)Then the target equation is equivalently

∑i=1Kpt​\(i\)​exp⁡\(ηt​gt​\(i\)−κt​ηt2\)=1\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)\\exp\\\!\\left\(\\eta\_\{t\}g\_\{t\}\(i\)\-\\kappa\_\{t\}\\eta\_\{t\}^\{2\}\\right\)=1\(61\)and for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

∑t=1Tηt​⟨ρ,gt⟩=∑t=1Tκt​ηt2\+KL​\(ρ∥p1\)−KL​\(ρ∥pT\+1\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\,\\langle\\rho,g\_\{t\}\\rangle=\\sum\_\{t=1\}^\{T\}\\kappa\_\{t\}\\eta\_\{t\}^\{2\}\+\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\(62\)Consequently, if\(κ¯t\)\(\\bar\{\\kappa\}\_\{t\}\)is any predictable sequence withκ¯t≥κt\\bar\{\\kappa\}\_\{t\}\\geq\\kappa\_\{t\}for alltt, then

∑t=1Tηt​⟨ρ,gt⟩≤KL​\(ρ∥p1\)\+∑t=1Tκ¯t​ηt2\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\,\\langle\\rho,g\_\{t\}\\rangle\\leq\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\+\\sum\_\{t=1\}^\{T\}\\bar\{\\kappa\}\_\{t\}\\eta\_\{t\}^\{2\}\(63\)

This is the clean repair of the conventional quadratic\-penalty proof strategy\. The coefficientκt\\kappa\_\{t\}is not an externally imposed surrogate but the realized finite\-temperature cumulant coefficient selected by the line search itself\. Replacing it by any predictable upper envelopeκ¯t\\bar\{\\kappa\}\_\{t\}converts the exact identity into the familiar supermartingale\-style inequality, but that surrogate is now visibly a proof device rather than the definition of the algorithm\.

##### Small\-temperature relation to second order\.

When the one\-step cgf ofgtg\_\{t\}is three times differentiable at the origin,

log⁡𝔼i∼pt​eη​gt​\(i\)=η22​Vari∼pt​\(gt​\(i\)\)\+O​\(η3\)\(η↓0\)\\log\\mathbb\{E\}\_\{i\\sim p\_\{t\}\}e^\{\\eta g\_\{t\}\(i\)\}=\\frac\{\\eta^\{2\}\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(g\_\{t\}\(i\)\)\+O\(\\eta^\{3\}\)\\qquad\(\\eta\\downarrow 0\)so the exact coefficient in Corollary[3\.17](https://arxiv.org/html/2607.08789#S3.Thmtheorem17)satisfies

κt=12​Vari∼pt​\(gt​\(i\)\)\+O​\(ηt\),bt=ηt2​Vari∼pt​\(gt​\(i\)\)\+O​\(ηt2\)\\kappa\_\{t\}=\\frac\{1\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(g\_\{t\}\(i\)\)\+O\(\\eta\_\{t\}\),\\qquad b\_\{t\}=\\frac\{\\eta\_\{t\}\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(g\_\{t\}\(i\)\)\+O\(\\eta\_\{t\}^\{2\}\)Thus the usual quadratic\-variation penalties are the small\-temperature shadow of the exact pressure\-target line search\. What the present section adds is that the same structure remains valid at finite temperature, without restrictingηt\\eta\_\{t\}to a bounded interval\.

##### How to choose the target\.

The targetata\_\{t\}is problem dependent\. For general score vectors with positive current mean, the unit\-potential ruleat=0a\_\{t\}=0is scale invariant and yieldsZt=1Z\_\{t\}=1exactly\. For centered excess scores with⟨pt,gt⟩=0\\langle p\_\{t\},g\_\{t\}\\rangle=0, however, the zero target collapses to the zero\-temperature limit, so the natural parameterization is instead the positive gain targetbt=−atb\_\{t\}=\-a\_\{t\}of Corollary[3\.16](https://arxiv.org/html/2607.08789#S3.Thmtheorem16)\. A fixed\-fraction targetat=λt​⟨pt,ct⟩a\_\{t\}=\\lambda\_\{t\}\\langle p\_\{t\},c\_\{t\}\\ranglewith0<λt<10<\\lambda\_\{t\}<1keeps a chosen fraction of the current mean score at the mix\-loss level\. In optimistic or model\-based settings one can instead target a predictable benchmark, so that⟨pt,ct⟩−at\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}measures only the residual part that the learner failed to explain away at roundtt\. The information\-targeted version of Section[3\.1](https://arxiv.org/html/2607.08789#S3.SS1)can also be used locally: for a candidateη\\eta, letpt\+1\(η\)​\(i\)∝pt​\(i\)​e−η​ct​\(i\)p\_\{t\+1\}^\{\(\\eta\)\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}and setItloc​\(η\):=KL​\(pt\+1\(η\)∥pt\)=η​\(mt​\(η\)−⟨pt\+1\(η\),ct⟩\)I\_\{t\}^\{\\rm loc\}\(\\eta\):=\\mathrm\{KL\}\(p\_\{t\+1\}^\{\(\\eta\)\}\\\|p\_\{t\}\)=\\eta\(m\_\{t\}\(\\eta\)\-\\langle p\_\{t\+1\}^\{\(\\eta\)\},c\_\{t\}\\rangle\); choosing a quotaβt\\beta\_\{t\}and solvingItloc​\(ηt\)=βtI\_\{t\}^\{\\rm loc\}\(\\eta\_\{t\}\)=\\beta\_\{t\}induces the pressure targetat=mt​\(ηt\)a\_\{t\}=m\_\{t\}\(\\eta\_\{t\}\)\.

##### Nonmonotone and other nonstandard rate schedules\.

The calibrated temperature in Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8)need not move monotonically in time\. Even at fixed weightspt=\(1/2,1/2\)p\_\{t\}=\(1/2,1/2\), the positive solutions of \([40](https://arxiv.org/html/2607.08789#S3.E40)\) for the zero targetat=0a\_\{t\}=0and the score vectors\(−1,2\)\(\-1,2\),\(−1,4\)\(\-1,4\), and\(−1,3/2\)\(\-1,3/2\)are approximately0\.480\.48,0\.660\.66, and0\.330\.33, respectively\. Thus an exact pressure\-target schedule can increase and later decrease in response to the realized score geometry\. This is fully compatible with Theorem[2\.8](https://arxiv.org/html/2607.08789#S2.Thmtheorem8): the exact chain holds for every predictable positive schedule, while monotonicity matters only when one wants to drop the reciprocal\-temperature variation term and obtain a one\-sided envelope\. Unlike the second\-order schedule, the pressure\-target family is defined implicitly and is designed to hit a local target rather than to maintain a monotone global budget\. The price of that flexibility is that one generally has to keep the reciprocal\-temperature variation term in \([44](https://arxiv.org/html/2607.08789#S3.E44)\) unless additional monotonicity is available\.

##### A finite\-scale regret spectrum\.

For a fixed horizonTTand temperatureη\\eta, the terminal weightsμi:=pT\+1​\(i\)\\mu\_\{i\}:=p\_\{T\+1\}\(i\)define a simple Rényi profileDq​\(T\):=−Hq​\(μ\)/log⁡TD\_\{q\}\(T\):=\-H\_\{q\}\(\\mu\)/\\log T, whereHqH\_\{q\}is the order\-qqRényi entropy\. Interpreted as a*regret spectrum*,D0​\(T\)D\_\{0\}\(T\)measures the effective number of experts still in play,D1​\(T\)D\_\{1\}\(T\)is entropy per unit log\-time, and curvature ofq↦Dq​\(T\)q\\mapsto D\_\{q\}\(T\)records how heterogeneous the loss landscape has become\. Flat spectra correspond to near\-indistinguishable experts; strongly curved spectra indicate a genuinely multiscale hierarchy of good experts\.

## 4Side information, comparators, and luckiness

Sections[2\.1](https://arxiv.org/html/2607.08789#S2.SS1)and[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)produced exact identities for composite losses, and Section[3](https://arxiv.org/html/2607.08789#S3)turned those identities into concrete rate rules\. We now return to the application layer\. Translating the chain back to the original losses records the exact side\-factor mismatch; varying the side information and the sufficient statistic fed to the Bayes update covers optimism, specialists, sparsity, and bounded\-influence transforms; enlarging the comparator class reaches shifting and quantile benchmarks; and overlaying stochastic structure yields fast\-rate, comparator\-centered Bernstein consequences\.

In particular, the bounded\-range/Hoeffding envelopes below are just the first relaxations of the exact identities, while the comparator\-centered second\-order results of Section[4\.5](https://arxiv.org/html/2607.08789#S4.SS5)are a later Bernstein\-style refinement of the same cumulant term rather than a different algorithm\. The section is therefore organized from exact translation, to statistic design, to richer comparators, and finally to luckiness\.

### 4\.1Returning to the original losses

Sections[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)–[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)treated the composite losses as the primary object\. We now translate those exact identities back to the original losses and the side factors from Section[2\.1](https://arxiv.org/html/2607.08789#S2.SS1)\. This step is purely algebraic, but it is what turns the reduction into application\-level regret statements\.

For every posteriorρ\\rho, the original\-loss and composite\-loss regrets are related pathwise by

∑t=1T⟨pt,ℓt⟩−⟨ρ,LT⟩=RTc​\(ρ\)\+∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}\\rangle=R\_\{T\}^\{c\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)\(64\)
###### Corollary 4\.1\(Exact original\-loss regret identity\)\.

Let the composite losses be of the formct​\(i\)=ℓt​\(i\)\+ut​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\+u\_\{t\}\(i\)withut​\(i\):=−ηt−1​log⁡st​\(i\)u\_\{t\}\(i\):=\-\\eta\_\{t\}^\{\-1\}\\log s\_\{t\}\(i\)\. Let\(pt\)\(p\_\{t\}\)be the variable\-temperature Bayes sequence \([11](https://arxiv.org/html/2607.08789#S2.E11)\)\. Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

∑t=1T⟨pt,ℓt⟩−⟨ρ,LT⟩=DT\+BT​\(ρ\)\+∑t=1Tηt​Qt​\(c\)\+∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}\\rangle=D\_\{T\}\+B\_\{T\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)\(65\)

The fixed\-rate specialization of the same identity yields the classical bounded\-range regret bound as a direct corollary\.

###### Corollary 4\.2\(Fixed\-rate bounded\-range regret bound\)\.

Under the same assumptions as Corollary[2\.4](https://arxiv.org/html/2607.08789#S2.Thmtheorem4), ifct​\(i\)∈\[at,bt\]c\_\{t\}\(i\)\\in\[a\_\{t\},b\_\{t\}\]for allii, then

∑t=1T⟨qt,ct⟩−⟨ρ,CT⟩≤η−1​KL​\(ρ∥q1\)\+η8​∑t=1T\(bt−at\)2\\sum\_\{t=1\}^\{T\}\\langle q\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,C\_\{T\}\\rangle\\leq\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|q\_\{1\}\)\+\\frac\{\\eta\}\{8\}\\sum\_\{t=1\}^\{T\}\(b\_\{t\}\-a\_\{t\}\)^\{2\}\(66\)and therefore

∑t=1T⟨qt,ℓt⟩−⟨ρ,LT⟩≤η−1​KL​\(ρ∥q1\)\+η8​∑t=1T\(bt−at\)2\+∑t=1T\(⟨ρ,ut⟩−⟨qt,ut⟩\)\\sum\_\{t=1\}^\{T\}\\langle q\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}\\rangle\\leq\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|q\_\{1\}\)\+\\frac\{\\eta\}\{8\}\\sum\_\{t=1\}^\{T\}\(b\_\{t\}\-a\_\{t\}\)^\{2\}\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle q\_\{t\},u\_\{t\}\\rangle\\right\)\(67\)

When the side factors have the decomposed form \([6](https://arxiv.org/html/2607.08789#S2.E6)\),

###### Corollary 4\.3\(Cross\-entropy form of the adaptive mismatch\)\.

The cross\-entropy form of the mismatch follows from the same calculation as Corollary[2\.5](https://arxiv.org/html/2607.08789#S2.Thmtheorem5), with the fixed rateη\\etareplaced by the round\-dependentηt\\eta\_\{t\}\. If

st​\(i\)=∏w=1Jtμt,w​\(i\)αt,ws\_\{t\}\(i\)=\\prod\_\{w=1\}^\{J\_\{t\}\}\\mu\_\{t,w\}\(i\)^\{\\alpha\_\{t,w\}\}\(68\)then the mismatch term in \([65](https://arxiv.org/html/2607.08789#S4.E65)\) is exactly

∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)=∑t=1T∑w=1Jtαt,wηt​\(H​\(ρ,μt,w\)−H​\(pt,μt,w\)\)\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)=\\sum\_\{t=1\}^\{T\}\\sum\_\{w=1\}^\{J\_\{t\}\}\\frac\{\\alpha\_\{t,w\}\}\{\\eta\_\{t\}\}\\left\(\\mathrm\{H\}\(\\rho,\\mu\_\{t,w\}\)\-\\mathrm\{H\}\(p\_\{t\},\\mu\_\{t,w\}\)\\right\)\(69\)

Substituting the two\-sided envelope of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)into the exact identity yields an explicit second\-order bound\.

###### Corollary 4\.4\(Second\-order original\-loss regret bounds\)\.

Under the schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\),

DT\+BT​\(ρ\)\+2​C​Γ​VT​\(c\)−C2​Γ\+∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)\\displaystyle D\_\{T\}\+B\_\{T\}\(\\rho\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\-C^\{2\}\\Gamma\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)≤∑t=1T⟨pt,ℓt⟩−⟨ρ,LT⟩\\displaystyle\\qquad\\leq\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}\\rangle≤DT\+BT​\(ρ\)\+C2​Γ\+Q∗T​\(c\)\+2​C​Γ​VT​\(c\)\+∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)\\displaystyle\\qquad\\leq D\_\{T\}\+B\_\{T\}\(\\rho\)\+C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)\(70\)IfKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, then

∑t=1T⟨pt,ℓt⟩−⟨ρ,LT⟩≤\(1\+C2\)​Γ\+Q∗T​\(c\)\+\(2​C\+C−1\)​Γ​VT​\(c\)\+∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}\\rangle\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)\(71\)For the optimized choiceC=1/2C=1/\\sqrt\{2\}, this becomes

∑t=1T⟨pt,ℓt⟩−⟨ρ,LT⟩≤32​Γ\+Q∗T​\(c\)\+2​2​Γ​VT​\(c\)\+∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}\\rangle\\leq\\frac\{3\}\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2\\sqrt\{2\\Gamma V\_\{T\}\(c\)\}\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)

##### How to read these first consequences\.

Corollary[4\.1](https://arxiv.org/html/2607.08789#S4.Thmtheorem1)is the exact translation back to the original losses\. Corollary[4\.2](https://arxiv.org/html/2607.08789#S4.Thmtheorem2)is the first Hoeffding\-style relaxation, obtained by upper bounding the exact cumulant by a range proxy\. Corollary[4\.4](https://arxiv.org/html/2607.08789#S4.Thmtheorem4)returns to the intrinsic\-time schedule and keeps the second\-order square\-root structure explicit\. Later, Section[4\.5](https://arxiv.org/html/2607.08789#S4.SS5)adds a comparator\-centered Bernstein conversion of the same term under stochastic low\-noise structure\.

##### Quantile corollaries\.

If the prior is uniform on\[K\]\[K\]andρ\\rhois uniform on a subsetA⊆\[K\]A\\subseteq\[K\], thenKL​\(ρ∥π\)=log⁡\(K/\|A\|\)\\mathrm\{KL\}\(\\rho\\\|\\pi\)=\\log\(K/\|A\|\)\. Every PAC\-Bayes theorem above therefore immediately implies the corresponding quantile statement by choosingAAto be a hindsight set of good experts\. The only additional term in the mixed\-prior setting is the explicit side\-factor mismatch from \([69](https://arxiv.org/html/2607.08789#S4.E69)\)\.

### 4\.2Compensators, side information, and sufficient\-statistic choices

With the original\-loss identity in hand, one can now modify the composite\-loss reduction in two structurally different ways\. One may change the predictable side factorssts\_\{t\}, which alters the composite losses by an exact algebraic correction\. Or one may keep the same comparator notion but replace the observed statistic by a transformed score before exponentiating it\. The examples below use both viewpoints\.

##### Vanilla Hedge\.

Takingst≡1s\_\{t\}\\equiv 1yieldsut≡0u\_\{t\}\\equiv 0\. Corollary[2\.4](https://arxiv.org/html/2607.08789#S2.Thmtheorem4)reduces to the standard entropic regret bound, while Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)gives a rigorous anytime second\-order bound for ordinary Hedge in terms ofVT​\(ℓ\)V\_\{T\}\(\\ell\)\.

##### Soft specialists and confidence\-rated experts\.

Letβt​\(i\)∈\(0,1\]\\beta\_\{t\}\(i\)\\in\(0,1\]encode the confidence or availability of expertiion roundtt, and setst​\(i\)=βt​\(i\)λts\_\{t\}\(i\)=\\beta\_\{t\}\(i\)^\{\\lambda\_\{t\}\}\. Thenut​\(i\)=−λtηt​log⁡βt​\(i\)u\_\{t\}\(i\)=\-\\frac\{\\lambda\_\{t\}\}\{\\eta\_\{t\}\}\\log\\beta\_\{t\}\(i\), so experts with low confidence receive an additive penalty\. This is a soft version of the specialist or sleeping\-expert setup in which zeros are replaced by small positive masses to keep all information terms finite\.

##### Predictable bonuses and variance penalties\.

Suppose a predictable bonusbt​\(i\)b\_\{t\}\(i\)and a variance proxyvt​\(i\)≥0v\_\{t\}\(i\)\\geq 0are available\. Takingst​\(i\)=exp⁡\(ηt​bt​\(i\)−γt​vt​\(i\)\)s\_\{t\}\(i\)=\\exp\\left\(\\eta\_\{t\}b\_\{t\}\(i\)\-\\gamma\_\{t\}v\_\{t\}\(i\)\\right\)givesct​\(i\)=ℓt​\(i\)−bt​\(i\)\+\(γt/ηt\)​vt​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\-b\_\{t\}\(i\)\+\(\\gamma\_\{t\}/\\eta\_\{t\}\)v\_\{t\}\(i\)\. Thus the same identity simultaneously handles optimism, pessimism, and second\-order damping: the algorithm is simply exponential weights on the composite lossesctc\_\{t\}\.

###### Corollary 4\.5\(Optimistic Hedge from predictable side information\)\.

Letmt∈ℝKm\_\{t\}\\in\\mathbb\{R\}^\{K\}be any predictable vector and choose the side informationut​\(i\):=−mt​\(i\)u\_\{t\}\(i\):=\-m\_\{t\}\(i\), equivalentlyst​\(i\)=exp⁡\(ηt​mt​\(i\)\)s\_\{t\}\(i\)=\\exp\(\\eta\_\{t\}m\_\{t\}\(i\)\)\. Thenct​\(i\)=ℓt​\(i\)−mt​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\-m\_\{t\}\(i\), and for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

∑t=1T⟨pt−ρ,ℓt⟩=DT\+BT​\(ρ\)\+∑t=1Tηt​Qt​\(ℓ−m\)\+∑t=1T⟨pt−ρ,mt⟩\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,\\ell\_\{t\}\\rangle=D\_\{T\}\+B\_\{T\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(\\ell\-m\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,m\_\{t\}\\ranglewhereQt​\(ℓ−m\)Q\_\{t\}\(\\ell\-m\)is the intrinsic increment computed from the residual vectorct=ℓt−mtc\_\{t\}=\\ell\_\{t\}\-m\_\{t\}\. Ifrange⁡\(ℓt−mt\)≤εt\\operatorname\{range\}\(\\ell\_\{t\}\-m\_\{t\}\)\\leq\\varepsilon\_\{t\}for everytt, then

∑t=1T⟨pt−ρ,ℓt⟩≤DT\+BT​\(ρ\)\+18​∑t=1Tηt​εt2\+∑t=1T⟨pt−ρ,mt⟩\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,\\ell\_\{t\}\\rangle\\leq D\_\{T\}\+B\_\{T\}\(\\rho\)\+\\frac\{1\}\{8\}\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\varepsilon\_\{t\}^\{2\}\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,m\_\{t\}\\rangleIfηt≡η\\eta\_\{t\}\\equiv\\eta, this simplifies to

∑t=1T⟨pt−ρ,ℓt⟩≤η−1​KL​\(ρ∥π\)\+η8​∑t=1Tεt2\+∑t=1T⟨pt−ρ,mt⟩\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,\\ell\_\{t\}\\rangle\\leq\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|\\pi\)\+\\frac\{\\eta\}\{8\}\\sum\_\{t=1\}^\{T\}\\varepsilon\_\{t\}^\{2\}\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,m\_\{t\}\\rangleUnder the schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\) and the complexity conditionKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, one also has

∑t=1T⟨pt−ρ,ℓt⟩≤\(1\+C2\)​Γ\+Q∗T​\(ℓ−m\)\+\(2​C\+C−1\)​Γ​VT​\(ℓ−m\)\+∑t=1T⟨pt−ρ,mt⟩\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,\\ell\_\{t\}\\rangle\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}^\{T\}\(\\ell\-m\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\(\\ell\-m\)\}\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,m\_\{t\}\\rangle

This is the standard optimistic\-Hedge mechanism, now written as a direct corollary of the side\-information calculus\. In repeated games or model\-based play, one can takemtm\_\{t\}to be the loss vector predicted from a forecast of the opponent’s move\. Then only the forecast residualsℓt−mt\\ell\_\{t\}\-m\_\{t\}contribute to the intrinsic\-time cost, while the term∑t⟨pt−ρ,mt⟩\\sum\_\{t\}\\langle p\_\{t\}\-\\rho,m\_\{t\}\\ranglerecords how the comparator and learner evaluate the predictable baseline\.

##### Perfect lookahead as a sanity check\.

In the degenerate casemt​\(i\)=ℓt​\(i\)m\_\{t\}\(i\)=\\ell\_\{t\}\(i\), the residual losses vanish:ct≡0c\_\{t\}\\equiv 0\. AccordinglyQt​\(c\)=0Q\_\{t\}\(c\)=0andVT​\(c\)=0V\_\{T\}\(c\)=0, so the online part of the theorem disappears\. Any remaining term in the original\-loss decomposition is purely the predictable planning term∑t⟨pt−ρ,ℓt⟩\\sum\_\{t\}\\langle p\_\{t\}\-\\rho,\\ell\_\{t\}\\rangle\. This is exactly the right intuition: if the full loss vector is known before acting, there is no irreducible online uncertainty left and one should not expect to pay adversarial regret\.

##### Slowly drifting losses\.

A particularly simple choice ismt​\(i\)=ℓt−1​\(i\)m\_\{t\}\(i\)=\\ell\_\{t\-1\}\(i\)\(with any convenient initialization fort=1t=1\), which yieldsct​\(i\)=ℓt​\(i\)−ℓt−1​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\-\\ell\_\{t\-1\}\(i\)\. If consecutive loss vectors satisfy\|ℓt​\(i\)−ℓt−1​\(i\)\|≤δt\|\\ell\_\{t\}\(i\)\-\\ell\_\{t\-1\}\(i\)\|\\leq\\delta\_\{t\}for allii, then againQt​\(c\)≤δt2/2Q\_\{t\}\(c\)\\leq\\delta\_\{t\}^\{2\}/2, soVT​\(c\)≤12​∑t=1Tδt2V\_\{T\}\(c\)\\leq\\frac\{1\}\{2\}\\sum\_\{t=1\}^\{T\}\\delta\_\{t\}^\{2\}\. The intrinsic\-time bound therefore scales with squared temporal drift, which matches the intuition behind variation\-based and gradual\-variation regret bounds\[[39](https://arxiv.org/html/2607.08789#bib.bib39),[15](https://arxiv.org/html/2607.08789#bib.bib15),[70](https://arxiv.org/html/2607.08789#bib.bib70)\]\.

##### Stochastic centering and compensators\.

Ifmt​\(i\)=𝔼​\[ℓt​\(i\)∣ℱt−1\]m\_\{t\}\(i\)=\\mathbb\{E\}\[\\ell\_\{t\}\(i\)\\mid\\mathcal\{F\}\_\{t\-1\}\]or a model\-based conditional mean, thenct​\(i\)=ℓt​\(i\)−mt​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\-m\_\{t\}\(i\)is the martingale\-difference residual\. The exact identities therefore separate the predictable mean component∑t⟨pt−ρ,mt⟩\\sum\_\{t\}\\langle p\_\{t\}\-\\rho,m\_\{t\}\\ranglefrom the noise componentRTc​\(ρ\)R\_\{T\}^\{c\}\(\\rho\)\. In favorable stochastic regimes the latter is controlled by the conditional noise scale, while the former is the genuine predictable excess\-risk process\. This is the martingale\-compensator interpretation of the same coincidence calculus\.

##### Positive\-part sufficient statistics and sparsity\.

Sparsity is orthogonal to the Bayesian prior calculus developed here\. The clean way to encode it is instead to change the observed sufficient statistic\. For a composite\-loss sequencectc\_\{t\}, letet​\(i\):=⟨pt,ct⟩−ct​\(i\)e\_\{t\}\(i\):=\\langle p\_\{t\},c\_\{t\}\\rangle\-c\_\{t\}\(i\)be the one\-round excess\-loss vector, and writeet\+​\(i\):=\(et​\(i\)\)\+e\_\{t\}^\{\+\}\(i\):=\(e\_\{t\}\(i\)\)\_\{\+\}\. Then for every posteriorρ\\rho,RTc​\(ρ\)=∑t=1T⟨ρ,et⟩≤∑t=1T⟨ρ,et\+⟩R\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\langle\\rho,e\_\{t\}\\rangle\\leq\\sum\_\{t=1\}^\{T\}\\langle\\rho,e\_\{t\}^\{\+\}\\rangle\. Thus the positive part already upper\-bounds the one\-sided regret: only experts that beat the learner on roundttcontribute\. Running the algorithm onet\+e\_\{t\}^\{\+\}\(or, more generally, on shortfalls of the form\(et−bt\)\+\(e\_\{t\}\-b\_\{t\}\)\_\{\+\}or\(ct−at\)\+\(c\_\{t\}\-a\_\{t\}\)\_\{\+\}\) therefore gives a sparsity\-seeking variant that deliberately forgets coordinates already on the favorable side of the comparison\. This construction is logically independent of the compensator choices above: it acts on the realized statistic after centering, not on the prior\. This is not an exact re\-expression of the original regret process; it is a conservative reduction that keeps only the part of the observation relevant to one\-sided excess loss\.

At the level of the crude intrinsic\-time envelopes used throughout the paper, truncation can only help\. Since the mapx↦x\+x\\mapsto x\_\{\+\}is11\-Lipschitz,range⁡\(et\+\)≤range⁡\(et\)\\operatorname\{range\}\(e\_\{t\}^\{\+\}\)\\leq\\operatorname\{range\}\(e\_\{t\}\), and therefore every range\-based bound such asQt\(e\+\)≤range\(et\+\)2/8Q\_\{t\}\(e^\{\+\}\)\\leq\\operatorname\{range\}\(e\_\{t\}^\{\+\}\)^\{2\}/8is no larger after truncation\. In that precise sense, sparsity should be viewed here as a change of sufficient statistic rather than as a modification of the prior\.

##### Bounded\-influence score transforms\.

The same sufficient\-statistic viewpoint also covers robust influence functions\. Instead of exponentiating a raw residualztz\_\{t\}, one may exponentiate a transformed scoreψ​\(zt\)\\psi\(z\_\{t\}\)chosen to clip or smooth rare but very large observations\. Positive\-part truncation for sparsity and Catoni\-type bounded\-influence transforms are two instances of the same move: they change the sufficient statistic fed to the Bayes update, not the exact information identity itself\. We postpone the explicit Catoni formulas to Section[6\.4](https://arxiv.org/html/2607.08789#S6.SS4), where this is especially useful for stabilizing importance\-weighted bandit estimates, but conceptually it belongs with the present discussion of statistic design rather than with a separate regret theorem\.

##### Several side priors\.

Ifst​\(i\)=∏wπt,w​\(i\)αt,ws\_\{t\}\(i\)=\\prod\_\{w\}\\pi\_\{t,w\}\(i\)^\{\\alpha\_\{t,w\}\}, thenut​\(i\)=−1ηt​∑wαt,w​log⁡πt,w​\(i\)u\_\{t\}\(i\)=\-\\frac\{1\}\{\\eta\_\{t\}\}\\sum\_\{w\}\\alpha\_\{t,w\}\\log\\pi\_\{t,w\}\(i\)\. The regret corrections in \([67](https://arxiv.org/html/2607.08789#S4.E67)\), \([64](https://arxiv.org/html/2607.08789#S4.E64)\), and \([65](https://arxiv.org/html/2607.08789#S4.E65)\) therefore become explicit sums of cross\-entropy gaps\. In particular, the scaling\-time quantity in Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)depends only on the resulting composite losses, not on how many priors were used to construct them\.

These examples all modify the composite\-loss reduction while keeping the comparator fixed\. We next change the comparator itself\.

### 4\.3Shifting comparators and sleeping experts

The same exact calculus also accommodates richer comparator objects\. A switching or tracking benchmark may be viewed either as a path of posteriors\(ρt\)\(\\rho\_\{t\}\)or, equivalently, as a structured prior on a parameter space of expert trajectories\. The theorem below records the pathwise identity first and only afterward bounds the path term in more familiar switching or sleeping language\.

###### Theorem 4\.6\(Exact shifting\-comparator identity\)\.

Letρ1,…,ρT∈Δ​\(\[K\]\)\\rho\_\{1\},\\dots,\\rho\_\{T\}\\in\\Delta\(\[K\]\)be any comparator path and define the dynamic composite\-loss regret

RTc,dyn​\(ρ1:T\):=∑t=1T⟨pt−ρt,ct⟩R\_\{T\}^\{c,\\mathrm\{dyn\}\}\(\\rho\_\{1:T\}\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho\_\{t\},c\_\{t\}\\rangleThen

RTc,dyn​\(ρ1:T\)=DT\+BT​\(ρT\)\+∑t=1Tηt​Qt​\(c\)\+∑t=1T−1⟨ρt\+1−ρt,Ct⟩R\_\{T\}^\{c,\\mathrm\{dyn\}\}\(\\rho\_\{1:T\}\)=D\_\{T\}\+B\_\{T\}\(\\rho\_\{T\}\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\+\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangle\(72\)Under the schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\),

DT\+BT​\(ρT\)\+2​C​Γ​VT​\(c\)−C2​Γ\+∑t=1T−1⟨ρt\+1−ρt,Ct⟩\\displaystyle D\_\{T\}\+B\_\{T\}\(\\rho\_\{T\}\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\-C^\{2\}\\Gamma\+\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangle≤RTc,dyn​\(ρ1:T\)\\displaystyle\\qquad\\leq R\_\{T\}^\{c,\\mathrm\{dyn\}\}\(\\rho\_\{1:T\}\)≤DT\+BT​\(ρT\)\+C2​Γ\+Q∗T​\(c\)\+2​C​Γ​VT​\(c\)\+∑t=1T−1⟨ρt\+1−ρt,Ct⟩\.\\displaystyle\\qquad\\leq D\_\{T\}\+B\_\{T\}\(\\rho\_\{T\}\)\+C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\+\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangle\.\(73\)If moreoverct​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\]for allttandii, then\|∑t=1T−1⟨ρt\+1−ρt,Ct⟩\|≤2​∑t=1T−1t​TV​\(ρt\+1,ρt\)\\left\|\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangle\\right\|\\leq 2\\sum\_\{t=1\}^\{T\-1\}t\\,\\mathrm\{TV\}\(\\rho\_\{t\+1\},\\rho\_\{t\}\)\. Consequently, wheneverKL​\(ρT∥π\)≤Γ\\mathrm\{KL\}\(\\rho\_\{T\}\\\|\\pi\)\\leq\\Gamma,

RTc,dyn​\(ρ1:T\)≤\(1\+C2\)​Γ\+Q∗T​\(c\)\+\(2​C\+C−1\)​Γ​VT​\(c\)\+2​∑t=1T−1t​TV​\(ρt\+1,ρt\)R\_\{T\}^\{c,\\mathrm\{dyn\}\}\(\\rho\_\{1:T\}\)\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\+2\\sum\_\{t=1\}^\{T\-1\}t\\,\\mathrm\{TV\}\(\\rho\_\{t\+1\},\\rho\_\{t\}\)\(74\)

The theorem contains several familiar special cases\. A comparator that switches among pure experts recovers a fixed\-share\-type tracking bound, in the same dynamic\-regret spirit as\[[40](https://arxiv.org/html/2607.08789#bib.bib40)\]; a comparator supported only on currently active experts gives a soft sleeping\-experts guarantee; and, because the additional term is exact before it is bounded by total variation, one can often exploit extra structure inCtC\_\{t\}beyond the crude range bound\|∑t=1T−1⟨ρt\+1−ρt,Ct⟩\|≤2​∑t=1T−1t​TV​\(ρt\+1,ρt\)\\left\|\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangle\\right\|\\leq 2\\sum\_\{t=1\}^\{T\-1\}t\\,\\mathrm\{TV\}\(\\rho\_\{t\+1\},\\rho\_\{t\}\)\.

### 4\.4A simultaneous quantile theorem by a logarithmic budget controller

The fixed\-budget theorem becomes simultaneous inε\\varepsilonafter one more layer of exponential weights over a dyadic grid of complexity budgets\. The construction is simpler than a potential\-specific single\-copy algorithm, but it stays entirely within the exact Hedge template of this paper\.

###### Theorem 4\.7\(Simultaneousε\\varepsilon\-quantile regret via a budget controller\)\.

Assumect​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\]for allttandii, and let the base prior on experts be uniform on\[K\]\[K\]\. FixJ:=⌈log2⁡log⁡\(max⁡\{e,K\}\)⌉J:=\\lceil\\log\_\{2\}\\log\(\\max\\\{e,K\\\}\)\\rceiland budgetsΓj:=2j\\Gamma\_\{j\}:=2^\{j\}forj=0,…,Jj=0,\\dots,J\. For eachjj, run a worker copy of the second\-order algorithm given by \([25](https://arxiv.org/html/2607.08789#S3.E25)\) with budgetΓj\\Gamma\_\{j\}, producing weightspt\(j\)p\_\{t\}^\{\(j\)\}\. Define the meta\-lossesmt​\(j\):=⟨pt\(j\),ct⟩∈\[0,1\]m\_\{t\}\(j\):=\\langle p\_\{t\}^\{\(j\)\},c\_\{t\}\\rangle\\in\[0,1\], run a controller copy of \([25](https://arxiv.org/html/2607.08789#S3.E25)\) on theJ\+1J\+1meta\-experts with uniform prior and budgetΓctl:=log⁡\(J\+1\)\\Gamma^\{\\mathrm\{ctl\}\}:=\\log\(J\+1\), obtaining weightsαt∈Δ​\(\[J\+1\]\)\\alpha\_\{t\}\\in\\Delta\(\[J\+1\]\), and playp¯t:=∑j=0Jαt​\(j\)​pt\(j\)\\bar\{p\}\_\{t\}:=\\sum\_\{j=0\}^\{J\}\\alpha\_\{t\}\(j\)p\_\{t\}^\{\(j\)\}\. LetQtctl,VTctl,QT∗,ctlQ\_\{t\}^\{\\mathrm\{ctl\}\},V\_\{T\}^\{\\mathrm\{ctl\}\},Q^\{\*,\\mathrm\{ctl\}\}\_\{T\}denote the controller’s intrinsic\-time quantities for the meta\-lossesmtm\_\{t\}, and letQt\(j\),VT\(j\),QT∗,jQ\_\{t\}^\{\(j\)\},V\_\{T\}^\{\(j\)\},Q^\{\*,j\}\_\{T\}denote the corresponding quantities for workerjjon the original lossesctc\_\{t\}\. Then for everyε∈\[K−1,e−1\]\\varepsilon\\in\[K^\{\-1\},e^\{\-1\}\]and every setA⊆\[K\]A\\subseteq\[K\]with\|A\|≥ε​K\|A\|\\geq\\varepsilon K, ifj​\(ε\):=min⁡\{j:Γj≥log⁡\(1/ε\)\}j\(\\varepsilon\):=\\min\\\{j:\\Gamma\_\{j\}\\geq\\log\(1/\\varepsilon\)\\\}, one has

∑t=1T⟨p¯t,ct⟩−1\|A\|​∑i∈ACT​\(i\)\\displaystyle\\sum\_\{t=1\}^\{T\}\\langle\\bar\{p\}\_\{t\},c\_\{t\}\\rangle\-\\frac\{1\}\{\|A\|\}\\sum\_\{i\\in A\}C\_\{T\}\(i\)≤\(1\+C2\)​Γctl\+QT∗,ctl\+\(2​C\+C−1\)​Γctl​VTctl\\displaystyle\\leq\(1\+C^\{2\}\)\\Gamma^\{\\mathrm\{ctl\}\}\+Q^\{\*,\\mathrm\{ctl\}\}\_\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma^\{\\mathrm\{ctl\}\}V\_\{T\}^\{\\mathrm\{ctl\}\}\}\+\(1\+C2\)​Γj​\(ε\)\+QT∗,j​\(ε\)\+\(2​C\+C−1\)​Γj​\(ε\)​VT\(j​\(ε\)\)\\displaystyle\\qquad\\quad\+\(1\+C^\{2\}\)\\Gamma\_\{j\(\\varepsilon\)\}\+Q^\{\*,j\(\\varepsilon\)\}\_\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma\_\{j\(\\varepsilon\)\}V\_\{T\}^\{\(j\(\\varepsilon\)\)\}\}\(75\)SinceΓj​\(ε\)≤2​log⁡\(1/ε\)\\Gamma\_\{j\(\\varepsilon\)\}\\leq 2\\log\(1/\\varepsilon\), the right\-hand side is the fixed\-budgetε\\varepsilon\-quantile bound at scalelog⁡\(1/ε\)\\log\(1/\\varepsilon\)plus the extra controller term depending only on theJ\+1J\+1budget copies\.

This is the simplest exact route to a simultaneous quantile theorem in the present notation\. It is not as sharp as a dedicated one\-copy potential such as Squint or the newer NormalHedge variants\[[45](https://arxiv.org/html/2607.08789#bib.bib45),[49](https://arxiv.org/html/2607.08789#bib.bib49),[30](https://arxiv.org/html/2607.08789#bib.bib30)\], but it already shows that the exact intrinsic\-time identities of this paper can be upgraded from fixed\-budget PAC\-Bayes statements to a single run that covers all quantiles\. On synthetic and real online sequences the controller meets every quantile simultaneously within the predicted overhead, and pays only a small fraction of it on real paths \(Appendices[C\.4\.6](https://arxiv.org/html/2607.08789#A3.SS4.SSS6)and[C\.4\.7](https://arxiv.org/html/2607.08789#A3.SS4.SSS7)\)\.

### 4\.5Stochastic luckiness and fast\-rate PAC\-Bayes control

The pathwise theorems above make no stochastic assumptions: they are valid for every realized sequence\. But one often wants a second layer of interpretation when the data happen to come from a benign law\. A stochastic\-luckiness bound does exactly this\. It identifies a regime in which the same quantity that drives the adversarial regret bound is itself controlled by the comparator’s excess risk\. Then the general square\-root behavior collapses to a constant or other fast expected rate, without changing the algorithm and without sacrificing the original pathwise guarantee\.

In the present framework, the relevant pathwise quantity is the intrinsic\-time cost∑t=1Tηt​Qt​\(c\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\. To turn that cost into a fast\-rate theorem, one needs a self\-bounding relation that makesQt​\(c\)Q\_\{t\}\(c\)comparable to the instantaneous gap to the comparator\. Because our regret identity is centered at a comparator distribution rather than only at a single best expert, the natural stochastic assumption is also comparator\-centered: the roundwise fluctuation around the comparator average should be controlled by the comparator’s mean excess loss\. This is why the resulting theorem is genuinely PAC\-Bayesian rather than merely a point\-comparator statement\.

This viewpoint is the online analogue of standard low\-noise fast\-rate conditions in statistical learning\. Such conditions do not alter the worst\-case guarantee; they say that on favorable problems the variance pays for the bias, so slow rates accelerate to fast ones\. Here the same idea is applied to the intrinsic\-time clock of the Bayes/exponential\-weights update\. When the process is lucky in that sense, the comparator\-centered intrinsic time largely pays for itself, and the expected regret becomes constant inTT\.

The relation with\[[58](https://arxiv.org/html/2607.08789#bib.bib58)\]is conceptually simple\. Their multiscale analysis combines a second\-order regret theorem with a stochastic self\-bounding argument to obtain constant expected pseudoregret under a low\-noise condition\. In the normalized composite\-loss regime studied here, the corresponding stochastic step is cleaner: one centers the one\-round mixability\-gap estimate at the comparator rather than at the learner mean, so the same self\-bounding idea plugs directly into the exact cumulant identity\.

###### Proposition 4\.8\(Comparator\-centered second\-order envelope\)\.

Assume thatct​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\]andηt∈\(0,1\]\\eta\_\{t\}\\in\(0,1\]for allt≤Tt\\leq Tandi∈\[K\]i\\in\[K\], and defineΨt​\(ρ\):=∑i=1Kpt​\(i\)​\(ct​\(i\)−⟨ρ,ct⟩\)2\\Psi\_\{t\}\(\\rho\):=\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)\\left\(c\_\{t\}\(i\)\-\\langle\\rho,c\_\{t\}\\rangle\\right\)^\{2\}\. Then the exact cumulant increment obeys

Qt​\(c\)≤\(e−2\)​Ψt​\(ρ\)for every​t≤TQ\_\{t\}\(c\)\\leq\(e\-2\)\\Psi\_\{t\}\(\\rho\)\\qquad\\text\{for every \}t\\leq T\(76\)If, in addition,\(ηt\)t=1T\(\\eta\_\{t\}\)\_\{t=1\}^\{T\}is nonincreasing, then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

RTc​\(ρ\)≤KL​\(ρ∥π\)ηT\+\(e−2\)​∑t=1Tηt​Ψt​\(ρ\)R\_\{T\}^\{c\}\(\\rho\)\\leq\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\_\{T\}\}\+\(e\-2\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\Psi\_\{t\}\(\\rho\)\(77\)

#### 4\.5\.1Fast learning under low comparator\-centered noise

The stochastic assumption we need for this result is the direct PAC\-Bayes analogue of the point\-comparator low\-noise condition\.

##### Definition\.

Letc1,c2,…c\_\{1\},c\_\{2\},\\dotsbe i\.i\.d\. random vectors in\[0,1\]K\[0,1\]^\{K\}and letμ:=𝔼​\[c1\]∈\[0,1\]K\\mu:=\\mathbb\{E\}\[c\_\{1\}\]\\in\[0,1\]^\{K\}\. Fixρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\)\. We say thatρ\\rhosatisfies the comparator\-centered low\-noise condition with constantκρ∈\[0,∞\)\\kappa\_\{\\rho\}\\in\[0,\\infty\)if for every experti∈\[K\]i\\in\[K\],

𝔼​\[\(ct​\(i\)−⟨ρ,ct⟩\)2\]≤κρ​\(μ​\(i\)−⟨ρ,μ⟩\)\\mathbb\{E\}\\bigl\[\(c\_\{t\}\(i\)\-\\langle\\rho,c\_\{t\}\\rangle\)^\{2\}\\bigr\]\\leq\\kappa\_\{\\rho\}\\left\(\\mu\(i\)\-\\langle\\rho,\\mu\\rangle\\right\)\(78\)
Condition \([78](https://arxiv.org/html/2607.08789#S4.E78)\) is exactly the standard point\-mass low\-noise condition whenρ=δk∗\\rho=\\delta\_\{k^\{\*\}\}, but it is highly restrictive for diffuse posteriors\. If some expert in the support ofρ\\rhohas the same mean loss as the posterior average, then \([78](https://arxiv.org/html/2607.08789#S4.E78)\) already forcesct​\(i\)=⟨ρ,ct⟩c\_\{t\}\(i\)=\\langle\\rho,c\_\{t\}\\ranglealmost surely\. This implies that for most non\-degenerate stochastic environments, the condition is practically vacuous for diffuse mixtures because it requires the losses of those experts to be perfectly deterministic\. While it algebraically closes the exact comparator\-centered second\-order proof, this fast expected rate is therefore meaningful primarily for point\-mass comparators \(single best experts\) or highly degenerate environments\. Constant regret for more general Bayes\-optimal posteriors may still be possible, but it would require a different argument\.

###### Theorem 4\.9\(Fixed\-rate stochastic luckiness\)\.

Assume thatc1,c2,…c\_\{1\},c\_\{2\},\\dotsare i\.i\.d\. random vectors in\[0,1\]K\[0,1\]^\{K\}and thatρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\)satisfies Condition \([78](https://arxiv.org/html/2607.08789#S4.E78)\) with constantκρ\\kappa\_\{\\rho\}\. Run fixed\-rate Hedge on the composite losses,pt​\(i\)∝π​\(i\)​e−η​Ct−1​\(i\),η∈\(0,1\]p\_\{t\}\(i\)\\propto\\pi\(i\)e^\{\-\\eta C\_\{t\-1\}\(i\)\},\\qquad\\eta\\in\(0,1\]\. Then for everyT≥1T\\geq 1and everyη\\etasuch that\(e−2\)​κρ​η<1\(e\-2\)\\kappa\_\{\\rho\}\\eta<1,

𝔼​\[RTc​\(ρ\)\]≤KL​\(ρ∥π\)η​\(1−\(e−2\)​κρ​η\)\\mathbb\{E\}\\bigl\[R\_\{T\}^\{c\}\(\\rho\)\\bigr\]\\leq\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\\left\(1\-\(e\-2\)\\kappa\_\{\\rho\}\\eta\\right\)\}\(79\)In particular, with the choiceη:=min⁡\{1,12​\(e−2\)​κρ\}\\displaystyle\\eta:=\\min\\\!\\left\\\{1,\\frac\{1\}\{2\(e\-2\)\\kappa\_\{\\rho\}\}\\right\\\}, we obtain the time\-uniform bound

𝔼​\[RTc​\(ρ\)\]≤2​\(1\+2​\(e−2\)​κρ\)​KL​\(ρ∥π\)\\mathbb\{E\}\\bigl\[R\_\{T\}^\{c\}\(\\rho\)\\bigr\]\\leq 2\\left\(1\+2\(e\-2\)\\kappa\_\{\\rho\}\\right\)\\mathrm\{KL\}\(\\rho\\\|\\pi\)\(80\)

###### Corollary 4\.10\(Point\-comparator low\-noise and gap corollaries\)\.

Letk∗∈\[K\]k^\{\*\}\\in\[K\]and setρ=δk∗\\rho=\\delta\_\{k^\{\*\}\}\.

1. \(i\)If the ordinary point\-comparator low\-noise condition holds, namely 𝔼​\[\(ct​\(i\)−ct​\(k∗\)\)2\]≤c∗​\(μ​\(i\)−μ​\(k∗\)\)for all​i∈\[K\]\\mathbb\{E\}\\bigl\[\(c\_\{t\}\(i\)\-c\_\{t\}\(k^\{\*\}\)\)^\{2\}\\bigr\]\\leq c\_\{\*\}\\left\(\\mu\(i\)\-\\mu\(k^\{\*\}\)\\right\)\\qquad\\text\{for all \}i\\in\[K\]\(81\)then Theorem[4\.9](https://arxiv.org/html/2607.08789#S4.Thmtheorem9)yields𝔼​\[RTc​\(k∗\)\]≤2​\(1\+2​\(e−2\)​c∗\)​log⁡1π​\(k∗\)\\mathbb\{E\}\\bigl\[R\_\{T\}^\{c\}\(k^\{\*\}\)\\bigr\]\\leq 2\\left\(1\+2\(e\-2\)c\_\{\*\}\\right\)\\log\\frac\{1\}\{\\pi\(k^\{\*\}\)\}\.
2. \(ii\)If, more concretely, there is a mean gapdmin:=mini≠k∗⁡\(μ​\(i\)−μ​\(k∗\)\)\>0d\_\{\\min\}:=\\min\_\{i\\neq k^\{\*\}\}\\left\(\\mu\(i\)\-\\mu\(k^\{\*\}\)\\right\)\>0, then \([81](https://arxiv.org/html/2607.08789#S4.E81)\) holds withc∗=1/dminc\_\{\*\}=1/d\_\{\\min\}, and therefore𝔼​\[RTc​\(k∗\)\]≤2​\(1\+2​\(e−2\)dmin\)​log⁡1π​\(k∗\)\\mathbb\{E\}\\bigl\[R\_\{T\}^\{c\}\(k^\{\*\}\)\\bigr\]\\leq 2\\left\(1\+\\frac\{2\(e\-2\)\}\{d\_\{\\min\}\}\\right\)\\log\\frac\{1\}\{\\pi\(k^\{\*\}\)\}\. For the uniform prior this isO​\(\(1\+dmin−1\)​log⁡K\)O\\left\(\(1\+d\_\{\\min\}^\{\-1\}\)\\log K\\right\)\.

The fixed\-rate theorem is a clean theoretical way to understand the luckiness mechanism, but in practice the variable\-rate algorithm from Sections[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)and[3\.2](https://arxiv.org/html/2607.08789#S3.SS2)admits an analogous conclusion, without requiring advance knowledge of the noise level parameterκρ\\kappa\_\{\\rho\}\.

###### Corollary 4\.11\(Predictable\-rate stochastic luckiness\)\.

Assume the same stochastic setting as in Theorem[4\.9](https://arxiv.org/html/2607.08789#S4.Thmtheorem9), and run the second\-order schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\) with budgetΓ\>0\\Gamma\>0\. IfKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, then for everyT≥1T\\geq 1,

𝔼​\[RTc​\(ρ\)\]≤2​\(1\+C2\)​Γ\+2​𝔼​\[Q∗T​\(c\)\]\+\(2​C\+C−1\)2​\(e−2\)​κρ​Γ\\mathbb\{E\}\\bigl\[R\_\{T\}^\{c\}\(\\rho\)\\bigr\]\\leq 2\(1\+C^\{2\}\)\\Gamma\+2\\,\\mathbb\{E\}\\bigl\[Q\_\{\*\}^\{T\}\(c\)\\bigr\]\+\(2C\+C^\{\-1\}\)^\{2\}\(e\-2\)\\kappa\_\{\\rho\}\\Gamma\(82\)In particular, ifct​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\]almost surely for allt,it,i, then Proposition[3\.1](https://arxiv.org/html/2607.08789#S3.Thmtheorem1)givesQ∗T​\(c\)≤1/8Q\_\{\*\}^\{T\}\(c\)\\leq 1/8and therefore, for example, ifC=1/2C=1/\\sqrt\{2\}, then𝔼​\[RTc​\(ρ\)\]≤3​Γ\+14\+8​\(e−2\)​κρ​Γ\\mathbb\{E\}\\bigl\[R\_\{T\}^\{c\}\(\\rho\)\\bigr\]\\leq 3\\Gamma\+\\frac\{1\}\{4\}\+8\(e\-2\)\\kappa\_\{\\rho\}\\Gamma\. So the same variable\-rate algorithm enjoys constant expected regret inTTunder the comparator\-centered low\-noise condition while retaining the exact cumulant intrinsic time in the pathwise theorem\.

#### 4\.5\.2Transfer back to the original losses

Combining Corollary[4\.1](https://arxiv.org/html/2607.08789#S4.Thmtheorem1)with any of the bounds above shows that the expected regret for the original lossesℓt\\ell\_\{t\}equals the composite\-loss constant plus the expected mismatch term𝔼​\[∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)\]\\mathbb\{E\}\\Biggl\[\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)\\Biggr\]\. Hence any application in which this term is uniformly bounded above inherits a constant expected actual\-loss regret bound as well\. When the side factors are decomposed as in \([68](https://arxiv.org/html/2607.08789#S4.E68)\), this mismatch is exactly the cumulative cross\-entropy difference in \([69](https://arxiv.org/html/2607.08789#S4.E69)\)\.

The relation with\[[58](https://arxiv.org/html/2607.08789#bib.bib58)\]is now quite transparent\. Their multiscale theorems show that a carefully tuned multiscale entropy regularizer can have both scale\-sensitive worst\-case regret and constant expected pseudoregret under the same low\-noise hypothesis\. The KL\-geometry results in the present section do*not*by themselves subsume that scale\-sensitive guarantee: standard Hedge on common\-scale composite losses does not automatically deliver regret that tracks expert\-dependent ranges\. What the present paper does show, first in the common\-scale setting here and then more directly in Section[5\.2](https://arxiv.org/html/2607.08789#S5.SS2), is that the exact accounting itself is not restricted to ordinary KL geometry\. The stochastic\-luckiness step becomes much simpler and more PAC\-Bayesian once one works with comparator\-centered intrinsic time, while the weighted\-entropy variant changes only the transport geometry and terminal potential\. In that sense, Proposition[4\.8](https://arxiv.org/html/2607.08789#S4.Thmtheorem8)is the common\-scale counterpart of the more delicate regularizer\-specific variance conversion carried out in\[[58](https://arxiv.org/html/2607.08789#bib.bib58)\]\.

## 5Full\-information extensions and applications

Once the composite\-loss reduction is in place, the same information accounting survives several changes of setting\. Rather than adding new proof tricks, we show that the same pattern reappears when one changes the geometry, the prior family, or the action domain: each update pays an immediate information\-based cost as regret, and transports the remaining information forward, simultaneously for any comparator until the algorithm runs out of information budget\.

### 5\.1Exact variational sharpenings of classical Hedge bounds

The usual fixed\-rate Hedge proof hides two sources of slack: it drops the terminal KL remainder and replaces the exact intrinsic increment by a Hoeffding proxy\. Keeping both terms visible yields the following sharper pathwise decomposition\.

###### Corollary 5\.1\(Classical bounded\-range bound with exact remainder\)\.

Under the hypotheses of Corollary[2\.9](https://arxiv.org/html/2607.08789#S2.Thmtheorem9), assume in addition thatct​\(i\)∈\[at,bt\]c\_\{t\}\(i\)\\in\[a\_\{t\},b\_\{t\}\]for alli,ti,t\. Define

ST:=∑t=1T\(bt−at\)2S\_\{T\}:=\\sum\_\{t=1\}^\{T\}\(b\_\{t\}\-a\_\{t\}\)^\{2\}and

ΔTclass​\(ρ,η\):=1η​KL​\(ρ∥pT\+1\)\+η​∑t=1T\(\(bt−at\)28−Qt​\(c\)\)\\Delta\_\{T\}^\{\\mathrm\{class\}\}\(\\rho,\\eta\):=\\frac\{1\}\{\\eta\}\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\+\\eta\\sum\_\{t=1\}^\{T\}\\left\(\\frac\{\(b\_\{t\}\-a\_\{t\}\)^\{2\}\}\{8\}\-Q\_\{t\}\(c\)\\right\)ThenΔTclass​\(ρ,η\)≥0\\Delta\_\{T\}^\{\\mathrm\{class\}\}\(\\rho,\\eta\)\\geq 0and, for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

RTc​\(ρ\)=KL​\(ρ∥π\)η\+η​ST8−ΔTclass​\(ρ,η\)R\_\{T\}^\{c\}\(\\rho\)=\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\}\+\\frac\{\\eta S\_\{T\}\}\{8\}\-\\Delta\_\{T\}^\{\\mathrm\{class\}\}\(\\rho,\\eta\)\(83\)Consequently,

RTc​\(ρ\)≤KL​\(ρ∥π\)η\+η​ST8R\_\{T\}^\{c\}\(\\rho\)\\leq\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\}\+\\frac\{\\eta S\_\{T\}\}\{8\}\(84\)and the classical bound is tight only on extremal paths where both pieces ofΔTclass​\(ρ,η\)\\Delta\_\{T\}^\{\\mathrm\{class\}\}\(\\rho,\\eta\)vanish\.

Equation \([83](https://arxiv.org/html/2607.08789#S5.E83)\) separates the usual pessimism into two pieces\. The termη−1​KL​\(ρ∥pT\+1\)\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)is a posterior\-mismatch penalty, which vanishes when the final Gibbs posterior is already close to the comparator\. The second term measures how much smaller the realized intrinsic time∑tQt​\(c\)\\sum\_\{t\}Q\_\{t\}\(c\)is than the worst\-case quadratic proxy∑t\(bt−at\)2/8\\sum\_\{t\}\(b\_\{t\}\-a\_\{t\}\)^\{2\}/8\. Thus the classical Hedge theorem is an exact envelope minus an explicit nonnegative slack\.

###### Corollary 5\.2\(Square\-root envelope with an exact gap\)\.

Under the assumptions of Corollary[5\.1](https://arxiv.org/html/2607.08789#S5.Thmtheorem1), letΓ\>0\\Gamma\>0satisfyΓ≥KL​\(ρ∥π\)\\Gamma\\geq\\mathrm\{KL\}\(\\rho\\\|\\pi\)and assumeST\>0S\_\{T\}\>0\. Set

ηΓ:=8​ΓST\\eta\_\{\\Gamma\}:=\\sqrt\{\\frac\{8\\Gamma\}\{S\_\{T\}\}\}Then

RTc​\(ρ\)≤Γ​ST2−\(Γ−KL​\(ρ∥π\)ηΓ\+ΔTclass​\(ρ,ηΓ\)\)R\_\{T\}^\{c\}\(\\rho\)\\leq\\sqrt\{\\frac\{\\Gamma S\_\{T\}\}\{2\}\}\-\\left\(\\frac\{\\Gamma\-\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\_\{\\Gamma\}\}\+\\Delta\_\{T\}^\{\\mathrm\{class\}\}\(\\rho,\\eta\_\{\\Gamma\}\)\\right\)\(85\)Thus the familiarΓ​ST/2\\sqrt\{\\Gamma S\_\{T\}/2\}fixed\-budget Hedge envelope is exact up to a nonnegative gap\. Applying the same corollary to bothctc\_\{t\}and−ct\-c\_\{t\}gives the corresponding absolute\-deviation versions as well\.

The same point also has a direct adaptive analogue that refers only to the realized schedule, with no comparison to a hindsight\-fixed temperature\.

###### Proposition 5\.3\(Direct pathwise identity for a realized adaptive schedule\)\.

Let\(qt\)\(q\_\{t\}\)be generated by the one\-step update \([4](https://arxiv.org/html/2607.08789#S2.E4)\) with any predictable positive schedule\(ηt\)t=1T\(\\eta\_\{t\}\)\_\{t=1\}^\{T\}\. Then for every posteriorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

∑t=1Tηt​\(⟨qt,ct⟩−⟨ρ,ct⟩\)=∑t=1Tηt​δt​\(c\)\+KL​\(ρ∥q1\)−KL​\(ρ∥qT\+1\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\langle q\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle\\bigr\)=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\delta\_\{t\}\(c\)\+\\mathrm\{KL\}\(\\rho\\\|q\_\{1\}\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{T\+1\}\)\(86\)Equivalently, ifQtstep​\(c\):=δt​\(c\)/ηtQ\_\{t\}^\{\\mathrm\{step\}\}\(c\):=\\delta\_\{t\}\(c\)/\\eta\_\{t\}, then

∑t=1Tηt​\(⟨qt,ct⟩−⟨ρ,ct⟩\)=∑t=1Tηt2​Qtstep​\(c\)\+KL​\(ρ∥q1\)−KL​\(ρ∥qT\+1\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\langle q\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle\\bigr\)=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}^\{2\}Q\_\{t\}^\{\\mathrm\{step\}\}\(c\)\+\\mathrm\{KL\}\(\\rho\\\|q\_\{1\}\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{T\+1\}\)\(87\)

This is the cleanest direct\-adaptive statement: once the learning\-rate path is fixed, the weighted regret already satisfies an exact identity\. The prior\-anchored retempered theorem is the matching decomposition for the anchored update, with the additional drift term recording the cost of changing temperature over time\.

### 5\.2Weighted negative entropy and multiscale experts

KL geometry is not special\. The same bookkeeping extends to other mirror maps, and the most useful example here is the weighted negative entropy used for multiscale experts\[[58](https://arxiv.org/html/2607.08789#bib.bib58)\]\. Fix scalesσ1,…,σK\>0\\sigma\_\{1\},\\dots,\\sigma\_\{K\}\>0and define

Fσ​\(p\):=∑i=1Kσi​p​\(i\)​log⁡p​\(i\),Dσ​\(q∥p\):=∑i=1Kσi​\(q​\(i\)​log⁡q​\(i\)p​\(i\)−q​\(i\)\+p​\(i\)\)F\_\{\\sigma\}\(p\):=\\sum\_\{i=1\}^\{K\}\\sigma\_\{i\}p\(i\)\\log p\(i\),\\qquad D\_\{\\sigma\}\(q\\\|p\):=\\sum\_\{i=1\}^\{K\}\\sigma\_\{i\}\\left\(q\(i\)\\log\\frac\{q\(i\)\}\{p\(i\)\}\-q\(i\)\+p\(i\)\\right\)Given lossesℓt∈ℝK\\ell\_\{t\}\\in\\mathbb\{R\}^\{K\}and a learning rateηt\>0\\eta\_\{t\}\>0, the corresponding weighted\-entropy mirror\-descent step takes the form

p~t\+1​\(i\):=pt​\(i\)​exp⁡\(−ηt​ℓt​\(i\)/σi\),pt\+1​\(i\):=p~t\+1​\(i\)​exp⁡\(−λt/σi\)\\widetilde\{p\}\_\{t\+1\}\(i\):=p\_\{t\}\(i\)\\exp\\\!\\left\(\-\\eta\_\{t\}\\ell\_\{t\}\(i\)/\\sigma\_\{i\}\\right\),\\qquad p\_\{t\+1\}\(i\):=\\widetilde\{p\}\_\{t\+1\}\(i\)\\exp\\\!\\left\(\-\\lambda\_\{t\}/\\sigma\_\{i\}\\right\)whereλt\\lambda\_\{t\}is the unique scalar making∑ipt\+1​\(i\)=1\\sum\_\{i\}p\_\{t\+1\}\(i\)=1\.

###### Proposition 5\.4\(Exact one\-step identity for weighted entropy\)\.

For every comparatorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\),

⟨pt,ℓt⟩−⟨ρ,ℓt⟩=Dσ​\(ρ∥pt\)−Dσ​\(ρ∥pt\+1\)ηt\+Dσ​\(pt∥pt\+1\)ηt\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,\\ell\_\{t\}\\rangle=\\frac\{D\_\{\\sigma\}\(\\rho\\\|p\_\{t\}\)\-D\_\{\\sigma\}\(\\rho\\\|p\_\{t\+1\}\)\}\{\\eta\_\{t\}\}\+\\frac\{D\_\{\\sigma\}\(p\_\{t\}\\\|p\_\{t\+1\}\)\}\{\\eta\_\{t\}\}\(88\)In particular, the immediate cost is the nonnegative weighted Bregman stepDσ​\(pt∥pt\+1\)/ηtD\_\{\\sigma\}\(p\_\{t\}\\\|p\_\{t\+1\}\)/\\eta\_\{t\}, and the rest is transport of comparator information in the same sense as \([8](https://arxiv.org/html/2607.08789#S2.E8)\)\.

This is the same one\-step balance in a different geometry: the comparator\-information term is now the weighted divergenceDσD\_\{\\sigma\}, and the immediate payment is the weighted Bregman step\. From there, the same telescoping and potential arguments as in Sections[2\.1](https://arxiv.org/html/2607.08789#S2.SS1)–[3](https://arxiv.org/html/2607.08789#S3)yield fixed\-rate and adaptive cumulative identities with the weighted terminal potentialAtσ​\(η\):=minq∈Δ​\(\[K\]\)⁡\{⟨q,Lt⟩\+η−1​Dσ​\(q∥π\)\}A\_\{t\}^\{\\sigma\}\(\\eta\):=\\min\_\{q\\in\\Delta\(\[K\]\)\}\\left\\\{\\langle q,L\_\{t\}\\rangle\+\\eta^\{\-1\}D\_\{\\sigma\}\(q\\\|\\pi\)\\right\\\}in place ofAt​\(η\)A\_\{t\}\(\\eta\)\. Thus multiscale experts still fit the paper’s theme: changing scales changes the geometry, not the information accounting\.

The link to Theorem[2\.1](https://arxiv.org/html/2607.08789#S2.Thmtheorem1)is static\-versus\-sequential\. There the geometric poolpα⋆p^\{\\star\}\_\{\\alpha\}is characterized variationally; here the weighted\-entropy update is the sequential analogue when expert\-specific scales determine the geometry\.

### 5\.3Multiscale forgetting by continuum prior mixtures

Continuum prior pooling also gives a simple interpretation of adaptive forgetting: a prior over discount factors becomes a memory kernel over lags\. Letβ∈\(0,1\)\\beta\\in\(0,1\)be a forgetting rate\. Define theβ\\beta\-discounted cumulative loss and the corresponding exponential\-weights prior

Lt,β​\(i\):=∑s=1t−1βt−1−s​ℓs​\(i\)πt,β​\(i\)∝exp⁡\(−η​Lt,β​\(i\)\)L\_\{t,\\beta\}\(i\):=\\sum\_\{s=1\}^\{t\-1\}\\beta^\{t\-1\-s\}\\ell\_\{s\}\(i\)\\qquad\\qquad\\pi\_\{t,\\beta\}\(i\)\\propto\\exp\(\-\\eta L\_\{t,\\beta\}\(i\)\)Now letαt\\alpha\_\{t\}be a finite positive measure onR⊂\(0,1\)R\\subset\(0,1\)such that∫R\|log⁡πt,β​\(i\)\|​αt​\(d​β\)<∞\\int\_\{R\}\|\\log\\pi\_\{t,\\beta\}\(i\)\|\\,\\alpha\_\{t\}\(d\\beta\)<\\inftyfor eachii\. We may then define the geometric pool by

qt​\(i\)∝exp⁡\(∫Rlog⁡πt,β​\(i\)​αt​\(d​β\)\)q\_\{t\}\(i\)\\propto\\exp\\\!\\left\(\\int\_\{R\}\\log\\pi\_\{t,\\beta\}\(i\)\\,\\alpha\_\{t\}\(d\\beta\)\\right\)
###### Proposition 5\.5\(Mixtures of forgetting rates produce mixtures of exponential kernels\)\.

With the above definitions,

qt​\(i\)∝exp⁡\(−η​∑s=1t−1kt​\(t−1−s\)​ℓs​\(i\)\)kt​\(u\):=∫Rβu​αt​\(d​β\)q\_\{t\}\(i\)\\propto\\exp\\\!\\left\(\-\\eta\\sum\_\{s=1\}^\{t\-1\}k\_\{t\}\(t\-1\-s\)\\ell\_\{s\}\(i\)\\right\)\\qquad\\qquad k\_\{t\}\(u\):=\\int\_\{R\}\\beta^\{u\}\\,\\alpha\_\{t\}\(d\\beta\)\(89\)In particular, the effective memory kernel is a mixture of exponential decays\.

In other words, mixing forgetting rates is exactly the same as mixing exponential decay kernels\. For the present paper, the value of the observation is conceptual: continuum pooling can be read directly as learned memory design\.

### 5\.4Repeated games and robust opponent modeling

In repeated zero\-sum games, the same calculus yields three natural objects: pooled opponent models, a robust center over model\-conditioned priors, and a scale\-sensitive regret matcher\. Consider a game with row actionsi∈\[K\]i\\in\[K\], column actionsj∈\[L\]j\\in\[L\], and loss matrixM∈\[0,1\]K×LM\\in\[0,1\]^\{K\\times L\}for the row player\. Classical multiplicative\-weights play against the sequenceℓt​\(i\)=Mi,jt\\ell\_\{t\}\(i\)=M\_\{i,j\_\{t\}\}gives no regret and even a constructive minimax theorem\[[28](https://arxiv.org/html/2607.08789#bib.bib28),[3](https://arxiv.org/html/2607.08789#bib.bib3)\]\.

#### 5\.4\.1Pooling several opponent models

Suppose that at roundttwe have candidate opponent mixed strategiesq~t,1,…,q~t,W∈Δ​\(\[L\]\)\\widetilde\{q\}\_\{t,1\},\\dots,\\widetilde\{q\}\_\{t,W\}\\in\\Delta\(\[L\]\)\. For each modelww, define the row\-response prior

πt,w​\(i\)∝exp⁡\(−η​\(M​q~t,w\)i\)\\pi\_\{t,w\}\(i\)\\propto\\exp\\left\(\-\\eta\(M\\widetilde\{q\}\_\{t,w\}\)\_\{i\}\\right\)\(90\)
###### Proposition 5\.6\(Pooling opponent models equals response to the averaged model\)\.

For any nonnegative weightsαt,1,…,αt,W\\alpha\_\{t,1\},\\dots,\\alpha\_\{t,W\},

pt​\(i\)∝∏w=1Wπt,w​\(i\)αt,w=exp⁡\(−η​∑w=1Wαt,w​\(M​q~t,w\)i\)p\_\{t\}\(i\)\\propto\\prod\_\{w=1\}^\{W\}\\pi\_\{t,w\}\(i\)^\{\\alpha\_\{t,w\}\}=\\exp\\\!\\left\(\-\\eta\\sum\_\{w=1\}^\{W\}\\alpha\_\{t,w\}\(M\\widetilde\{q\}\_\{t,w\}\)\_\{i\}\\right\)\(91\)Ifλt:=∑w=1Wαt,w\>0\\lambda\_\{t\}:=\\sum\_\{w=1\}^\{W\}\\alpha\_\{t,w\}\>0andq¯t:=λt−1​∑w=1Wαt,w​q~t,w\\bar\{q\}\_\{t\}:=\\lambda\_\{t\}^\{\-1\}\\sum\_\{w=1\}^\{W\}\\alpha\_\{t,w\}\\widetilde\{q\}\_\{t,w\}, then

pt​\(i\)∝exp⁡\(−η​λt​\(M​q¯t\)i\)\.p\_\{t\}\(i\)\\propto\\exp\\left\(\-\\eta\\lambda\_\{t\}\(M\\bar\{q\}\_\{t\}\)\_\{i\}\\right\)\.\(92\)

So geometric pooling of several opponent models is equivalent to responding to their weighted average\. If the models come from different windows or discount scales, the pool simply forms a multiscale fictitious\-play prior\.

#### 5\.4\.2Robust strategy centers from coincidence divergence

Now supposeπ1,…,πW∈Δ​\(\[K\]\)\\pi\_\{1\},\\dots,\\pi\_\{W\}\\in\\Delta\(\[K\]\)are strictly positive strategy priors induced by different game models, different states of nature, or different opponent types\. The finite mixed coincidence identity already gives the corresponding robust\-center characterization; the proof, via the minimax theorem in\[[69](https://arxiv.org/html/2607.08789#bib.bib69)\], is deferred to the appendix\.

###### Proposition 5\.7\(Robust center over strategy priors\)\.

Define𝒞α​\(π1:W\):=−log​∑i=1K∏w=1Wπw​\(i\)αw\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\):=\-\\log\\sum\_\{i=1\}^\{K\}\\prod\_\{w=1\}^\{W\}\\pi\_\{w\}\(i\)^\{\\alpha\_\{w\}\}and𝒞​\(π1:W\):=maxα∈Δ​\(\[W\]\)⁡𝒞α​\(π1:W\)\\mathcal\{C\}\(\\pi\_\{1:W\}\):=\\max\_\{\\alpha\\in\\Delta\(\[W\]\)\}\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\)\. Then

𝒞​\(π1:W\)=minp∈Δ​\(\[K\]\)⁡maxw∈\[W\]⁡KL​\(p∥πw\)\.\\mathcal\{C\}\(\\pi\_\{1:W\}\)=\\min\_\{p\\in\\Delta\(\[K\]\)\}\\max\_\{w\\in\[W\]\}\\mathrm\{KL\}\(p\\\|\\pi\_\{w\}\)\.\(93\)Moreover, any optimizerp⋆p^\{\\star\}has the geometric\-mixture formp⋆=pα⋆⋆p^\{\\star\}=p^\{\\star\}\_\{\\alpha^\{\\star\}\}for some maximizingα⋆∈Δ​\(\[W\]\)\\alpha^\{\\star\}\\in\\Delta\(\[W\]\)\.

This is the right robust object when the learner does not trust a single opponent model\. The value𝒞​\(π1:W\)\\mathcal\{C\}\(\\pi\_\{1:W\}\)is the minimum worst\-case reverse\-KL distance to the model\-conditioned priors, and the optimizer is itself a geometric mixture\.

#### 5\.4\.3Second\-order equalizers and regret matching

Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)’s second\-order schedule already equalizes the complexity termΓ/η\\Gamma/\\etaagainst the intrinsic\-time termη​V\\eta V\. In a repeated zero\-sum game, that same balance becomes an entropic regret matcher\.

###### Proposition 5\.8\(Exact second\-order regret matching\)\.

Letℓt∈\[0,1\]K\\ell\_\{t\}\\in\[0,1\]^\{K\}be the row player’s loss vector and define the instantaneous regret vectorgt​\(i\):=ℓt​\(i\)−⟨pt,ℓt⟩g\_\{t\}\(i\):=\\ell\_\{t\}\(i\)\-\\langle p\_\{t\},\\ell\_\{t\}\\rangle\. Then⟨pt,gt⟩=0\\langle p\_\{t\},g\_\{t\}\\rangle=0\. Run predictable\-rate Hedge on the sequencegtg\_\{t\}with priorπ\\piand schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\)\. For every actioni∈\[K\]i\\in\[K\],

∑t=1T\(⟨pt,ℓt⟩−ℓt​\(i\)\)=DTg\+BTg​\(ei\)\+∑t=1Tηt​Qt​\(g\)\\sum\_\{t=1\}^\{T\}\\left\(\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\ell\_\{t\}\(i\)\\right\)=D\_\{T\}^\{g\}\+B\_\{T\}^\{g\}\(e\_\{i\}\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(g\)\(94\)whereDTgD\_\{T\}^\{g\}andBTgB\_\{T\}^\{g\}are the drift and terminal terms for the centered lossesgtg\_\{t\}\. Hence, ifKL​\(ei∥π\)≤Γ\\mathrm\{KL\}\(e\_\{i\}\\\|\\pi\)\\leq\\Gamma,

∑t=1T\(⟨pt,ℓt⟩−ℓt​\(i\)\)≤\(1\+C2\)​Γ\+Q∗T​\(g\)\+\(2​C\+C−1\)​Γ​VT​\(g\)\\sum\_\{t=1\}^\{T\}\\left\(\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\ell\_\{t\}\(i\)\\right\)\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}^\{T\}\(g\)\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\(g\)\}\(95\)Moreover, in the unclipped regimeηt=C​Γ/Vt−1​\(g\)\\eta\_\{t\}=C\\sqrt\{\\Gamma/V\_\{t\-1\}\(g\)\}, scaling every loss vector by a constanta\>0a\>0scalesQt​\(g\)Q\_\{t\}\(g\)andVT​\(g\)V\_\{T\}\(g\)bya2a^\{2\}and therefore rescales the second\-order learning rate by1/a1/a\. In that precise intrinsic\-time sense the schedule is scale invariant; the clipped initial rounds contribute exactly the sameC2​ΓC^\{2\}\\Gammainitialization tax as before\.

###### Corollary 5\.9\(Self\-play gap from intrinsic\-time equalizers\)\.

Consider a repeated zero\-sum matrix game, and let both players run the second\-order update on their own instantaneous regret vectors\. Ifp¯T\\bar\{p\}\_\{T\}andq¯T\\bar\{q\}\_\{T\}are the average row and column plays, writePTrow:=∑tηtrow​QtrowP\_\{T\}^\{\\mathrm\{row\}\}:=\\sum\_\{t\}\\eta\_\{t\}^\{\\mathrm\{row\}\}Q\_\{t\}^\{\\mathrm\{row\}\}andPTcol:=∑tηtcol​QtcolP\_\{T\}^\{\\mathrm\{col\}\}:=\\sum\_\{t\}\\eta\_\{t\}^\{\\mathrm\{col\}\}Q\_\{t\}^\{\\mathrm\{col\}\}for the two intrinsic\-time payments, and writeDTrow,BTrowD\_\{T\}^\{\\mathrm\{row\}\},B\_\{T\}^\{\\mathrm\{row\}\}andDTcol,BTcolD\_\{T\}^\{\\mathrm\{col\}\},B\_\{T\}^\{\\mathrm\{col\}\}for the corresponding drift and terminal\-information terms\. Then the exploitability gap has the exact informational form

T​\(maxq⁡p¯T⊤​M​q−minp⁡p⊤​M​q¯T\)\\displaystyle T\\left\(\\max\_\{q\}\\,\\bar\{p\}\_\{T\}^\{\\top\}Mq\-\\min\_\{p\}\\,p^\{\\top\}M\\bar\{q\}\_\{T\}\\right\)=maxj⁡\{DTcol\+BTcol​\(ej\)\+PTcol\}\+maxi⁡\{DTrow\+BTrow​\(ei\)\+PTrow\}\.\\displaystyle\\qquad=\\max\_\{j\}\\left\\\{D\_\{T\}^\{\\mathrm\{col\}\}\+B\_\{T\}^\{\\mathrm\{col\}\}\(e\_\{j\}\)\+P\_\{T\}^\{\\mathrm\{col\}\}\\right\\\}\+\\max\_\{i\}\\left\\\{D\_\{T\}^\{\\mathrm\{row\}\}\+B\_\{T\}^\{\\mathrm\{row\}\}\(e\_\{i\}\)\+P\_\{T\}^\{\\mathrm\{row\}\}\\right\\\}\.\(96\)Consequently,

maxq⁡p¯T⊤​M​q−minp⁡p⊤​M​q¯T≤UTrow\+UTcolT,\\max\_\{q\}\\,\\bar\{p\}\_\{T\}^\{\\top\}Mq\-\\min\_\{p\}\\,p^\{\\top\}M\\bar\{q\}\_\{T\}\\leq\\frac\{U\_\{T\}^\{\\mathrm\{row\}\}\+U\_\{T\}^\{\\mathrm\{col\}\}\}\{T\},\(97\)whereUTrowU\_\{T\}^\{\\mathrm\{row\}\}andUTcolU\_\{T\}^\{\\mathrm\{col\}\}are the corresponding upper bounds from \([95](https://arxiv.org/html/2607.08789#S5.E95)\) for the two players\.

The interpretation is simple: each player runs a scale\-adaptive regret matcher whose effective time is its own intrinsic time\. In self\-play, the exploitability gap is just the average of those two information budgets\.

The same information accounting also survives when the expert class is infinite\. We record three examples: continuum prior mixtures, continuous\-action online convex optimization, and boosting\.

### 5\.5Continuum priors, hyperparameter mixtures, and PAC\-Bayes

#### 5\.5\.1Continuum\-indexed prior families

The finite geometric\-pooling identity extends whenever the logarithmic average is well defined\. Concretely, let\{πθ\}θ∈Θ\\\{\\pi\_\{\\theta\}\\\}\_\{\\theta\\in\\Theta\}be a measurable family of positive measures and letα\\alphabe a finite positive measure onΘ\\Thetasuch that∫Θ\|log⁡πθ​\(x\)\|​α​\(d​θ\)<∞for each​x\\int\_\{\\Theta\}\|\\log\\pi\_\{\\theta\}\(x\)\|\\,\\alpha\(d\\theta\)<\\infty\\qquad\\text\{for each \}x\. Then one may define the pooled measure by

pα⋆​\(x\)∝exp⁡\(∫Θlog⁡πθ​\(x\)​α​\(d​θ\)\)p^\{\\star\}\_\{\\alpha\}\(x\)\\propto\\exp\\\!\\left\(\\int\_\{\\Theta\}\\log\\pi\_\{\\theta\}\(x\)\\,\\alpha\(d\\theta\)\\right\)For online learning, this means one can mix over a continuum of temperatures, window lengths, forgetting rates, or model classes without discretizing first; Proposition[5\.5](https://arxiv.org/html/2607.08789#S5.Thmtheorem5)is the simplest example\. The point is conceptual: hyperparameter averaging becomes part of the same variational update, not an outer model\-selection layer\.

#### 5\.5\.2A direct multi\-prior PAC\-Bayes reduction

One immediate consequence is a multi\-prior PAC\-Bayes penalty\. If a variational formula or bound contains a single termKL​\(ρ∥π\)\\mathrm\{KL\}\(\\rho\\\|\\pi\), replacingπ\\piby a pooled priorpα⋆p^\{\\star\}\_\{\\alpha\}and applying Theorem[2\.1](https://arxiv.org/html/2607.08789#S2.Thmtheorem1)gives an exact decomposition into several prior penalties minus a coincidence bonus\.

###### Proposition 5\.10\(Multi\-prior PAC\-Bayes penalty\)\.

Letμ\\mube aσ\\sigma\-finite reference measure on a hypothesis spaceℋ\\mathcal\{H\}, letπ1,…,πW\\pi\_\{1\},\\dots,\\pi\_\{W\}be probability measures with strictly positiveμ\\mu\-densitiespw:=d​πw/d​μp\_\{w\}:=d\\pi\_\{w\}/d\\mu, and letα∈Δ​\(\[W\]\)\\alpha\\in\\Delta\(\[W\]\)\. Define

gα​\(h\):=∏w=1Wpw​\(h\)αw,Zα:=∫ℋgα​\(h\)​μ​\(d​h\),d​pα⋆d​μ​\(h\):=gα​\(h\)Zαg\_\{\\alpha\}\(h\):=\\prod\_\{w=1\}^\{W\}p\_\{w\}\(h\)^\{\\alpha\_\{w\}\},\\qquad Z\_\{\\alpha\}:=\\int\_\{\\mathcal\{H\}\}g\_\{\\alpha\}\(h\)\\,\\mu\(dh\),\\qquad\\frac\{dp^\{\\star\}\_\{\\alpha\}\}\{d\\mu\}\(h\):=\\frac\{g\_\{\\alpha\}\(h\)\}\{Z\_\{\\alpha\}\}Then for every posteriorρ\\rhosuch thatρ≪πw\\rho\\ll\\pi\_\{w\}for everyww,

KL​\(ρ∥pα⋆\)=∑w=1Wαw​KL​\(ρ∥πw\)−𝒞α​\(π1:W\),\\mathrm\{KL\}\(\\rho\\\|p^\{\\star\}\_\{\\alpha\}\)=\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{KL\}\(\\rho\\\|\\pi\_\{w\}\)\-\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\),\(98\)where𝒞α​\(π1:W\):=−log⁡Zα\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\):=\-\\log Z\_\{\\alpha\}\. Hence every PAC\-Bayes variational formula or inequality whose dependence on the prior enters only through the termKL​\(ρ∥π\)\\mathrm\{KL\}\(\\rho\\\|\\pi\)admits an exact multi\-prior version with penalty∑wαw​KL​\(ρ∥πw\)−𝒞α​\(π1:W\)\\sum\_\{w\}\\alpha\_\{w\}\\mathrm\{KL\}\(\\rho\\\|\\pi\_\{w\}\)\-\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\)\.

This is useful whenever several source priors are plausible, as in domain adaptation, meta\-learning, or transfer\. The KL terms measure mismatch to each source, while the coincidence bonus rewards prior overlap\. Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)provides the dynamic analogue through side priors and explicit mismatch bookkeeping\.

### 5\.6Continuous\-action online convex optimization

The same chain passes to compact convex action sets by replacing sums with integrals and playing the barycenter of the exponential\-weights density\[[70](https://arxiv.org/html/2607.08789#bib.bib70)\]\. LetS⊂ℝdS\\subset\\mathbb\{R\}^\{d\}be compact and convex, letπ\\pibe a probability measure onSSwith full support, and letft:S→\[0,1\]f\_\{t\}:S\\to\[0,1\]be convex and continuous\. For predictable positive ratesηt\\eta\_\{t\}, define

pt​\(d​x\):=e−ηt​Ft−1​\(x\)​π​\(d​x\)∫Se−ηt​Ft−1​\(u\)​π​\(d​u\),Ft​\(x\):=∑s=1tfs​\(x\),xt:=∫Sx​pt​\(d​x\)∈Sp\_\{t\}\(dx\):=\\frac\{e^\{\-\\eta\_\{t\}F\_\{t\-1\}\(x\)\}\\,\\pi\(dx\)\}\{\\int\_\{S\}e^\{\-\\eta\_\{t\}F\_\{t\-1\}\(u\)\}\\,\\pi\(du\)\},\\qquad F\_\{t\}\(x\):=\\sum\_\{s=1\}^\{t\}f\_\{s\}\(x\),\\qquad x\_\{t\}:=\\int\_\{S\}x\\,p\_\{t\}\(dx\)\\in SBecausextx\_\{t\}is the barycenter ofptp\_\{t\}, Jensen’s inequality givesft​\(xt\)≤∫Sft​𝑑ptf\_\{t\}\(x\_\{t\}\)\\leq\\int\_\{S\}f\_\{t\}\\,dp\_\{t\}on each round\.

###### Theorem 5\.11\(Exact continuous PAC\-Bayes chain and OCO corollary\)\.

For any posterior measureρ≪π\\rho\\ll\\pi, define the density regret

RTdens​\(ρ\):=∑t=1T∫Sft​\(x\)​pt​\(d​x\)−∫SFT​\(x\)​ρ​\(d​x\)R\_\{T\}^\{\\mathrm\{dens\}\}\(\\rho\):=\\sum\_\{t=1\}^\{T\}\\int\_\{S\}f\_\{t\}\(x\)\\,p\_\{t\}\(dx\)\-\\int\_\{S\}F\_\{T\}\(x\)\\,\\rho\(dx\)Also define the following quantities:Atoco​\(η\):=−η−1​log​∫Se−η​Ft​\(x\)​π​\(d​x\),qt,η​\(d​x\):=e−η​Ft​\(x\)​π​\(d​x\)∫Se−η​Ft​\(u\)​π​\(d​u\)\\displaystyle A\_\{t\}^\{\\mathrm\{oco\}\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\\!\\int\_\{S\}e^\{\-\\eta F\_\{t\}\(x\)\}\\,\\pi\(dx\),\\qquad q\_\{t,\\eta\}\(dx\):=\\frac\{e^\{\-\\eta F\_\{t\}\(x\)\}\\,\\pi\(dx\)\}\{\\int\_\{S\}e^\{\-\\eta F\_\{t\}\(u\)\}\\,\\pi\(du\)\}, and defineψtoco​\(λ\):=log​∫Sexp⁡\(λ​\(ft​\(x\)−∫Sft​𝑑pt\)\)​pt​\(d​x\),ϕtoco​\(η\):=ψtoco​\(−η\)η,Qtoco:=ϕtoco​\(ηt\)ηt\\psi\_\{t\}^\{\\mathrm\{oco\}\}\(\\lambda\):=\\log\\\!\\int\_\{S\}\\exp\\\!\\left\(\\lambda\\left\(f\_\{t\}\(x\)\-\\int\_\{S\}f\_\{t\}\\,dp\_\{t\}\\right\)\\right\)\\,p\_\{t\}\(dx\),\\phi\_\{t\}^\{\\mathrm\{oco\}\}\(\\eta\):=\\frac\{\\psi\_\{t\}^\{\\mathrm\{oco\}\}\(\-\\eta\)\}\{\\eta\},Q\_\{t\}^\{\\mathrm\{oco\}\}:=\\frac\{\\phi\_\{t\}^\{\\mathrm\{oco\}\}\(\\eta\_\{t\}\)\}\{\\eta\_\{t\}\}\.

Then withBToco​\(ρ\):=KL​\(ρ∥π\)−KL​\(ρ∥qT,ηT\)ηTDToco:=∑t=1T−1\(Atoco​\(ηt\)−Atoco​\(ηt\+1\)\)B\_\{T\}^\{\\mathrm\{oco\}\}\(\\rho\):=\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{T,\\eta\_\{T\}\}\)\}\{\\eta\_\{T\}\}\\qquad D\_\{T\}^\{\\mathrm\{oco\}\}:=\\sum\_\{t=1\}^\{T\-1\}\\left\(A\_\{t\}^\{\\mathrm\{oco\}\}\(\\eta\_\{t\}\)\-A\_\{t\}^\{\\mathrm\{oco\}\}\(\\eta\_\{t\+1\}\)\\right\), we have

RTdens​\(ρ\)=DToco\+BToco​\(ρ\)\+∑t=1Tηt​QtocoR\_\{T\}^\{\\mathrm\{dens\}\}\(\\rho\)=D\_\{T\}^\{\\mathrm\{oco\}\}\+B\_\{T\}^\{\\mathrm\{oco\}\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}^\{\\mathrm\{oco\}\}\(99\)Consequently the ordinary OCO regret satisfies

∑t=1Tft​\(xt\)−∫SFT​\(x\)​ρ​\(d​x\)=DToco\+BToco​\(ρ\)\+∑t=1Tηt​Qtoco\\sum\_\{t=1\}^\{T\}f\_\{t\}\(x\_\{t\}\)\-\\int\_\{S\}F\_\{T\}\(x\)\\,\\rho\(dx\)=D\_\{T\}^\{\\mathrm\{oco\}\}\+B\_\{T\}^\{\\mathrm\{oco\}\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}^\{\\mathrm\{oco\}\}\(100\)If the second\-order schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\) is run with budgetΓ\\GammaandKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, then withVToco:=∑t=1TQtocoV\_\{T\}^\{\\mathrm\{oco\}\}:=\\sum\_\{t=1\}^\{T\}Q\_\{t\}^\{\\mathrm\{oco\}\}andQ∗,ocoT:=maxt≤T⁡QtocoQ\_\{\*,\\mathrm\{oco\}\}^\{T\}:=\\max\_\{t\\leq T\}Q\_\{t\}^\{\\mathrm\{oco\}\},

∑t=1Tft​\(xt\)−∫SFT​𝑑ρ≤\(1\+C2\)​Γ\+Q∗,ocoT\+\(2​C\+C−1\)​Γ​VToco\\sum\_\{t=1\}^\{T\}f\_\{t\}\(x\_\{t\}\)\-\\int\_\{S\}F\_\{T\}\\,d\\rho\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*,\\mathrm\{oco\}\}^\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}^\{\\mathrm\{oco\}\}\}\(101\)

Nothing fundamentally new happens algebraically\. The exact density\-regret chain is the same as in the finite case, and Jensen’s inequality is the only step needed to pass to standard OCO regret\.

###### Corollary 5\.12\(Point\-comparator regret by geometric shrinking\)\.

Assume now thatSShas nonempty interior and thatπ\\piis the normalized Lebesgue measure onSS\. Letx⋆∈arg⁡minx∈S⁡FT​\(x\)x^\{\\star\}\\in\\arg\\min\_\{x\\in S\}F\_\{T\}\(x\), fixϵ∈\(0,1\]\\epsilon\\in\(0,1\], and writeα:=ϵ1/d\\alpha:=\\epsilon^\{1/d\}\. Then letρϵ,x⋆\\rho\_\{\\epsilon,x^\{\\star\}\}be the uniform measure on the shrunken copy\(1−α\)​x⋆\+α​S\(1\-\\alpha\)x^\{\\star\}\+\\alpha S\. ThenKL​\(ρϵ,x⋆∥π\)=log⁡\(1/ϵ\)\\mathrm\{KL\}\(\\rho\_\{\\epsilon,x^\{\\star\}\}\\\|\\pi\)=\\log\(1/\\epsilon\)and

∑t=1Tft​\(xt\)−FT​\(x⋆\)≤DToco\+BToco​\(ρϵ,x⋆\)\+∑t=1Tηt​Qtoco\+T​ϵ1/d\\sum\_\{t=1\}^\{T\}f\_\{t\}\(x\_\{t\}\)\-F\_\{T\}\(x^\{\\star\}\)\\leq D\_\{T\}^\{\\mathrm\{oco\}\}\+B\_\{T\}^\{\\mathrm\{oco\}\}\(\\rho\_\{\\epsilon,x^\{\\star\}\}\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}^\{\\mathrm\{oco\}\}\+T\\epsilon^\{1/d\}\(102\)IfRegT​\(x⋆\):=∑t=1Tft​\(xt\)−FT​\(x⋆\)\\mathrm\{Reg\}\_\{T\}\(x^\{\\star\}\):=\\sum\_\{t=1\}^\{T\}f\_\{t\}\(x\_\{t\}\)\-F\_\{T\}\(x^\{\\star\}\)and the second\-order schedule is run withΓ=log⁡\(1/ϵ\)\\Gamma=\\log\(1/\\epsilon\), then

RegT​\(x⋆\)≤\(1\+C2\)​log⁡\(1/ϵ\)\+Q∗,ocoT\+\(2​C\+C−1\)​VToco​log⁡\(1/ϵ\)\+T​ϵ1/d\\mathrm\{Reg\}\_\{T\}\(x^\{\\star\}\)\\leq\(1\+C^\{2\}\)\\log\(1/\\epsilon\)\+Q\_\{\*,\\mathrm\{oco\}\}^\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{V\_\{T\}^\{\\mathrm\{oco\}\}\\log\(1/\\epsilon\)\}\+T\\epsilon^\{1/d\}\(103\)In particular, takingϵ=T−d\\epsilon=T^\{\-d\}andC=1/2C=1/\\sqrt\{2\}yieldsRegT​\(x⋆\)≤32​d​log⁡T\+Q∗,ocoT\+2​2​d​VToco​log⁡T\+1\\mathrm\{Reg\}\_\{T\}\(x^\{\\star\}\)\\leq\\frac\{3\}\{2\}d\\log T\+Q\_\{\*,\\mathrm\{oco\}\}^\{T\}\+2\\sqrt\{2dV\_\{T\}^\{\\mathrm\{oco\}\}\\log T\}\+1\. Sinceft∈\[0,1\]f\_\{t\}\\in\[0,1\], the standard bounded\-range exponential\-moment bound givesQtoco≤1/8Q\_\{t\}^\{\\mathrm\{oco\}\}\\leq 1/8, soVToco≤T/8V\_\{T\}^\{\\mathrm\{oco\}\}\\leq T/8and thereforeRegT​\(x⋆\)=O​\(d​T​log⁡T\)\\mathrm\{Reg\}\_\{T\}\(x^\{\\star\}\)=O\(\\sqrt\{dT\\log T\}\)\.

This extension is mainly structural\. It shows that the information accounting survives the passage from finite experts to continuous actions, but it is not meant as a competitive practical OCO method\. In high dimensions the exact density update and its barycenter are expensive to compute, and the entropic geometry carries ad\\sqrt\{d\}penalty that projection\-based methods such as Online Gradient Descent avoid\.

### 5\.7Boosting: regret identities, schedules, and exponential loss

Boosting fits the framework in two complementary ways\. The first treats examples as experts and turns regret identities into quantile\-margin guarantees\. The second works directly with signed margins and recovers the usual exponential\-loss variational formula\. The first viewpoint highlights intrinsic time; the second is closer to standard boosting analyses such as the drifting\-games treatment in\[[70](https://arxiv.org/html/2607.08789#bib.bib70)\]\.

Consider binary\-labeled data\(xi,yi\)i=1N\(x\_\{i\},y\_\{i\}\)\_\{i=1\}^\{N\}withyi∈\{−1,\+1\}y\_\{i\}\\in\\\{\-1,\+1\\\}\. At roundttthe booster maintains weightspt∈Δ​\(\[N\]\)p\_\{t\}\\in\\Delta\(\[N\]\)over examples, calls a weak learner, and receives a hypothesishth\_\{t\}with edgeγt:=12​∑i=1Npt​\(i\)​yi​ht​\(xi\)\\gamma\_\{t\}:=\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{N\}p\_\{t\}\(i\)y\_\{i\}h\_\{t\}\(x\_\{i\}\)\. For the regret\-based reduction we feed the learner the correctness scoreℓt​\(i\):=𝟏​\{ht​\(xi\)=yi\}=\(1\+yi​ht​\(xi\)\)/2∈\{0,1\}\\ell\_\{t\}\(i\):=\\mathbf\{1\}\\\{h\_\{t\}\(x\_\{i\}\)=y\_\{i\}\\\}=\(1\+y\_\{i\}h\_\{t\}\(x\_\{i\}\)\)/2\\in\\\{0,1\\\}rather than a literal misclassification loss\. AfterTTrounds output the majority voteHT​\(x\):=sign​\(∑t=1Tht​\(x\)\)H\_\{T\}\(x\):=\\mathrm\{sign\}\\\!\\left\(\\sum\_\{t=1\}^\{T\}h\_\{t\}\(x\)\\right\)\. The margin of exampleiiismi:=T−1​∑t=1Tyi​ht​\(xi\)m\_\{i\}:=T^\{\-1\}\\sum\_\{t=1\}^\{T\}y\_\{i\}h\_\{t\}\(x\_\{i\}\), soLTℓ​\(i\)=T2​\(1\+mi\)L\_\{T\}^\{\\ell\}\(i\)=\\frac\{T\}\{2\}\(1\+m\_\{i\}\)\.

###### Theorem 5\.13\(Generic regret\-to\-margin conversion\)\.

Letγ¯T:=T−1​∑t=1Tγt\\bar\{\\gamma\}\_\{T\}:=T^\{\-1\}\\sum\_\{t=1\}^\{T\}\\gamma\_\{t\}\. For any posteriorρ∈Δ​\(\[N\]\)\\rho\\in\\Delta\(\[N\]\),

⟨ρ,m⟩=2​γ¯T−2T​RTℓ​\(ρ\)\\langle\\rho,m\\rangle=2\\bar\{\\gamma\}\_\{T\}\-\\frac\{2\}\{T\}R\_\{T\}^\{\\ell\}\(\\rho\)\(104\)whereRTℓ​\(ρ\):=∑t=1T⟨pt,ℓt⟩−⟨ρ,LTℓ⟩R\_\{T\}^\{\\ell\}\(\\rho\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}^\{\\ell\}\\rangle\. Equivalently, any regret upper boundRTℓ​\(ρ\)≤UT​\(ρ\)R\_\{T\}^\{\\ell\}\(\\rho\)\\leq U\_\{T\}\(\\rho\)immediately implies the average\-margin lower bound⟨ρ,m⟩≥2​γ¯T−2​UT​\(ρ\)/T\\langle\\rho,m\\rangle\\geq 2\\bar\{\\gamma\}\_\{T\}\-2U\_\{T\}\(\\rho\)/T\.

Now assume the prior is uniform on the examples and letAϵ⊂\[N\]A\_\{\\epsilon\}\\subset\[N\]be a set of⌈ϵ​N⌉\\lceil\\epsilon N\\rceilexamples with the smallest margins\. Ifm\[ϵ\]m\_\{\[\\epsilon\]\}denotes theϵ\\epsilon\-quantile margin, namely the largest margin insideAϵA\_\{\\epsilon\}, then

m\[ϵ\]≥2​γ¯T−2T​RTℓ​\(ρAϵ\)m\_\{\[\\epsilon\]\}\\geq 2\\bar\{\\gamma\}\_\{T\}\-\\frac\{2\}\{T\}R\_\{T\}^\{\\ell\}\(\\rho\_\{A\_\{\\epsilon\}\}\)\(105\)whereρAϵ\\rho\_\{A\_\{\\epsilon\}\}is uniform onAϵA\_\{\\epsilon\}\. Hence any boundRTℓ​\(ρAϵ\)≤UT​\(ϵ\)R\_\{T\}^\{\\ell\}\(\\rho\_\{A\_\{\\epsilon\}\}\)\\leq U\_\{T\}\(\\epsilon\)yields

m\[ϵ\]≥2​γ¯T−2T​UT​\(ϵ\)m\_\{\[\\epsilon\]\}\\geq 2\\bar\{\\gamma\}\_\{T\}\-\\frac\{2\}\{T\}U\_\{T\}\(\\epsilon\)\(106\)Consequently, ifθ<2​γ¯T−2​UT​\(ϵ\)/T\\theta<2\\bar\{\\gamma\}\_\{T\}\-2U\_\{T\}\(\\epsilon\)/T, then at most anϵ\\epsilonfraction of the training examples have margin at mostθ\\theta\. In particular, the training error ofHTH\_\{T\}is at mostϵ\\epsilonwhenever2​γ¯T\>2​UT​\(ϵ\)/T2\\bar\{\\gamma\}\_\{T\}\>2U\_\{T\}\(\\epsilon\)/T\.

The point is immediate: any regret guarantee over examples is automatically a quantile\-margin guarantee\. Plugging in the intrinsic\-time Hedge theorem gives the direct boosting counterpart of the main results of this paper\.

###### Corollary 5\.14\(Intrinsic\-time boosting via predictable\-rate Hedge\)\.

Run the predictable\-rate Hedge update on the example lossesℓt\\ell\_\{t\}with uniform prior and the second\-order schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\)\. Then

m\[ϵ\]≥2​γ¯T−2T​\(DTℓ\+BTℓ​\(ρAϵ\)\+∑t=1Tηt​Qt​\(ℓ\)\)m\_\{\[\\epsilon\]\}\\geq 2\\bar\{\\gamma\}\_\{T\}\-\\frac\{2\}\{T\}\\\!\\left\(D\_\{T\}^\{\\ell\}\+B\_\{T\}^\{\\ell\}\(\\rho\_\{A\_\{\\epsilon\}\}\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(\\ell\)\\right\)\(107\)whereDTℓD\_\{T\}^\{\\ell\},BTℓB\_\{T\}^\{\\ell\}, andQt​\(ℓ\)Q\_\{t\}\(\\ell\)are the exact objects from Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)evaluated on the correctness lossesℓt\\ell\_\{t\}\. SinceKL​\(ρAϵ∥uN\)=log⁡\(N/\|Aϵ\|\)≤log⁡\(1/ϵ\)\\mathrm\{KL\}\(\\rho\_\{A\_\{\\epsilon\}\}\\\|u\_\{N\}\)=\\log\(N/\|A\_\{\\epsilon\}\|\)\\leq\\log\(1/\\epsilon\), the second\-order bound gives

m\[ϵ\]≥2​γ¯T−2T​\(\(1\+C2\)​log⁡\(1/ϵ\)\+Q∗T​\(ℓ\)\+\(2​C\+C−1\)​VT​\(ℓ\)​log⁡\(1/ϵ\)\)m\_\{\[\\epsilon\]\}\\geq 2\\bar\{\\gamma\}\_\{T\}\-\\frac\{2\}\{T\}\\\!\\left\(\(1\+C^\{2\}\)\\log\(1/\\epsilon\)\+Q\_\{\*\}^\{T\}\(\\ell\)\+\(2C\+C^\{\-1\}\)\\sqrt\{V\_\{T\}\(\\ell\)\\log\(1/\\epsilon\)\}\\right\)\(108\)Becauseℓt∈\{0,1\}\\ell\_\{t\}\\in\\\{0,1\\\}, one hasQt​\(ℓ\)≤1/8Q\_\{t\}\(\\ell\)\\leq 1/8andVT​\(ℓ\)≤T/8V\_\{T\}\(\\ell\)\\leq T/8\. Thus the margin shortfall is of orderlog⁡\(1/ϵ\)/T\\sqrt\{\\log\(1/\\epsilon\)/T\}, and if the weak learner has a uniform edge lower boundγt≥γ\>0\\gamma\_\{t\}\\geq\\gamma\>0then choosingϵ=e−c​T​γ2\\epsilon=e^\{\-cT\\gamma^\{2\}\}withc\>0c\>0small enough forces the right\-hand side above zero and therefore gives exponentially small training error\.

The second\-order schedule therefore yields a margin bound whose data\-dependent part is the realized intrinsic time on the correctness scores\. The next result gives the complementary, more classical viewpoint based directly on exponential loss\.

###### Theorem 5\.15\(Variational identity for exponential\-loss boosting\)\.

Letgt​\(i\):=yi​ht​\(xi\)∈\[−1,1\]g\_\{t\}\(i\):=y\_\{i\}h\_\{t\}\(x\_\{i\}\)\\in\[\-1,1\], choose arbitrary coefficientsαt\>0\\alpha\_\{t\}\>0, and runpt\+1​\(i\)=pt​\(i\)​e−αt​gt​\(i\)Zt,p1=uNp\_\{t\+1\}\(i\)=\\frac\{p\_\{t\}\(i\)e^\{\-\\alpha\_\{t\}g\_\{t\}\(i\)\}\}\{Z\_\{t\}\},\\qquad p\_\{1\}=u\_\{N\}, whereuNu\_\{N\}is the uniform distribution on theNNtraining examples\. Define

FT​\(x\):=∑t=1Tαt​ht​\(x\),MT​\(i\):=yi​FT​\(xi\)=∑t=1Tαt​gt​\(i\),AT:=∑t=1TαtF\_\{T\}\(x\):=\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}h\_\{t\}\(x\),\\qquad M\_\{T\}\(i\):=y\_\{i\}F\_\{T\}\(x\_\{i\}\)=\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}g\_\{t\}\(i\),\\qquad A\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}and the exponential training lossℒTexp:=1N​∑i=1Ne−MT​\(i\)=∏t=1TZt\\mathcal\{L\}\_\{T\}^\{\\exp\}:=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}e^\{\-M\_\{T\}\(i\)\}=\\prod\_\{t=1\}^\{T\}Z\_\{t\}\. Then for every posteriorρ∈Δ​\(\[N\]\)\\rho\\in\\Delta\(\[N\]\),

−log⁡ℒTexp\+KL​\(ρ∥pT\+1\)=KL​\(ρ∥uN\)\+⟨ρ,MT⟩\-\\log\\mathcal\{L\}\_\{T\}^\{\\exp\}\+\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)=\\mathrm\{KL\}\(\\rho\\\|u\_\{N\}\)\+\\langle\\rho,M\_\{T\}\\rangle\(109\)Equivalently,

−log⁡ℒTexp=supρ∈Δ​\(\[N\]\)\{⟨ρ,MT⟩−KL​\(ρ∥uN\)\}\-\\log\\mathcal\{L\}\_\{T\}^\{\\exp\}=\\sup\_\{\\rho\\in\\Delta\(\[N\]\)\}\\Bigl\\\{\\langle\\rho,M\_\{T\}\\rangle\-\\mathrm\{KL\}\(\\rho\\\|u\_\{N\}\)\\Bigr\\\}\(110\)and the supremum is attained atρ=pT\+1\\rho=p\_\{T\+1\}\. Consequently, ifm\[ϵ\]αm\_\{\[\\epsilon\]\}^\{\\alpha\}denotes theϵ\\epsilon\-quantile of the normalized weighted marginsMT​\(i\)/ATM\_\{T\}\(i\)/A\_\{T\}, then

m\[ϵ\]α≥−log⁡ℒTexp−log⁡\(1/ϵ\)ATm\_\{\[\\epsilon\]\}^\{\\alpha\}\\geq\\frac\{\-\\log\\mathcal\{L\}\_\{T\}^\{\\exp\}\-\\log\(1/\\epsilon\)\}\{A\_\{T\}\}\(111\)

This is the familiar exponential\-loss duality written in the paper’s language:−log⁡ℒTexp\-\\log\\mathcal\{L\}\_\{T\}^\{\\exp\}is exactly the best tradeoff between weighted margin and KL complexity over example reweightings\.

###### Corollary 5\.16\(Training error and margin tails in standard boosting form\)\.

Let

ErrT:=1N​\|\{i:MT​\(i\)≤0\}\|,ℰT​\(θ\):=1N​\|\{i:MT​\(i\)/AT≤θ\}\|\\mathrm\{Err\}\_\{T\}:=\\frac\{1\}\{N\}\\bigl\|\\\{i:M\_\{T\}\(i\)\\leq 0\\\}\\bigr\|,\\qquad\\mathcal\{E\}\_\{T\}\(\\theta\):=\\frac\{1\}\{N\}\\bigl\|\\\{i:M\_\{T\}\(i\)/A\_\{T\}\\leq\\theta\\\}\\bigr\|Under Theorem[5\.15](https://arxiv.org/html/2607.08789#S5.Thmtheorem15),

ErrT≤ℒTexp\\mathrm\{Err\}\_\{T\}\\leq\\mathcal\{L\}\_\{T\}^\{\\exp\}\(112\)and for everyθ∈ℝ\\theta\\in\\mathbb\{R\},

ℰT​\(θ\)≤eθ​AT​ℒTexp\\mathcal\{E\}\_\{T\}\(\\theta\)\\leq e^\{\\theta A\_\{T\}\}\\,\\mathcal\{L\}\_\{T\}^\{\\exp\}\(113\)Equivalently,

ℰT​\(θ\)≤exp⁡\(θ​AT−supρ∈Δ​\(\[N\]\)\{⟨ρ,MT⟩−KL​\(ρ∥uN\)\}\)\\mathcal\{E\}\_\{T\}\(\\theta\)\\leq\\exp\\\!\\left\(\\theta A\_\{T\}\-\\sup\_\{\\rho\\in\\Delta\(\[N\]\)\}\\Bigl\\\{\\langle\\rho,M\_\{T\}\\rangle\-\\mathrm\{KL\}\(\\rho\\\|u\_\{N\}\)\\Bigr\\\}\\right\)\(114\)

###### Corollary 5\.17\(Pressure\-targeted boosting\)\.

Assumegtg\_\{t\}is nonconstant and letμt:=⟨pt,gt⟩\\mu\_\{t\}:=\\langle p\_\{t\},g\_\{t\}\\rangle\. Fix a predictable target sequenceata\_\{t\}withmini⁡gt​\(i\)<at<μt\\min\_\{i\}g\_\{t\}\(i\)<a\_\{t\}<\\mu\_\{t\}\. Letαt\>0\\alpha\_\{t\}\>0be the unique solution of

−αt−1​log⁡\(∑i=1Npt​\(i\)​e−αt​gt​\(i\)\)=at\-\\alpha\_\{t\}^\{\-1\}\\log\\\!\\left\(\\sum\_\{i=1\}^\{N\}p\_\{t\}\(i\)e^\{\-\\alpha\_\{t\}g\_\{t\}\(i\)\}\\right\)=a\_\{t\}\(115\)Then

−log⁡ℒTexp=∑t=1Tαt​at\-\\log\\mathcal\{L\}\_\{T\}^\{\\exp\}=\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}a\_\{t\}\(116\)and, for every posteriorρ∈Δ​\(\[N\]\)\\rho\\in\\Delta\(\[N\]\),

∑t=1Tαt​\(⟨ρ,gt⟩−at\)=KL​\(ρ∥pT\+1\)−KL​\(ρ∥uN\)\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}\\left\(\\langle\\rho,g\_\{t\}\\rangle\-a\_\{t\}\\right\)=\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\-\\mathrm\{KL\}\(\\rho\\\|u\_\{N\}\)\(117\)Consequently the standard boosting performance measures satisfy

ErrT≤exp⁡\(−∑t=1Tαt​at\)\\mathrm\{Err\}\_\{T\}\\leq\\exp\\\!\\left\(\-\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}a\_\{t\}\\right\)\(118\)and, for everyθ∈ℝ\\theta\\in\\mathbb\{R\},

ℰT​\(θ\)≤exp⁡\(−∑t=1Tαt​\(at−θ\)\)\\mathcal\{E\}\_\{T\}\(\\theta\)\\leq\\exp\\\!\\left\(\-\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}\(a\_\{t\}\-\\theta\)\\right\)\(119\)Equivalently,

m\[ϵ\]α≥∑t=1Tαt​at−log⁡\(1/ϵ\)ATm\_\{\[\\epsilon\]\}^\{\\alpha\}\\geq\\frac\{\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}a\_\{t\}\-\\log\(1/\\epsilon\)\}\{A\_\{T\}\}\(120\)In particular, any fixed\-fraction targetat=λ​μta\_\{t\}=\\lambda\\mu\_\{t\}with0<λ<10<\\lambda<1is feasible wheneverμt\>0\\mu\_\{t\}\>0; the resulting weighted\-margin guarantee is then exact\.

###### Corollary 5\.18\(Classical exponential\-loss step as a special case\)\.

Assume now that each weak hypothesis is binary, sogt​\(i\)∈\{−1,\+1\}g\_\{t\}\(i\)\\in\\\{\-1,\+1\\\}, and that the weak learner is oriented so0<μt<10<\\mu\_\{t\}<1on each round\. Letμt:=⟨pt,gt⟩=2​γt,εt:=¶i∼pt​\{gt​\(i\)=−1\}=1−μt2\\mu\_\{t\}:=\\langle p\_\{t\},g\_\{t\}\\rangle=2\\gamma\_\{t\},\\qquad\\varepsilon\_\{t\}:=\\P\_\{i\\sim p\_\{t\}\}\\\{g\_\{t\}\(i\)=\-1\\\}=\\frac\{1\-\\mu\_\{t\}\}\{2\}\. The one\-step normalizerZt​\(α\)=∑i=1Npt​\(i\)​e−α​gt​\(i\)=1\+μt2​e−α\+1−μt2​eαZ\_\{t\}\(\\alpha\)=\\sum\_\{i=1\}^\{N\}p\_\{t\}\(i\)e^\{\-\\alpha g\_\{t\}\(i\)\}=\\frac\{1\+\\mu\_\{t\}\}\{2\}e^\{\-\\alpha\}\+\\frac\{1\-\\mu\_\{t\}\}\{2\}e^\{\\alpha\}is minimized at

αt⋆=12​log⁡1\+μt1−μt=12​log⁡1−εtεt\\alpha\_\{t\}^\{\\star\}=\\frac\{1\}\{2\}\\log\\frac\{1\+\\mu\_\{t\}\}\{1\-\\mu\_\{t\}\}=\\frac\{1\}\{2\}\\log\\frac\{1\-\\varepsilon\_\{t\}\}\{\\varepsilon\_\{t\}\}\(121\)for which

Zt​\(αt⋆\)=1−μt2=2​εt​\(1−εt\)Z\_\{t\}\(\\alpha\_\{t\}^\{\\star\}\)=\\sqrt\{1\-\\mu\_\{t\}^\{2\}\}=2\\sqrt\{\\varepsilon\_\{t\}\(1\-\\varepsilon\_\{t\}\)\}\(122\)Therefore

−log⁡ℒTexp=12​∑t=1Tlog⁡11−μt2\-\\log\\mathcal\{L\}\_\{T\}^\{\\exp\}=\\frac\{1\}\{2\}\\sum\_\{t=1\}^\{T\}\\log\\frac\{1\}\{1\-\\mu\_\{t\}^\{2\}\}\(123\)ErrT≤exp⁡\(−12​∑t=1Tlog⁡11−μt2\)\\mathrm\{Err\}\_\{T\}\\leq\\exp\\\!\\left\(\-\\frac\{1\}\{2\}\\sum\_\{t=1\}^\{T\}\\log\\frac\{1\}\{1\-\\mu\_\{t\}^\{2\}\}\\right\)\(124\)and, for everyθ∈ℝ\\theta\\in\\mathbb\{R\},

ℰT​\(θ\)≤exp⁡\(θ​AT⋆−12​∑t=1Tlog⁡11−μt2\),AT⋆:=∑t=1Tαt⋆\\mathcal\{E\}\_\{T\}\(\\theta\)\\leq\\exp\\\!\\left\(\\theta A\_\{T\}^\{\\star\}\-\\frac\{1\}\{2\}\\sum\_\{t=1\}^\{T\}\\log\\frac\{1\}\{1\-\\mu\_\{t\}^\{2\}\}\\right\),\\qquad A\_\{T\}^\{\\star\}:=\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}^\{\\star\}\(125\)Ifμt≥2​γ\>0\\mu\_\{t\}\\geq 2\\gamma\>0for alltt, then

ErrT≤e−2​γ2​T\\mathrm\{Err\}\_\{T\}\\leq e^\{\-2\\gamma^\{2\}T\}\(126\)

##### What the unit\-potential rule means in boosting\.

The choiceat=0a\_\{t\}=0is the unit\-potential rule from Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)applied to signed margins\. It gives an exact weighted\-margin identity but keepsℒTexp=1\\mathcal\{L\}\_\{T\}^\{\\exp\}=1\. Thus it is best read as a scale\-free calibration rule, not as the default loss\-decreasing coefficient choice\. Positive targetsat\>0a\_\{t\}\>0are what turn the same algebra into genuine exponential\-loss descent\.

##### One\-sided shortfalls and sparse example weights\.

A closely related route to sparse boosting is to replace the signed scoregt​\(i\)g\_\{t\}\(i\)by a one\-sided shortfall such as\(at−gt​\(i\)\)\+\(a\_\{t\}\-g\_\{t\}\(i\)\)\_\{\+\}or\(−gt​\(i\)\)\+\(\-g\_\{t\}\(i\)\)\_\{\+\}\. Then examples already above the target receive zero instantaneous cost, so the update concentrates only on margin violations\. This is the boosting analogue of the positive\-part sufficient\-statistic reduction: sparsity appears because the statistic itself is one\-sided\. The construction is compatible with either pressure\-targeted or classical coefficient choices and is close in spirit to the sparse behavior emphasized for NH\-Boost\.DT in\[[70](https://arxiv.org/html/2607.08789#bib.bib70)\]\.

##### Mixed\-prior boosting\.

Nothing forces the booster to use raw example losses only\. If one instead runs the mixed\-prior update of Section[4\.1](https://arxiv.org/html/2607.08789#S4.SS1)with composite example lossesct​\(i\)=ℓt​\(i\)\+ut​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\+u\_\{t\}\(i\), then the same derivation gives

⟨ρ,m⟩=2​γ¯T−2T​\(DT\+BT​\(ρ\)\+∑t=1Tηt​Qt​\(c\)\+∑t=1T\(⟨ρ,ut⟩−⟨pt,ut⟩\)\)\\langle\\rho,m\\rangle=2\\bar\{\\gamma\}\_\{T\}\-\\frac\{2\}\{T\}\\left\(D\_\{T\}\+B\_\{T\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)\\right\)So curricula, example reliabilities, or domain\-weighting side factors enter boosting through the same explicit mismatch terms that appear throughout the paper\.

## 6Partial information with contextual bandits

Partial feedback does not change the core accounting, but it inserts three new pieces: action\-sampling noise, estimator noise, and any predictable estimator bias\. The intrinsic\-time term itself is still computed from the learner’s realized estimated losses\.

In contextual bandits this dependence is especially sharp because intrinsic time is evaluated under the current posterior on policies and along the realized context sequence\. We therefore begin with finite policy classes\. The ordinary multi\-armed setting is recovered by taking the policy class to consist of the constant policies\. A final, explicitly speculative application then returns to latent model identification\.

### 6\.1Policy experts and an exact EXP4\-IX chain

Let𝒳\\mathcal\{X\}be a context space, let\[A\]\[A\]be a finite action set, and letΠ\\Pibe a finite class of deterministic policiesπ:𝒳→\[A\]\\pi:\\mathcal\{X\}\\to\[A\]\. On roundttthe learner observes the contextxtx\_\{t\}, maintains a posteriorpt∈Δ​\(Π\)p\_\{t\}\\in\\Delta\(\\Pi\), and induces the action distributionμt​\(a∣xt\):=∑π∈Π:π​\(xt\)=apt​\(π\)\\mu\_\{t\}\(a\\mid x\_\{t\}\):=\\sum\_\{\\pi\\in\\Pi:\\,\\pi\(x\_\{t\}\)=a\}p\_\{t\}\(\\pi\)\. It samplesAt∼μt\(⋅∣xt\)A\_\{t\}\\sim\\mu\_\{t\}\(\\cdot\\mid x\_\{t\}\), observes the lossℓt​\(At\)\\ell\_\{t\}\(A\_\{t\}\), forms an estimated policy\-loss vectorℓ^t∈ℝΠ\\hat\{\\ell\}\_\{t\}\\in\\mathbb\{R\}^\{\\Pi\}, and updates

pt\+1​\(π\)=π0​\(π\)​exp⁡\(−ηt​L^t​\(π\)\)∑π′∈Ππ0​\(π′\)​exp⁡\(−ηt​L^t​\(π′\)\),L^t​\(π\):=∑s=1tℓ^s​\(π\)p\_\{t\+1\}\(\\pi\)=\\frac\{\\pi\_\{0\}\(\\pi\)\\exp\(\-\\eta\_\{t\}\\widehat\{L\}\_\{t\}\(\\pi\)\)\}\{\\sum\_\{\\pi^\{\\prime\}\\in\\Pi\}\\pi\_\{0\}\(\\pi^\{\\prime\}\)\\exp\(\-\\eta\_\{t\}\\widehat\{L\}\_\{t\}\(\\pi^\{\\prime\}\)\)\},\\qquad\\widehat\{L\}\_\{t\}\(\\pi\):=\\sum\_\{s=1\}^\{t\}\\hat\{\\ell\}\_\{s\}\(\\pi\)This is again Bayes\-rule updating, now on a policy class\. Predictable policywise side penalties may be added coordinatewise exactly as in the armwise theorem below, but we suppress them here to keep the contextual notation readable\.

For the realized estimate sequence define

ψ^tΠ​\(λ\)\\displaystyle\\hat\{\\psi\}\_\{t\}^\{\\Pi\}\(\\lambda\):=log⁡𝔼π∼pt​exp⁡\(λ​\(ℓ^t​\(π\)−⟨pt,ℓ^t⟩\)\),\\displaystyle:=\\log\\mathbb\{E\}\_\{\\pi\\sim p\_\{t\}\}\\exp\\\!\\left\(\\lambda\\left\(\\hat\{\\ell\}\_\{t\}\(\\pi\)\-\\langle p\_\{t\},\\hat\{\\ell\}\_\{t\}\\rangle\\right\)\\right\),ϕ^tΠ​\(η\)\\displaystyle\\hat\{\\phi\}\_\{t\}^\{\\Pi\}\(\\eta\):=ψ^tΠ​\(−η\)ηQ^tΠ:=ϕ^tΠ​\(ηt\)ηtV^TΠ:=∑t=1TQ^tΠ\\displaystyle:=\\frac\{\\hat\{\\psi\}\_\{t\}^\{\\Pi\}\(\-\\eta\)\}\{\\eta\}\\qquad\\hat\{Q\}\_\{t\}^\{\\Pi\}:=\\frac\{\\hat\{\\phi\}\_\{t\}^\{\\Pi\}\(\\eta\_\{t\}\)\}\{\\eta\_\{t\}\}\\qquad\\hat\{V\}\_\{T\}^\{\\Pi\}:=\\sum\_\{t=1\}^\{T\}\\hat\{Q\}\_\{t\}^\{\\Pi\}Also letA^tΠ​\(η\)\\hat\{A\}\_\{t\}^\{\\Pi\}\(\\eta\),q^t,ηΠ\\hat\{q\}\_\{t,\\eta\}^\{\\Pi\},D^TΠ\\hat\{D\}\_\{T\}^\{\\Pi\}, andB^TΠ​\(u\)\\hat\{B\}\_\{T\}^\{\\Pi\}\(u\)be the terminal\-potential objects of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)formed on the policy classΠ\\Piwith lossesℓ^t\\hat\{\\ell\}\_\{t\}in place ofctc\_\{t\}\.

###### Theorem 6\.1\(Exact contextual policy\-expert identity\)\.

For every comparator distributionu∈Δ​\(Π\)u\\in\\Delta\(\\Pi\),

∑t=1T⟨pt,ℓ^t⟩−⟨u,L^T⟩=D^TΠ\+B^TΠ​\(u\)\+∑t=1Tηt​Q^tΠ\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\hat\{\\ell\}\_\{t\}\\rangle\-\\langle u,\\widehat\{L\}\_\{T\}\\rangle=\\hat\{D\}\_\{T\}^\{\\Pi\}\+\\hat\{B\}\_\{T\}^\{\\Pi\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}^\{\\Pi\}\(127\)Under the second\-order scheduleηt=min⁡\{1,C​ΓV^t−1Π\}\\eta\_\{t\}=\\min\\\!\\left\\\{1,\\,C\\sqrt\{\\frac\{\\Gamma\}\{\\hat\{V\}\_\{t\-1\}^\{\\Pi\}\}\}\\right\\\}, one has

2​C​Γ​V^TΠ−C2​Γ≤∑t=1Tηt​Q^tΠ≤C2​Γ\+Q^∗,ΠT\+2​C​Γ​V^TΠ2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\Pi\}\}\-C^\{2\}\\Gamma\\leq\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}^\{\\Pi\}\\leq C^\{2\}\\Gamma\+\\hat\{Q\}\_\{\*,\\Pi\}^\{T\}\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\Pi\}\}\(128\)whereQ^∗,ΠT:=maxt≤T⁡Q^tΠ\\hat\{Q\}\_\{\*,\\Pi\}^\{T\}:=\\max\_\{t\\leq T\}\\hat\{Q\}\_\{t\}^\{\\Pi\}\. Consequently, ifKL​\(u∥π0\)≤Γ\\mathrm\{KL\}\(u\\\|\\pi\_\{0\}\)\\leq\\Gamma, then

∑t=1T⟨pt,ℓ^t⟩−⟨u,L^T⟩≤\(1\+C2\)​Γ\+Q^∗,ΠT\+\(2​C\+C−1\)​Γ​V^TΠ\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\hat\{\\ell\}\_\{t\}\\rangle\-\\langle u,\\widehat\{L\}\_\{T\}\\rangle\\leq\(1\+C^\{2\}\)\\Gamma\+\\hat\{Q\}\_\{\*,\\Pi\}^\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\Pi\}\}\(129\)

This is the full\-information part of contextual bandits: once estimated policy losses are chosen, the usual intrinsic\-time theorem applies without modification\.

To return to the realized action losses, letℓtΠ​\(π\):=ℓt​\(π​\(xt\)\)\\ell\_\{t\}^\{\\Pi\}\(\\pi\):=\\ell\_\{t\}\(\\pi\(x\_\{t\}\)\)be the true policy\-loss vector, writeLTΠ:=∑t=1TℓtΠL\_\{T\}^\{\\Pi\}:=\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}^\{\\Pi\}, and let𝒢t\\mathcal\{G\}\_\{t\}denote theσ\\sigma\-field obtained after the contextxtx\_\{t\}is revealed and before the learner randomizes on roundtt\. Defineℓ¯t:=𝔼​\[ℓ^t∣𝒢t\]\\bar\{\\ell\}\_\{t\}:=\\mathbb\{E\}\[\\hat\{\\ell\}\_\{t\}\\mid\\mathcal\{G\}\_\{t\}\]\.

###### Theorem 6\.2\(Generic pathwise contextual\-bandit decomposition\)\.

Foru∈Δ​\(Π\)u\\in\\Delta\(\\Pi\), define the sampled policy regret

R~TΠ​\(u\):=∑t=1Tℓt​\(At\)−⟨u,LTΠ⟩\\widetilde\{R\}\_\{T\}^\{\\Pi\}\(u\):=\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}\(A\_\{t\}\)\-\\langle u,L\_\{T\}^\{\\Pi\}\\ranglethe play martingale

MTctx:=∑t=1T\(ℓt​\(At\)−⟨pt,ℓtΠ⟩\)M\_\{T\}^\{\\mathrm\{ctx\}\}:=\\sum\_\{t=1\}^\{T\}\\left\(\\ell\_\{t\}\(A\_\{t\}\)\-\\langle p\_\{t\},\\ell\_\{t\}^\{\\Pi\}\\rangle\\right\)the estimation martingale

M^TΠ​\(u\):=∑t=1T⟨pt−u,ℓ^t−ℓ¯t⟩\\hat\{M\}\_\{T\}^\{\\Pi\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{\\ell\}\_\{t\}\-\\bar\{\\ell\}\_\{t\}\\rangleand the predictable bias

BiasTΠ​\(u\):=∑t=1T⟨pt−u,ℓtΠ−ℓ¯t⟩\\mathrm\{Bias\}\_\{T\}^\{\\Pi\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\ell\_\{t\}^\{\\Pi\}\-\\bar\{\\ell\}\_\{t\}\\rangleThen, pathwise,

R~TΠ​\(u\)=MTctx−M^TΠ​\(u\)\+BiasTΠ​\(u\)\+D^TΠ\+B^TΠ​\(u\)\+∑t=1Tηt​Q^tΠ\\widetilde\{R\}\_\{T\}^\{\\Pi\}\(u\)=M\_\{T\}^\{\\mathrm\{ctx\}\}\-\\hat\{M\}\_\{T\}^\{\\Pi\}\(u\)\+\\mathrm\{Bias\}\_\{T\}^\{\\Pi\}\(u\)\+\\hat\{D\}\_\{T\}^\{\\Pi\}\+\\hat\{B\}\_\{T\}^\{\\Pi\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}^\{\\Pi\}\(130\)Under the same second\-order schedule,

MTctx−M^TΠ​\(u\)\+BiasTΠ​\(u\)\+D^TΠ\+B^TΠ​\(u\)\+2​C​Γ​V^TΠ−C2​Γ\\displaystyle M\_\{T\}^\{\\mathrm\{ctx\}\}\-\\hat\{M\}\_\{T\}^\{\\Pi\}\(u\)\+\\mathrm\{Bias\}\_\{T\}^\{\\Pi\}\(u\)\+\\hat\{D\}\_\{T\}^\{\\Pi\}\+\\hat\{B\}\_\{T\}^\{\\Pi\}\(u\)\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\Pi\}\}\-C^\{2\}\\Gamma≤R~TΠ​\(u\)\\displaystyle\\qquad\\leq\\widetilde\{R\}\_\{T\}^\{\\Pi\}\(u\)≤MTctx−M^TΠ​\(u\)\+BiasTΠ​\(u\)\+D^TΠ\+B^TΠ​\(u\)\+C2​Γ\+Q^∗,ΠT\+2​C​Γ​V^TΠ\.\\displaystyle\\qquad\\leq M\_\{T\}^\{\\mathrm\{ctx\}\}\-\\hat\{M\}\_\{T\}^\{\\Pi\}\(u\)\+\\mathrm\{Bias\}\_\{T\}^\{\\Pi\}\(u\)\+\\hat\{D\}\_\{T\}^\{\\Pi\}\+\\hat\{B\}\_\{T\}^\{\\Pi\}\(u\)\+C^\{2\}\\Gamma\+\\hat\{Q\}\_\{\*,\\Pi\}^\{T\}\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\Pi\}\}\.\(131\)BothMTctxM\_\{T\}^\{\\mathrm\{ctx\}\}andM^TΠ​\(u\)\\hat\{M\}\_\{T\}^\{\\Pi\}\(u\)are martingales\.

The decomposition cleanly isolates the new contextual\-bandit ingredients: action\-sampling noiseMTctxM\_\{T\}^\{\\mathrm\{ctx\}\}, estimator noiseM^TΠ​\(u\)\\hat\{M\}\_\{T\}^\{\\Pi\}\(u\), estimator biasBiasTΠ​\(u\)\\mathrm\{Bias\}\_\{T\}^\{\\Pi\}\(u\), and the same intrinsic\-time term as before\.

A particularly clean instance is the implicit\-exploration estimator

ℓ^tIX​\(π\):=ℓt​\(At\)​𝟏​\{π​\(xt\)=At\}μt​\(At∣xt\)\+γt,γt\>0\\hat\{\\ell\}\_\{t\}^\{\\mathrm\{IX\}\}\(\\pi\):=\\frac\{\\ell\_\{t\}\(A\_\{t\}\)\\mathbf\{1\}\\\{\\pi\(x\_\{t\}\)=A\_\{t\}\\\}\}\{\\mu\_\{t\}\(A\_\{t\}\\mid x\_\{t\}\)\+\\gamma\_\{t\}\},\\qquad\\gamma\_\{t\}\>0
###### Proposition 6\.3\(EXP4\-IX as an exact intrinsic\-time tracker\)\.

Assumeℓt​\(a\)∈\[0,1\]\\ell\_\{t\}\(a\)\\in\[0,1\]and letℓ^t=ℓ^tIX\\hat\{\\ell\}\_\{t\}=\\hat\{\\ell\}\_\{t\}^\{\\mathrm\{IX\}\}\. DefinedtIX​\(π\):=γtμt​\(π​\(xt\)∣xt\)\+γt​ℓt​\(π​\(xt\)\)d\_\{t\}^\{\\mathrm\{IX\}\}\(\\pi\):=\\frac\{\\gamma\_\{t\}\}\{\\mu\_\{t\}\(\\pi\(x\_\{t\}\)\\mid x\_\{t\}\)\+\\gamma\_\{t\}\}\\,\\ell\_\{t\}\(\\pi\(x\_\{t\}\)\)\. Thenℓ¯tIX=ℓtΠ−dtIX\\bar\{\\ell\}\_\{t\}^\{\\mathrm\{IX\}\}=\\ell\_\{t\}^\{\\Pi\}\-d\_\{t\}^\{\\mathrm\{IX\}\}\. Hence

R~TΠ​\(u\)=MTctx−M^TIX​\(u\)\+∑t=1T⟨pt−u,dtIX⟩\+D^TIX\+B^TIX​\(u\)\+∑t=1Tηt​Q^tIX\\widetilde\{R\}\_\{T\}^\{\\Pi\}\(u\)=M\_\{T\}^\{\\mathrm\{ctx\}\}\-\\hat\{M\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,d\_\{t\}^\{\\mathrm\{IX\}\}\\rangle\+\\hat\{D\}\_\{T\}^\{\\mathrm\{IX\}\}\+\\hat\{B\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}^\{\\mathrm\{IX\}\}\(132\)where

M^TIX​\(u\):=∑t=1T⟨pt−u,ℓ^tIX−ℓ¯tIX⟩\\hat\{M\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{\\ell\}\_\{t\}^\{\\mathrm\{IX\}\}\-\\bar\{\\ell\}\_\{t\}^\{\\mathrm\{IX\}\}\\rangleUnder the second\-order schedule built fromV^t−1IX\\hat\{V\}\_\{t\-1\}^\{\\mathrm\{IX\}\},

MTctx−M^TIX​\(u\)\+∑t=1T⟨pt−u,dtIX⟩\+D^TIX\+B^TIX​\(u\)\+2​C​Γ​V^TIX−C2​Γ\\displaystyle M\_\{T\}^\{\\mathrm\{ctx\}\}\-\\hat\{M\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,d\_\{t\}^\{\\mathrm\{IX\}\}\\rangle\+\\hat\{D\}\_\{T\}^\{\\mathrm\{IX\}\}\+\\hat\{B\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\mathrm\{IX\}\}\}\-C^\{2\}\\Gamma≤R~TΠ​\(u\)\\displaystyle\\qquad\\leq\\widetilde\{R\}\_\{T\}^\{\\Pi\}\(u\)≤MTctx−M^TIX​\(u\)\+∑t=1T⟨pt−u,dtIX⟩\+D^TIX\+B^TIX​\(u\)\+C2​Γ\+Q^∗,ΠT,IX\+2​C​Γ​V^TIX\.\\displaystyle\\qquad\\leq M\_\{T\}^\{\\mathrm\{ctx\}\}\-\\hat\{M\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,d\_\{t\}^\{\\mathrm\{IX\}\}\\rangle\+\\hat\{D\}\_\{T\}^\{\\mathrm\{IX\}\}\+\\hat\{B\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+C^\{2\}\\Gamma\+\\hat\{Q\}\_\{\*,\\Pi\}^\{T,\\mathrm\{IX\}\}\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\mathrm\{IX\}\}\}\.\(133\)In particular, EXP4\-IX does not merely upper bound its contextual regret by a convenient second moment\. Along every realized context sequence and action trajectory it pays the exact intrinsic\-time term∑tηt​Q^tIX\\sum\_\{t\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}^\{\\mathrm\{IX\}\}, computed from the learner’s own posterior on policies\.

##### Why not posterior sampling here?

Posterior sampling fits the same Bayes\-rule template at the model level, especially under logarithmic loss\[[57](https://arxiv.org/html/2607.08789#bib.bib57)\], but not the exact linear contextual\-regret identity above\. The action is produced only after an additional argmax/probability\-matching map, so the realized loss is not generally a linear functional of the model posterior\. Converting posterior evidence to contextual\-bandit regret therefore requires an extra surrogate or information\-ratio argument\[[64](https://arxiv.org/html/2607.08789#bib.bib64),[47](https://arxiv.org/html/2607.08789#bib.bib47)\]\. Recent optimal contextual\-bandit policy\-optimization methods are likewise of a different oracle/optimization flavor rather than Bayes\-rule updating\. For the exact identity\-driven program here, EXP4\-IX\-style policy updates are the cleaner fit\.

TakingΠ=\{πa:πa​\(x\)≡a,a∈\[A\]\}\\Pi=\\\{\\pi\_\{a\}:\\pi\_\{a\}\(x\)\\equiv a,\\ a\\in\[A\]\\\}reduces the contextual formulas above to the ordinary multi\-armed setting\. We record that armwise specialization separately because the notation is lighter and the estimator templates are easiest to compare in arm coordinates\.

### 6\.2Armwise specialization: Exp3\-type updates

Letct=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}, whereℓt​\(a\)∈\[0,1\]\\ell\_\{t\}\(a\)\\in\[0,1\]is the adversarial loss andut​\(a\)u\_\{t\}\(a\)is a predictable side offset\. An Exp3\-type algorithm\[[5](https://arxiv.org/html/2607.08789#bib.bib5)\]first forms an estimated composite lossc^t\\hat\{c\}\_\{t\}and then runs exponential weights onc^t\\hat\{c\}\_\{t\}\. We therefore write

pt​\(a\)∝π​\(a\)​e−ηt​C^t−1​\(a\)C^t​\(a\):=∑s=1tc^s​\(a\)p\_\{t\}\(a\)\\propto\\pi\(a\)e^\{\-\\eta\_\{t\}\\hat\{C\}\_\{t\-1\}\(a\)\}\\qquad\\hat\{C\}\_\{t\}\(a\):=\\sum\_\{s=1\}^\{t\}\\hat\{c\}\_\{s\}\(a\)\(134\)with predictable positive ratesηt\>0\\eta\_\{t\}\>0\. Define

ψ^t​\(λ\):=log​∑a=1Kpt​\(a\)​exp⁡\(λ​\(c^t​\(a\)−⟨pt,c^t⟩\)\),ϕ^t​\(η\):=ψ^t​\(−η\)/η,Q^t:=ϕ^t​\(ηt\)/ηt\\hat\{\\psi\}\_\{t\}\(\\lambda\):=\\log\\sum\_\{a=1\}^\{K\}p\_\{t\}\(a\)\\exp\\\!\\left\(\\lambda\(\\hat\{c\}\_\{t\}\(a\)\-\\langle p\_\{t\},\\hat\{c\}\_\{t\}\\rangle\)\\right\),\\qquad\\hat\{\\phi\}\_\{t\}\(\\eta\):=\\hat\{\\psi\}\_\{t\}\(\-\\eta\)/\\eta,\\qquad\\hat\{Q\}\_\{t\}:=\\hat\{\\phi\}\_\{t\}\(\\eta\_\{t\}\)/\\eta\_\{t\}and letV^T:=∑t=1TQ^t\\hat\{V\}\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\hat\{Q\}\_\{t\}\.

###### Theorem 6\.4\(Exact PAC\-Bayes identity for estimated\-loss Exp3\)\.

Let

A^t​\(η\):=−η−1​log​∑a=1Kπ​\(a\)​e−η​C^t​\(a\),D^T:=∑t=1T−1\(A^t​\(ηt\)−A^t​\(ηt\+1\)\)\\hat\{A\}\_\{t\}\(\\eta\):=\-\\eta^\{\-1\}\\log\\sum\_\{a=1\}^\{K\}\\pi\(a\)e^\{\-\\eta\\hat\{C\}\_\{t\}\(a\)\},\\qquad\\hat\{D\}\_\{T\}:=\\sum\_\{t=1\}^\{T\-1\}\(\\hat\{A\}\_\{t\}\(\\eta\_\{t\}\)\-\\hat\{A\}\_\{t\}\(\\eta\_\{t\+1\}\)\)and let

B^T​\(u\):=KL​\(u∥π\)−KL​\(u∥q^T,ηT\)ηT,q^T,ηT​\(a\)∝π​\(a\)​e−ηT​C^T​\(a\)\\hat\{B\}\_\{T\}\(u\):=\\frac\{\\mathrm\{KL\}\(u\\\|\\pi\)\-\\mathrm\{KL\}\(u\\\|\\hat\{q\}\_\{T,\\eta\_\{T\}\}\)\}\{\\eta\_\{T\}\},\\qquad\\hat\{q\}\_\{T,\\eta\_\{T\}\}\(a\)\\propto\\pi\(a\)e^\{\-\\eta\_\{T\}\\hat\{C\}\_\{T\}\(a\)\}Then for every comparatoru∈Δ​\(\[K\]\)u\\in\\Delta\(\[K\]\),

R^Tc​\(u\):=∑t=1T⟨pt−u,c^t⟩=D^T\+B^T​\(u\)\+∑t=1Tηt​Q^t\\widehat\{R\}\_\{T\}^\{c\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}\\rangle=\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}\(135\)Ifηt≡η\\eta\_\{t\}\\equiv\\eta, thenD^T=0\\hat\{D\}\_\{T\}=0andR^Tc​\(u\)=B^T​\(u\)\+η​V^T\\widehat\{R\}\_\{T\}^\{c\}\(u\)=\\hat\{B\}\_\{T\}\(u\)\+\\eta\\hat\{V\}\_\{T\}\.

###### Corollary 6\.5\(Second\-order envelope for estimated\-loss Exp3\)\.

Run the second\-order schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\), but withV^t−1\\hat\{V\}\_\{t\-1\}in place ofVt−1V\_\{t\-1\}; equivalently,ηt=min⁡\{1,C​Γ/V^t−1\}\\eta\_\{t\}=\\min\\left\\\{1,C\\sqrt\{\\Gamma/\\hat\{V\}\_\{t\-1\}\}\\right\\\}with the usual conventionηt=1\\eta\_\{t\}=1whenV^t−1=0\\hat\{V\}\_\{t\-1\}=0\. Also setQ^∗T:=maxt≤T⁡Q^t\\hat\{Q\}\_\{\*\}^\{T\}:=\\max\_\{t\\leq T\}\\hat\{Q\}\_\{t\}\. Then

D^T\+B^T​\(u\)\+2​C​Γ​V^T−C2​Γ≤R^Tc​\(u\)≤D^T\+B^T​\(u\)\+C2​Γ\+Q^∗T\+2​C​Γ​V^T\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}\}\-C^\{2\}\\Gamma\\leq\\widehat\{R\}\_\{T\}^\{c\}\(u\)\\leq\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+C^\{2\}\\Gamma\+\\hat\{Q\}\_\{\*\}^\{T\}\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}\}\(136\)IfKL​\(u∥π\)≤Γ\\mathrm\{KL\}\(u\\\|\\pi\)\\leq\\Gamma, thenR^Tc​\(u\)≤\(1\+C2\)​Γ\+Q^∗T\+\(2​C\+C−1\)​Γ​V^T\\widehat\{R\}\_\{T\}^\{c\}\(u\)\\leq\(1\+C^\{2\}\)\\Gamma\+\\hat\{Q\}\_\{\*\}^\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}\}\.

Up to this point the statement is purely algorithmic: any loss estimate plugged into Exp3 satisfies the same exact estimated\-loss identity\. The genuinely bandit\-specific step is to relate the realized estimatec^t\\hat\{c\}\_\{t\}back to the played losses\. Let\(ℱt\)t≥0\(\\mathcal\{F\}\_\{t\}\)\_\{t\\geq 0\}be the natural filtration of the learner’s randomization and observations\.

###### Theorem 6\.6\(Generic pathwise bandit decomposition\)\.

On roundtt, suppose the learner samplesAt∼μtA\_\{t\}\\sim\\mu\_\{t\}, whereμt∈Δ​\(\[K\]\)\\mu\_\{t\}\\in\\Delta\(\[K\]\)isℱt−1\\mathcal\{F\}\_\{t\-1\}\-measurable, suffers composite lossct​\(At\)c\_\{t\}\(A\_\{t\}\), forms anℱt\\mathcal\{F\}\_\{t\}\-measurable estimatec^t\\hat\{c\}\_\{t\}, and updatespt\+1p\_\{t\+1\}via \([134](https://arxiv.org/html/2607.08789#S6.E134)\)\. Letc¯t:=𝔼​\[c^t∣ℱt−1\]\\bar\{c\}\_\{t\}:=\\mathbb\{E\}\[\\hat\{c\}\_\{t\}\\mid\\mathcal\{F\}\_\{t\-1\}\]\. For a comparatoru∈Δ​\(\[K\]\)u\\in\\Delta\(\[K\]\), define the sampled composite\-loss regretR~Tc​\(u\):=∑t=1Tct​\(At\)−⟨u,CT⟩\\widetilde\{R\}\_\{T\}^\{c\}\(u\):=\\sum\_\{t=1\}^\{T\}c\_\{t\}\(A\_\{t\}\)\-\\langle u,C\_\{T\}\\rangle, the play martingale

MTplay:=∑t=1T\(ct​\(At\)−⟨μt,ct⟩\)M\_\{T\}^\{\\mathrm\{play\}\}:=\\sum\_\{t=1\}^\{T\}\\left\(c\_\{t\}\(A\_\{t\}\)\-\\langle\\mu\_\{t\},c\_\{t\}\\rangle\\right\)the estimation martingale

MTest​\(u\):=∑t=1T⟨pt−u,c^t−c¯t⟩M\_\{T\}^\{\\mathrm\{est\}\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}\-\\bar\{c\}\_\{t\}\\rangleand the predictable correction terms

ΞT:=∑t=1T⟨μt−pt,ct⟩,BiasT​\(u\):=∑t=1T⟨pt−u,ct−c¯t⟩\\Xi\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\langle\\mu\_\{t\}\-p\_\{t\},c\_\{t\}\\rangle,\\qquad\\mathrm\{Bias\}\_\{T\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,c\_\{t\}\-\\bar\{c\}\_\{t\}\\rangleThen, pathwise,

R~Tc​\(u\)=MTplay\+ΞT−MTest​\(u\)\+BiasT​\(u\)\+D^T\+B^T​\(u\)\+∑t=1Tηt​Q^t\\widetilde\{R\}\_\{T\}^\{c\}\(u\)=M\_\{T\}^\{\\mathrm\{play\}\}\+\\Xi\_\{T\}\-M\_\{T\}^\{\\mathrm\{est\}\}\(u\)\+\\mathrm\{Bias\}\_\{T\}\(u\)\+\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}\(137\)Under the second\-order schedule of Corollary[6\.5](https://arxiv.org/html/2607.08789#S6.Thmtheorem5),

MTplay\+ΞT−MTest​\(u\)\+BiasT​\(u\)\+D^T\+B^T​\(u\)\+2​C​Γ​V^T−C2​Γ\\displaystyle M\_\{T\}^\{\\mathrm\{play\}\}\+\\Xi\_\{T\}\-M\_\{T\}^\{\\mathrm\{est\}\}\(u\)\+\\mathrm\{Bias\}\_\{T\}\(u\)\+\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}\}\-C^\{2\}\\Gamma≤R~Tc​\(u\)\\displaystyle\\qquad\\leq\\widetilde\{R\}\_\{T\}^\{c\}\(u\)≤MTplay\+ΞT−MTest​\(u\)\+BiasT​\(u\)\+D^T\+B^T​\(u\)\+C2​Γ\+Q^∗T\+2​C​Γ​V^T\.\\displaystyle\\qquad\\leq M\_\{T\}^\{\\mathrm\{play\}\}\+\\Xi\_\{T\}\-M\_\{T\}^\{\\mathrm\{est\}\}\(u\)\+\\mathrm\{Bias\}\_\{T\}\(u\)\+\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+C^\{2\}\\Gamma\+\\hat\{Q\}\_\{\*\}^\{T\}\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}\}\.\(138\)Also,

∑t=1Tℓt​\(At\)−⟨u,LT⟩=R~Tc​\(u\)\+∑t=1T\(⟨u,ut⟩−ut​\(At\)\)\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}\(A\_\{t\}\)\-\\langle u,L\_\{T\}\\rangle=\\widetilde\{R\}\_\{T\}^\{c\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle u,u\_\{t\}\\rangle\-u\_\{t\}\(A\_\{t\}\)\\right\)\(139\)BothMTplayM\_\{T\}^\{\\mathrm\{play\}\}andMTest​\(u\)M\_\{T\}^\{\\mathrm\{est\}\}\(u\)are martingales, so

𝔼​\[R~Tc​\(u\)\]=𝔼​\[ΞT\+BiasT​\(u\)\+D^T\+B^T​\(u\)\+∑t=1Tηt​Q^t\]\\mathbb\{E\}\[\\widetilde\{R\}\_\{T\}^\{c\}\(u\)\]=\\mathbb\{E\}\\\!\\left\[\\Xi\_\{T\}\+\\mathrm\{Bias\}\_\{T\}\(u\)\+\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}\\right\]

This is the master formula for the bandit section\. It shows exactly where partial feedback enters:ΞT\\Xi\_\{T\}records exploration,MTplayM\_\{T\}^\{\\mathrm\{play\}\}andMTest​\(u\)M\_\{T\}^\{\\mathrm\{est\}\}\(u\)are the two martingale fluctuations,BiasT​\(u\)\\mathrm\{Bias\}\_\{T\}\(u\)captures estimator bias, and the intrinsic\-time term is unchanged\.

Two estimator templates make the usefulness of \([137](https://arxiv.org/html/2607.08789#S6.E137)\) explicit\.

###### Corollary 6\.7\(Explicit\-exploration inverse propensity scoring \(IPS\)\)\.

For the standard explicit\-exploration estimatorμt:=\(1−γt\)​pt\+γt​uK\\mu\_\{t\}:=\(1\-\\gamma\_\{t\}\)p\_\{t\}\+\\gamma\_\{t\}u\_\{K\},c^t​\(a\):=ut​\(a\)\+ℓt​\(At\)​𝟏​\{At=a\}/μt​\(a\)\\hat\{c\}\_\{t\}\(a\):=u\_\{t\}\(a\)\+\\ell\_\{t\}\(A\_\{t\}\)\\mathbf\{1\}\\\{A\_\{t\}=a\\\}/\\mu\_\{t\}\(a\), andAt∼μtA\_\{t\}\\sim\\mu\_\{t\}, whereuKu\_\{K\}is uniform on\[K\]\[K\]andγt∈\[0,1\]\\gamma\_\{t\}\\in\[0,1\]is predictable, one hasc¯t=ct\\bar\{c\}\_\{t\}=c\_\{t\}\. HenceBiasT​\(u\)=0\\mathrm\{Bias\}\_\{T\}\(u\)=0and

R~Tc​\(u\)=MTplay\+ΞT−MTIPS​\(u\)\+D^T\+B^T​\(u\)\+∑t=1Tηt​Q^t\\widetilde\{R\}\_\{T\}^\{c\}\(u\)=M\_\{T\}^\{\\mathrm\{play\}\}\+\\Xi\_\{T\}\-M\_\{T\}^\{\\mathrm\{IPS\}\}\(u\)\+\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}\(140\)whereMTIPS​\(u\):=∑t=1T⟨pt−u,c^t−ct⟩M\_\{T\}^\{\\mathrm\{IPS\}\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}\-c\_\{t\}\\rangleis a martingale\. Under the second\-order schedule,

MTplay\+ΞT−MTIPS​\(u\)\+D^T\+B^T​\(u\)\+2​C​Γ​V^T−C2​Γ\\displaystyle M\_\{T\}^\{\\mathrm\{play\}\}\+\\Xi\_\{T\}\-M\_\{T\}^\{\\mathrm\{IPS\}\}\(u\)\+\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}\}\-C^\{2\}\\Gamma≤R~Tc​\(u\)\\displaystyle\\qquad\\leq\\widetilde\{R\}\_\{T\}^\{c\}\(u\)≤MTplay\+ΞT−MTIPS​\(u\)\+D^T\+B^T​\(u\)\+C2​Γ\+Q^∗T\+2​C​Γ​V^T\.\\displaystyle\\qquad\\leq M\_\{T\}^\{\\mathrm\{play\}\}\+\\Xi\_\{T\}\-M\_\{T\}^\{\\mathrm\{IPS\}\}\(u\)\+\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\(u\)\+C^\{2\}\\Gamma\+\\hat\{Q\}\_\{\*\}^\{T\}\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}\}\.\(141\)

###### Proposition 6\.8\(Implicit exploration / EXP3\-IX\)\.

The implicit\-exploration update of\[[53](https://arxiv.org/html/2607.08789#bib.bib53),[44](https://arxiv.org/html/2607.08789#bib.bib44)\]keeps the sampling ruleAt∼ptA\_\{t\}\\sim p\_\{t\}but replaces the IPS denominatorpt​\(a\)p\_\{t\}\(a\)bypt​\(a\)\+γtp\_\{t\}\(a\)\+\\gamma\_\{t\}with a predictable bias parameterγt≥0\\gamma\_\{t\}\\geq 0\. Definec^tIX​\(a\):=ut​\(a\)\+ℓt​\(At\)​𝟏​\{At=a\}/\(pt​\(a\)\+γt\)\\hat\{c\}\_\{t\}^\{\\mathrm\{IX\}\}\(a\):=u\_\{t\}\(a\)\+\\ell\_\{t\}\(A\_\{t\}\)\\mathbf\{1\}\\\{A\_\{t\}=a\\\}/\(p\_\{t\}\(a\)\+\\gamma\_\{t\}\)withAt∼ptA\_\{t\}\\sim p\_\{t\}, and let\(D^TIX,B^TIX​\(u\),Q^tIX,V^TIX\)\(\\hat\{D\}\_\{T\}^\{\\mathrm\{IX\}\},\\hat\{B\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\),\\hat\{Q\}\_\{t\}^\{\\mathrm\{IX\}\},\\hat\{V\}\_\{T\}^\{\\mathrm\{IX\}\}\)be the corresponding objects from Theorem[6\.4](https://arxiv.org/html/2607.08789#S6.Thmtheorem4)\. Set

dtIX​\(a\):=γt​ℓt​\(a\)pt​\(a\)\+γt,βtIX:=⟨pt,dtIX⟩d\_\{t\}^\{\\mathrm\{IX\}\}\(a\):=\\frac\{\\gamma\_\{t\}\\ell\_\{t\}\(a\)\}\{p\_\{t\}\(a\)\+\\gamma\_\{t\}\},\\qquad\\beta\_\{t\}^\{\\mathrm\{IX\}\}:=\\langle p\_\{t\},d\_\{t\}^\{\\mathrm\{IX\}\}\\rangleThenc¯tIX=ct−dtIX\\bar\{c\}\_\{t\}^\{\\mathrm\{IX\}\}=c\_\{t\}\-d\_\{t\}^\{\\mathrm\{IX\}\}, the processMTIX​\(u\):=∑t=1T⟨pt−u,c^tIX−c¯tIX⟩M\_\{T\}^\{\\mathrm\{IX\}\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}^\{\\mathrm\{IX\}\}\-\\bar\{c\}\_\{t\}^\{\\mathrm\{IX\}\}\\rangleis a martingale, and pathwise

R~Tc​\(u\)=MTplay−MTIX​\(u\)\+BiasTIX​\(u\)\+D^TIX\+B^TIX​\(u\)\+∑t=1Tηt​Q^tIX\\widetilde\{R\}\_\{T\}^\{c\}\(u\)=M\_\{T\}^\{\\mathrm\{play\}\}\-M\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\mathrm\{Bias\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\hat\{D\}\_\{T\}^\{\\mathrm\{IX\}\}\+\\hat\{B\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}^\{\\mathrm\{IX\}\}\(142\)where

BiasTIX​\(u\):=∑t=1T⟨pt−u,dtIX⟩\\mathrm\{Bias\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,d\_\{t\}^\{\\mathrm\{IX\}\}\\rangleUnder the second\-order schedule built fromV^t−1IX\\hat\{V\}\_\{t\-1\}^\{\\mathrm\{IX\}\},

MTplay−MTIX​\(u\)\+BiasTIX​\(u\)\+D^TIX\+B^TIX​\(u\)\+2​C​Γ​V^TIX−C2​Γ\\displaystyle M\_\{T\}^\{\\mathrm\{play\}\}\-M\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\mathrm\{Bias\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\hat\{D\}\_\{T\}^\{\\mathrm\{IX\}\}\+\\hat\{B\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\mathrm\{IX\}\}\}\-C^\{2\}\\Gamma≤R~Tc​\(u\)\\displaystyle\\qquad\\leq\\widetilde\{R\}\_\{T\}^\{c\}\(u\)≤MTplay−MTIX​\(u\)\+BiasTIX​\(u\)\+D^TIX\+B^TIX​\(u\)\+C2​Γ\+Q^∗,IXT\+2​C​Γ​V^TIX,\\displaystyle\\qquad\\leq M\_\{T\}^\{\\mathrm\{play\}\}\-M\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\mathrm\{Bias\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+\\hat\{D\}\_\{T\}^\{\\mathrm\{IX\}\}\+\\hat\{B\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\+C^\{2\}\\Gamma\+\\hat\{Q\}\_\{\*,\\mathrm\{IX\}\}^\{T\}\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{\\mathrm\{IX\}\}\},\(143\)whereQ^∗,IXT:=maxt≤T⁡Q^tIX\\hat\{Q\}\_\{\*,\\mathrm\{IX\}\}^\{T\}:=\\max\_\{t\\leq T\}\\hat\{Q\}\_\{t\}^\{\\mathrm\{IX\}\}\. SincedtIX≥0d\_\{t\}^\{\\mathrm\{IX\}\}\\geq 0coordinatewise,

BiasTIX​\(u\)≤∑t=1TβtIX,βtIX≤K​γt1\+K​γt≤min⁡\{1,K​γt\}\\mathrm\{Bias\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)\\leq\\sum\_\{t=1\}^\{T\}\\beta\_\{t\}^\{\\mathrm\{IX\}\},\\qquad\\beta\_\{t\}^\{\\mathrm\{IX\}\}\\leq\\frac\{K\\gamma\_\{t\}\}\{1\+K\\gamma\_\{t\}\}\\leq\\min\\\{1,K\\gamma\_\{t\}\\\}

###### Proposition 6\.9\(Predictable\-offset estimator\)\.

Assumept​\(a\)\>0p\_\{t\}\(a\)\>0for every arm and round, letAt∼ptA\_\{t\}\\sim p\_\{t\}, and letmt∈\[0,1\]Km\_\{t\}\\in\[0,1\]^\{K\}be any predictable offset vector\. Define the optimistic/control\-variate estimatec^tm​\(a\):=ut​\(a\)\+mt​\(a\)\+\(ℓt​\(At\)−mt​\(At\)\)​𝟏​\{At=a\}/pt​\(a\)\\hat\{c\}\_\{t\}^\{m\}\(a\):=u\_\{t\}\(a\)\+m\_\{t\}\(a\)\+\\left\(\\ell\_\{t\}\(A\_\{t\}\)\-m\_\{t\}\(A\_\{t\}\)\\right\)\\mathbf\{1\}\\\{A\_\{t\}=a\\\}/p\_\{t\}\(a\)\. Thenc¯tm=ct\\bar\{c\}\_\{t\}^\{m\}=c\_\{t\}\. If\(D^Tm,B^Tm​\(u\),Q^tm,V^Tm\)\(\\hat\{D\}\_\{T\}^\{m\},\\hat\{B\}\_\{T\}^\{m\}\(u\),\\hat\{Q\}\_\{t\}^\{m\},\\hat\{V\}\_\{T\}^\{m\}\)are built fromc^tm\\hat\{c\}\_\{t\}^\{m\}, the martingaleMTm​\(u\):=∑t=1T⟨pt−u,c^tm−ct⟩M\_\{T\}^\{m\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}^\{m\}\-c\_\{t\}\\ranglesatisfies

R~Tc​\(u\)=MTplay−MTm​\(u\)\+D^Tm\+B^Tm​\(u\)\+∑t=1Tηt​Q^tm\\widetilde\{R\}\_\{T\}^\{c\}\(u\)=M\_\{T\}^\{\\mathrm\{play\}\}\-M\_\{T\}^\{m\}\(u\)\+\\hat\{D\}\_\{T\}^\{m\}\+\\hat\{B\}\_\{T\}^\{m\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}^\{m\}\(144\)Under the second\-order schedule built fromV^t−1m\\hat\{V\}\_\{t\-1\}^\{m\},

MTplay−MTm​\(u\)\+D^Tm\+B^Tm​\(u\)\+2​C​Γ​V^Tm−C2​Γ\\displaystyle M\_\{T\}^\{\\mathrm\{play\}\}\-M\_\{T\}^\{m\}\(u\)\+\\hat\{D\}\_\{T\}^\{m\}\+\\hat\{B\}\_\{T\}^\{m\}\(u\)\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{m\}\}\-C^\{2\}\\Gamma≤R~Tc​\(u\)\\displaystyle\\qquad\\leq\\widetilde\{R\}\_\{T\}^\{c\}\(u\)≤MTplay−MTm​\(u\)\+D^Tm\+B^Tm​\(u\)\+C2​Γ\+Q^∗,mT\+2​C​Γ​V^Tm,\\displaystyle\\qquad\\leq M\_\{T\}^\{\\mathrm\{play\}\}\-M\_\{T\}^\{m\}\(u\)\+\\hat\{D\}\_\{T\}^\{m\}\+\\hat\{B\}\_\{T\}^\{m\}\(u\)\+C^\{2\}\\Gamma\+\\hat\{Q\}\_\{\*,m\}^\{T\}\+2C\\sqrt\{\\Gamma\\hat\{V\}\_\{T\}^\{m\}\},\(145\)whereQ^∗,mT:=maxt≤T⁡Q^tm\\hat\{Q\}\_\{\*,m\}^\{T\}:=\\max\_\{t\\leq T\}\\hat\{Q\}\_\{t\}^\{m\}\. The point is that changing the predictormtm\_\{t\}affects only the exact intrinsic timeV^Tm\\hat\{V\}\_\{T\}^\{m\}and the jump termQ^∗,mT\\hat\{Q\}\_\{\*,m\}^\{T\}; the variational reduction is unchanged\.

These examples already show the point\. IPS, explicit exploration, implicit exploration, and optimistic offsets all share the same variational backbone\. What changes is only the conditional meanc¯t\\bar\{c\}\_\{t\}, the resulting intrinsic timeV^T\\hat\{V\}\_\{T\}, the jump termQ^∗T\\hat\{Q\}\_\{\*\}^\{T\}, and, for biased estimators such as EXP3\-IX, one explicit predictable bias term\.

### 6\.3Online feedback graphs and Best\-of\-Both\-Worlds

LetGt=\(V,Et\)G\_\{t\}=\(V,E\_\{t\}\)be a directed feedback graph on the action setV=\[K\]V=\[K\], revealed before acting, and assume self\-loops\(i,i\)∈Et\(i,i\)\\in E\_\{t\}\. On roundttthe learner samplesAt∼μt∈Δ​\(\[K\]\)A\_\{t\}\\sim\\mu\_\{t\}\\in\\Delta\(\[K\]\), whereμt\\mu\_\{t\}isℱt−1\\mathcal\{F\}\_\{t\-1\}\-measurable and may include explicit exploration, so in generalμt≠pt\\mu\_\{t\}\\neq p\_\{t\}\. The loss of armaais observed wheneverAt∈Nin,t​\(a\):=\{j:\(j,a\)∈Et\}A\_\{t\}\\in N\_\{\\mathrm\{in\},t\}\(a\):=\\\{j:\(j,a\)\\in E\_\{t\}\\\}\. The conditional probability of observing armaais therefore

ot​\(a\):=ℙ​\(At∈Nin,t​\(a\)∣ℱt−1\)=∑j∈Nin,t​\(a\)μt​\(j\)o\_\{t\}\(a\):=\\mathbb\{P\}\\left\(A\_\{t\}\\in N\_\{\\mathrm\{in\},t\}\(a\)\\mid\\mathcal\{F\}\_\{t\-1\}\\right\)=\\sum\_\{j\\in N\_\{\\mathrm\{in\},t\}\(a\)\}\\mu\_\{t\}\(j\)For composite lossesct​\(a\)=ℓt​\(a\)\+ut​\(a\)c\_\{t\}\(a\)=\\ell\_\{t\}\(a\)\+u\_\{t\}\(a\), define the graph\-aware implicit\-exploration estimate

c^tG​\-​I​X​\(a\):=ut​\(a\)\+ℓt​\(a\)​𝟏​\{At∈Nin,t​\(a\)\}ot​\(a\)\+γt,γt≥0\\hat\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\(a\):=u\_\{t\}\(a\)\+\\frac\{\\ell\_\{t\}\(a\)\\mathbf\{1\}\\\{A\_\{t\}\\in N\_\{\\mathrm\{in\},t\}\(a\)\\\}\}\{o\_\{t\}\(a\)\+\\gamma\_\{t\}\},\\qquad\\gamma\_\{t\}\\geq 0Its conditional mean is

c¯tG​\-​I​X​\(a\):=𝔼​\[c^tG​\-​I​X​\(a\)∣ℱt−1\]=ut​\(a\)\+ℓt​\(a\)​ot​\(a\)ot​\(a\)\+γt=ct​\(a\)−dtG​\-​I​X​\(a\)\\bar\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\(a\):=\\mathbb\{E\}\[\\hat\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\(a\)\\mid\\mathcal\{F\}\_\{t\-1\}\]=u\_\{t\}\(a\)\+\\ell\_\{t\}\(a\)\\frac\{o\_\{t\}\(a\)\}\{o\_\{t\}\(a\)\+\\gamma\_\{t\}\}=c\_\{t\}\(a\)\-d\_\{t\}^\{G\\text\{\-\}IX\}\(a\)wheredtG​\-​I​X​\(a\):=γt​ℓt​\(a\)/\(ot​\(a\)\+γt\)d\_\{t\}^\{G\\text\{\-\}IX\}\(a\):=\\gamma\_\{t\}\\ell\_\{t\}\(a\)/\(o\_\{t\}\(a\)\+\\gamma\_\{t\}\)is a predictable downward bias\.

###### Proposition 6\.10\(Exact feedback\-graph identity\)\.

Feed the estimatesc^tG​\-​I​X\\hat\{c\}\_\{t\}^\{G\\text\{\-\}IX\}into the predictable\-rate update \([134](https://arxiv.org/html/2607.08789#S6.E134)\) to produce weightsptp\_\{t\}, and let\(D^TG​\-​I​X,B^TG​\-​I​X​\(u\),Q^tG​\-​I​X,V^TG​\-​I​X\)\(\\hat\{D\}\_\{T\}^\{G\\text\{\-\}IX\},\\hat\{B\}\_\{T\}^\{G\\text\{\-\}IX\}\(u\),\\hat\{Q\}\_\{t\}^\{G\\text\{\-\}IX\},\\hat\{V\}\_\{T\}^\{G\\text\{\-\}IX\}\)denote the corresponding estimated\-loss objects of Theorem[6\.4](https://arxiv.org/html/2607.08789#S6.Thmtheorem4)\. Then, pathwise,

R~Tc​\(u\)=MTplay−MTG​\-​I​X​\(u\)\+ΞT\+BiasTG​\-​I​X​\(u\)\+D^TG​\-​I​X\+B^TG​\-​I​X​\(u\)\+∑t=1Tηt​Q^tG​\-​I​X\\widetilde\{R\}\_\{T\}^\{c\}\(u\)=M\_\{T\}^\{\\mathrm\{play\}\}\-M\_\{T\}^\{G\\text\{\-\}IX\}\(u\)\+\\Xi\_\{T\}\+\\mathrm\{Bias\}\_\{T\}^\{G\\text\{\-\}IX\}\(u\)\+\\hat\{D\}\_\{T\}^\{G\\text\{\-\}IX\}\+\\hat\{B\}\_\{T\}^\{G\\text\{\-\}IX\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}^\{G\\text\{\-\}IX\}\(146\)where

MTG​\-​I​X​\(u\):=∑t=1T⟨pt−u,c^tG​\-​I​X−c¯tG​\-​I​X⟩,BiasTG​\-​I​X​\(u\):=∑t=1T⟨pt−u,dtG​\-​I​X⟩M\_\{T\}^\{G\\text\{\-\}IX\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\-\\bar\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\\rangle,\\qquad\\mathrm\{Bias\}\_\{T\}^\{G\\text\{\-\}IX\}\(u\):=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,d\_\{t\}^\{G\\text\{\-\}IX\}\\rangleUnder the second\-order schedule built fromV^t−1G​\-​I​X\\hat\{V\}\_\{t\-1\}^\{G\\text\{\-\}IX\}, the same two\-sided intrinsic\-time envelope as in Theorem[6\.6](https://arxiv.org/html/2607.08789#S6.Thmtheorem6)applies to the final term of \([146](https://arxiv.org/html/2607.08789#S6.E146)\)\.

##### Feedback\-graph topology and observability ratios\.

The graph affects the bound only through observability ratios\. BecausedtG​\-​I​X≥0d\_\{t\}^\{G\\text\{\-\}IX\}\\geq 0coordinatewise andℓt​\(a\)≤1\\ell\_\{t\}\(a\)\\leq 1,

∑t=1T⟨pt,dtG​\-​I​X⟩≤∑t=1Tγt​∑a=1Kpt​\(a\)ot​\(a\)\+γt≤∑t=1Tγt​∑a=1Kpt​\(a\)ot​\(a\)\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},d\_\{t\}^\{G\\text\{\-\}IX\}\\rangle\\leq\\sum\_\{t=1\}^\{T\}\\gamma\_\{t\}\\sum\_\{a=1\}^\{K\}\\frac\{p\_\{t\}\(a\)\}\{o\_\{t\}\(a\)\+\\gamma\_\{t\}\}\\leq\\sum\_\{t=1\}^\{T\}\\gamma\_\{t\}\\sum\_\{a=1\}^\{K\}\\frac\{p\_\{t\}\(a\)\}\{o\_\{t\}\(a\)\}The same observation ratio controls any quadratic surrogate for intrinsic time, because conditionally

𝔼​\[\(c^tG​\-​I​X​\(a\)−ut​\(a\)\)2∣ℱt−1\]=ℓt​\(a\)2​ot​\(a\)\(ot​\(a\)\+γt\)2≤1ot​\(a\)\\mathbb\{E\}\\\!\\left\[\\left\(\\hat\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\(a\)\-u\_\{t\}\(a\)\\right\)^\{2\}\\mid\\mathcal\{F\}\_\{t\-1\}\\right\]=\\frac\{\\ell\_\{t\}\(a\)^\{2\}o\_\{t\}\(a\)\}\{\(o\_\{t\}\(a\)\+\\gamma\_\{t\}\)^\{2\}\}\\leq\\frac\{1\}\{o\_\{t\}\(a\)\}So both the explicit bias and the natural second\-order scale are governed by∑apt​\(a\)/ot​\(a\)\\sum\_\{a\}p\_\{t\}\(a\)/o\_\{t\}\(a\)\. For undirected graphs withμt=pt\\mu\_\{t\}=p\_\{t\},\[[2](https://arxiv.org/html/2607.08789#bib.bib2)\]bounds this ratio by the independence numberα​\(Gt\)\\alpha\(G\_\{t\}\), so the graph complexity appears directly in the exact decomposition\.

##### Isolating Best\-of\-Both\-Worlds mechanics\.

Recent feedback\-graph best\-of\-both\-worlds algorithms combine explicit exploration with fast stochastic rates while retaining adversarial guarantees\[[62](https://arxiv.org/html/2607.08789#bib.bib62)\]\. The exact decomposition clarifies the mechanism term by term: exploration costsΞT\\Xi\_\{T\}, graph\-aware bias costsBiasTG​\-​I​X​\(u\)\\mathrm\{Bias\}\_\{T\}^\{G\\text\{\-\}IX\}\(u\), and once stochastic gaps make the relevant observation ratios favorable, the realized intrinsic timeV^TG​\-​I​X\\hat\{V\}\_\{T\}^\{G\\text\{\-\}IX\}slows or stops\. In adversarial regimes it grows only at the graph\-topological rate\.

### 6\.4Robust score transforms and unbounded observations

A useful feature of the exact bandit calculus is that boundedness is not needed for the identity itself\. Boundedness enters only when one later upper boundsQ^t\\hat\{Q\}\_\{t\}or concentrates martingales\. As long as the current one\-step log\-normalizer is finite at the chosen temperature, one may plug in robust score transforms directly\.

Letgtg\_\{t\}denote any armwise or policywise target vector, letztz\_\{t\}be any raw score vector built from the bandit observations, and letψCat\\psi\_\{\\mathrm\{Cat\}\}be the bounded\-influence score transform from\[[8](https://arxiv.org/html/2607.08789#bib.bib8)\]

ψCat​\(x\):=\{log⁡\(1\+x\+12​x2\),x≥0,−log⁡\(1−x\+12​x2\),x<0\.\\psi\_\{\\mathrm\{Cat\}\}\(x\):=\\begin\{cases\}\\log\(1\+x\+\\tfrac\{1\}\{2\}x^\{2\}\),&x\\geq 0,\\\\\[1\.72218pt\] \-\\log\(1\-x\+\\tfrac\{1\}\{2\}x^\{2\}\),&x<0\.\\end\{cases\}For a predictable scaleαt\>0\\alpha\_\{t\}\>0, define the robustified score coordinatewise byz^tCat:=αt−1​ψCat​\(αt​zt\)\\hat\{z\}\_\{t\}^\{\\mathrm\{Cat\}\}:=\\alpha\_\{t\}^\{\-1\}\\psi\_\{\\mathrm\{Cat\}\}\(\\alpha\_\{t\}z\_\{t\}\)\.

###### Proposition 6\.11\(Bounded\-influence robust scores fit the exact chain\)\.

Every exact identity in this section applies verbatim after replacing a raw estimate by the transformed scorez^tCat\\hat\{z\}\_\{t\}^\{\\mathrm\{Cat\}\}\. In particular, letℋt\\mathcal\{H\}\_\{t\}denote the appropriate pre\-decisionσ\\sigma\-field \(namelyℱt−1\\mathcal\{F\}\_\{t\-1\}in the armwise case and𝒢t\\mathcal\{G\}\_\{t\}in the contextual case\)\. Ifz¯tCat:=𝔼​\[z^tCat∣ℋt\]=gt−bt\\bar\{z\}\_\{t\}^\{\\mathrm\{Cat\}\}:=\\mathbb\{E\}\[\\hat\{z\}\_\{t\}^\{\\mathrm\{Cat\}\}\\mid\\mathcal\{H\}\_\{t\}\]=g\_\{t\}\-b\_\{t\}for a predictable bias vectorbtb\_\{t\}, then the corresponding pathwise regret decomposition is the same as before with one explicit predictable correction term∑t⟨pt−u,bt⟩\\sum\_\{t\}\\langle p\_\{t\}\-u,b\_\{t\}\\rangle, and the intrinsic time is the exact cumulant sum computed from the robustified scores themselves\.

This is why bounded\-influence heavy\-tailed bandit methods fit naturally here\[[8](https://arxiv.org/html/2607.08789#bib.bib8)\]: robustification changes the chosen score vector and perhaps adds predictable bias, but it does not change the underlying Bayes\-rule accounting\.

### 6\.5A proper contextual\-bandit wrapper

The duplication construction of\[[51](https://arxiv.org/html/2607.08789#bib.bib51),[30](https://arxiv.org/html/2607.08789#bib.bib30)\]turns a full\-information experts guarantee into a proper contextual\-bandit guarantee\. Our exact PAC\-Bayes theorem plugs into that wrapper without modification\.

###### Theorem 6\.12\(Oracle proper contextual\-bandit wrapper from exact PAC\-Bayes regret\)\.

Fix a finite action set\[A\]\[A\], a finite policy classΠ2=\{π0,π1,…,πk\}\\Pi\_\{2\}=\\\{\\pi\_\{0\},\\pi\_\{1\},\\dots,\\pi\_\{k\}\\\}, and the duplicated multiset of expertsℰ:=\{π0,…,π0,π1,…,πk\}\\mathcal\{E\}:=\\\{\\pi\_\{0\},\\dots,\\pi\_\{0\},\\pi\_\{1\},\\dots,\\pi\_\{k\}\\\}, withkkcopies ofπ0\\pi\_\{0\}\. Let an experts algorithm onℰ\\mathcal\{E\}output weightsptp\_\{t\}, and define the proper action distribution

qt​\(a∣xt\)\\displaystyle q\_\{t\}\(a\\mid x\_\{t\}\):=γA\+\(1−γ\)​∑π∈ℰ:π​\(xt\)=apt​\(π\),\\displaystyle=\\frac\{\\gamma\}\{A\}\+\(1\-\\gamma\)\\sum\_\{\\pi\\in\\mathcal\{E\}:\\,\\pi\(x\_\{t\}\)=a\}p\_\{t\}\(\\pi\),At\\displaystyle A\_\{t\}∼qt\(⋅∣xt\)\.\\displaystyle\\sim q\_\{t\}\(\\cdot\\mid x\_\{t\}\)\.With exploration parameterγ∈\(0,1\)\\gamma\\in\(0,1\), feed the experts algorithm the scaled IPS losses

ℓ~t​\(π\):=γA​𝟏​\{π​\(xt\)=At\}​ℓt​\(At\)qt​\(At∣xt\)∈\[0,1\]\\tilde\{\\ell\}\_\{t\}\(\\pi\):=\\frac\{\\gamma\}\{A\}\\frac\{\\mathbf\{1\}\\\{\\pi\(x\_\{t\}\)=A\_\{t\}\\\}\\ell\_\{t\}\(A\_\{t\}\)\}\{q\_\{t\}\(A\_\{t\}\\mid x\_\{t\}\)\}\\in\[0,1\]For a policy classΠ\\Pi, let

Reg⁡\(T,Π\):=𝔼​\[∑t=1Tℓt​\(At\)\]−minπ∈Π​∑t=1Tℓt​\(π​\(xt\)\)\\operatorname\{Reg\}\(T,\\Pi\):=\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}\(A\_\{t\}\)\\right\]\-\\min\_\{\\pi\\in\\Pi\}\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}\(\\pi\(x\_\{t\}\)\)Then

Reg⁡\(T,\{π0\}\)≤γ​T\+A​\(1−γ\)γ​𝔼​\[∑t=1T⟨pt−u\(0\),ℓ~t⟩\]\\operatorname\{Reg\}\(T,\\\{\\pi\_\{0\}\\\}\)\\leq\\gamma T\+\\frac\{A\(1\-\\gamma\)\}\{\\gamma\}\\,\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u^\{\(0\)\},\\tilde\{\\ell\}\_\{t\}\\rangle\\right\]\(147\)whereu\(0\)u^\{\(0\)\}is the uniform distribution over thekkcopies ofπ0\\pi\_\{0\}, and

Reg⁡\(T,Π2\)≤γ​T\+A​\(1−γ\)γ​𝔼​\[∑t=1T⟨pt−u⋆,ℓ~t⟩\]\\operatorname\{Reg\}\(T,\\Pi\_\{2\}\)\\leq\\gamma T\+\\frac\{A\(1\-\\gamma\)\}\{\\gamma\}\\,\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u^\{\\star\},\\tilde\{\\ell\}\_\{t\}\\rangle\\right\]\(148\)whereu⋆u^\{\\star\}is any point mass on a best policy inΠ2\\Pi\_\{2\}\. Consequently, if the experts algorithm onℰ\\mathcal\{E\}is the second\-order algorithm of \([25](https://arxiv.org/html/2607.08789#S3.E25)\) applied to the lossesℓ~t\\tilde\{\\ell\}\_\{t\}, then withQ~∗T:=maxt≤T⁡Q~t\\tilde\{Q\}\_\{\*\}^\{T\}:=\\max\_\{t\\leq T\}\\tilde\{Q\}\_\{t\}andV~T:=∑t=1TQ~t\\tilde\{V\}\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\tilde\{Q\}\_\{t\},

Reg⁡\(T,\{π0\}\)\\displaystyle\\operatorname\{Reg\}\(T,\\\{\\pi\_\{0\}\\\}\)≤γ​T\+A​\(1−γ\)γ​𝔼​\[\(1\+C2\)​log⁡2\+Q~∗T\+\(2​C\+C−1\)​V~T​log⁡2\],\\displaystyle\\leq\\gamma T\+\\frac\{A\(1\-\\gamma\)\}\{\\gamma\}\\,\\mathbb\{E\}\\\!\\left\[\(1\+C^\{2\}\)\\log 2\+\\tilde\{Q\}\_\{\*\}^\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\tilde\{V\}\_\{T\}\\log 2\}\\right\],\(149\)Reg⁡\(T,Π2\)\\displaystyle\\operatorname\{Reg\}\(T,\\Pi\_\{2\}\)≤γ​T\+A​\(1−γ\)γ​𝔼​\[\(1\+C2\)​log⁡\|ℰ\|\+Q~∗T\+\(2​C\+C−1\)​V~T​log⁡\|ℰ\|\]\.\\displaystyle\\leq\\gamma T\+\\frac\{A\(1\-\\gamma\)\}\{\\gamma\}\\,\\mathbb\{E\}\\\!\\left\[\(1\+C^\{2\}\)\\log\|\\mathcal\{E\}\|\+\\tilde\{Q\}\_\{\*\}^\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\tilde\{V\}\_\{T\}\\log\|\\mathcal\{E\}\|\}\\right\]\.\(150\)

##### Remark\.

The duplication construction assumes access to the distinguished policyπ0\\pi\_\{0\}when forming the multisetℰ\\mathcal\{E\}\. The theorem should therefore be read as an oracle reduction that isolates the information\-theoretic price of proper contextual\-bandit prediction, not as a deployable algorithm\.

The duplication effect is visible in \([149](https://arxiv.org/html/2607.08789#S6.E149)\): spreading the comparator distribution uniformly over the copies ofπ0\\pi\_\{0\}lowers comparator complexity fromlog⁡\|ℰ\|\\log\|\\mathcal\{E\}\|tolog⁡2\\log 2\. In that oracle sense, the wrapper inherits the same intrinsic\-time structure, with only the explicit exploration costγ​T\\gamma Tand the usualA/γA/\\gammaIPS rescaling added\.

### 6\.6A speculative link: testing\-exponent\-guided identification scores

This subsection is only motivational\. Suppose there areWWlatent environment hypotheses\. Under hypothesiswwand armaa, an observation has lawPwaP\_\{w\}^\{a\}\. If one repeatedly samples a fixed armaaand performs maximum a posteriori decoding under positive priors, then classical i\.i\.d\. hypothesis\-testing theory identifies the minimum pairwise Chernoff exponent as the governing error rate\. This suggests the scoreΓ​\(a\):=mini≠j⁡maxs∈\[0,1\]⁡𝒞s​\(Pia,Pja\)\\Gamma\(a\):=\\min\_\{i\\neq j\}\\max\_\{s\\in\[0,1\]\}\\mathcal\{C\}\_\{s\}\(P\_\{i\}^\{a\},P\_\{j\}^\{a\}\), where

𝒞s​\(P,Q\):=−log​∫\(d​Pd​μ\)s​\(d​Qd​μ\)1−s​𝑑μ\\mathcal\{C\}\_\{s\}\(P,Q\):=\-\\log\\int\\left\(\\frac\{dP\}\{d\\mu\}\\right\)^\{s\}\\left\(\\frac\{dQ\}\{d\\mu\}\\right\)^\{1\-s\}d\\muis the usual coincidence/testing divergence, independent of the dominating measureμ\\mu\[[14](https://arxiv.org/html/2607.08789#bib.bib14),[55](https://arxiv.org/html/2607.08789#bib.bib55)\]\. Thus the same geometric mean that underlies our mixed\-prior formulas also underlies classical testing exponents\.

What we are claiming is modest:Γ​\(a\)\\Gamma\(a\)is a natural fixed\-arm informativeness score\. What we are not claiming is a new sequential bandit theorem\. Turning these fixed\-arm exponents into an online exploration policy remains open\.

## 7Empirical diagnostics and numerical results

Because Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)is an exact identity rather than an upper bound, the natural empirical object is the cumulative split of the regret into its three accounting terms, not the terminal regret number alone\. The same low terminal regret can be reached for very different reasons: a path may have been easy \(small intrinsic time\), the comparator may have been simple relative to the prior \(small comparator\-information term\), or the schedule itself may have been forgiving \(favorable temperature drift\)\. The exact decomposition exposes which of these mechanisms each algorithm was actually using on a given run\.

### 7\.1Reading conventions and the prefix decomposition

Fix a horizonTTand a comparatorρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\)\. For the prior\-retempered update of §[2\.2](https://arxiv.org/html/2607.08789#S2.SS2), define for every prefixt≤Tt\\leq T

Pt​\(c\):=∑s=1tηs​Qs​\(c\)Dt:=∑s=1t−1\(As​\(ηs\)−As​\(ηs\+1\)\)Bt​\(ρ\):=KL​\(ρ∥π\)−KL​\(ρ∥qt,ηt\)ηt\.P\_\{t\}\(c\):=\\sum\_\{s=1\}^\{t\}\\eta\_\{s\}Q\_\{s\}\(c\)\\quad D\_\{t\}:=\\sum\_\{s=1\}^\{t\-1\}\\bigl\(A\_\{s\}\(\\eta\_\{s\}\)\-A\_\{s\}\(\\eta\_\{s\+1\}\)\\bigr\)\\quad B\_\{t\}\(\\rho\):=\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{t,\\eta\_\{t\}\}\)\}\{\\eta\_\{t\}\}\.\(151\)The three terms are the prefix*intrinsic\-time payment*,*temperature drift*, and*terminal comparator information*\. Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)states that they sum exactly to the composite\-loss prefix regret:

Rtc​\(ρ\)=Pt​\(c\)\+Dt\+Bt​\(ρ\)\.R\_\{t\}^\{c\}\(\\rho\)\\;=\\;P\_\{t\}\(c\)\\;\+\\;D\_\{t\}\\;\+\\;B\_\{t\}\(\\rho\)\.\(152\)With side informationutu\_\{t\}, the original\-loss regret carries an additional predictable\-mismatch termMt​\(ρ\)=∑s≤t\(⟨ρ,us⟩−⟨ps,us⟩\)M\_\{t\}\(\\rho\)=\\sum\_\{s\\leq t\}\(\\langle\\rho,u\_\{s\}\\rangle\-\\langle p\_\{s\},u\_\{s\}\\rangle\)\. The normalized shares

St​\(ρ\):=Pt​\(c\)\+\|Dt\|\+Bt​\(ρ\),ωtpay:=Pt​\(c\)St​\(ρ\),ωtdrift:=\|Dt\|St​\(ρ\),ωtinfo:=Bt​\(ρ\)St​\(ρ\)S\_\{t\}\(\\rho\):=P\_\{t\}\(c\)\+\|D\_\{t\}\|\+B\_\{t\}\(\\rho\),\\quad\\omega\_\{t\}^\{\\mathrm\{pay\}\}\\\!:=\\\!\\frac\{P\_\{t\}\(c\)\}\{S\_\{t\}\(\\rho\)\},\\;\\omega\_\{t\}^\{\\mathrm\{drift\}\}\\\!:=\\\!\\frac\{\|D\_\{t\}\|\}\{S\_\{t\}\(\\rho\)\},\\;\\omega\_\{t\}^\{\\mathrm\{info\}\}\\\!:=\\\!\\frac\{B\_\{t\}\(\\rho\)\}\{S\_\{t\}\(\\rho\)\}\(153\)lie in\[0,1\]\[0,1\]and sum to11at everytt\. We plot them stacked on a primary axis and overlay the signed prefix regretRtc​\(ρ\)R\_\{t\}^\{c\}\(\\rho\)on a secondary axis to obtain the*regret\-decomposition plot*used throughout\. We report seed\-aggregated quantities with BCa95%95\\%bootstrap confidence intervals \(1000 resamples\); identity verifications report the maximum absolute residual over the grid\. Figures use the Okabe–Ito categorical palette and the cividis sequential colormap; we do not use jet/rainbow palettes anywhere\. The local update of §[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)satisfies a parallel exact identity \(Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)\) withDtD\_\{t\}replaced by a transport driftDtloc=∑s≥2KL​\(ρ∥ps\)​\(1/ηs−1/ηs−1\)D\_\{t\}^\{\\mathrm\{loc\}\}=\\sum\_\{s\\geq 2\}\\mathrm\{KL\}\(\\rho\\\|p\_\{s\}\)\\,\(1/\\eta\_\{s\}\-1/\\eta\_\{s\-1\}\)and the same numerical residual envelope; both forms underlie the figures below\.

### 7\.2Decomposition signatures across rates and comparators

Figure[1](https://arxiv.org/html/2607.08789#S7.F1)shows the three\-share decomposition simultaneously across learning\-rate schedules and comparators on a regime\-switching mixed\-character sequence\. The dominant share switches at every regime boundary, but for any fixed sequence the family\-determined signature is reproduced across\(η,ρ\)\(\\eta,\\rho\)pairs in each row: the decomposition is a property of the path, not of any one schedule choice\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig_decomp_combined.png)Figure 1:Regret decomposition on a regime\-switching mixed\-character sequence \(K=8K=8,T=2000T=2000; four blocks of length500500alternating i\.i\.d\. stochastic / cycling\-adversarial / i\.i\.d\. stochastic / cycling\-adversarial\)\. Three rows correspond to the cumulative sharesωtinfo\\omega^\{\\mathrm\{info\}\}\_\{t\},ωtpay\\omega^\{\\mathrm\{pay\}\}\_\{t\},ωtdrift\\omega^\{\\mathrm\{drift\}\}\_\{t\}on a shared time axis\. Within each row, color encodes the learning\-rate schedule \(Ret\-Sqrt, fixedη=0\.1\\eta=0\.1, fixedη=1\.0\\eta=1\.0\) and line style encodes the comparatorρ\\rho\(best single expert, softmax\-quantile, uniform mixture\)\. Vertical dotted lines mark regime boundaries\. The pathwise prefix identity \([152](https://arxiv.org/html/2607.08789#S7.E152)\) holds at machine precision on every\(η,ρ\)\(\\eta,\\rho\)trace; BCa95%95\\%bootstrap bands across1212seeds shown for theRet\-Sqrtschedule\.
### 7\.3The same regret can decompose very differently

A direct demonstration of why the decomposition matters: Figure[2](https://arxiv.org/html/2607.08789#S7.F2)pairs three schedules whose cumulative regret curves are visually close throughout \(top panel\) with their per\-schedule\(ωpay,ωdrift,ωinfo\)\(\\omega^\{\\mathrm\{pay\}\},\\omega^\{\\mathrm\{drift\}\},\\omega^\{\\mathrm\{info\}\}\)stacks \(bottom row\)\. The three stacks differ substantially even though the regret traces coincide\. The aggregate regret number alone hides this: only the decomposition reveals what each algorithm was actually doing on the same path\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig_same_regret_different_decomp.png)Figure 2:Same regret, different decomposition\.Top:cumulative composite\-loss prefix regretRtc​\(ρ\)R^\{c\}\_\{t\}\(\\rho\)for three schedules—Ret\-Sqrt, the pressure\-target line search, and fixedη=0\.43\\eta=0\.43—on the martingale family with optimistic side information \(K=8K=8,T=2000T=2000,1212seeds\)\. All three reach terminal regret within1\.9%1\.9\\%of each other \(RTc=\+7\.84,\+7\.99,\+7\.80R\_\{T\}^\{c\}=\+7\.84,\+7\.99,\+7\.80\)\.Bottom:stacked\-share decomposition\(ωpay,ωdrift,ωinfo\)\(\\omega^\{\\mathrm\{pay\}\},\\omega^\{\\mathrm\{drift\}\},\\omega^\{\\mathrm\{info\}\}\)for each schedule on the same path; the secondary axis carries the signed regret as a thin black line\. The three stacks differ substantially:Ret\-Sqrtand fixed\-η\\etaare pay\-dominated \(ωTpay≈0\.85\\omega^\{\\mathrm\{pay\}\}\_\{T\}\\approx 0\.85,ωTdrift≈0\\omega^\{\\mathrm\{drift\}\}\_\{T\}\\approx 0,ωTinfo≈0\.13\\omega^\{\\mathrm\{info\}\}\_\{T\}\\approx 0\.13\), whereas the pressure\-target schedule reaches the same regret with substantial drift \(ωTpay≈0\.50\\omega^\{\\mathrm\{pay\}\}\_\{T\}\\approx 0\.50,ωTdrift≈0\.30\\omega^\{\\mathrm\{drift\}\}\_\{T\}\\approx 0\.30,ωTinfo≈0\.21\\omega^\{\\mathrm\{info\}\}\_\{T\}\\approx 0\.21\) because its temperature varies locally with the data\.
### 7\.4Eight properties of the framework

The synthetic experiments in Appendix[C](https://arxiv.org/html/2607.08789#A3)confirm eight properties of the framework \(the benchmark\-trace cells use synthetic substitutes, not the original traces; see §[C\.6\.1](https://arxiv.org/html/2607.08789#A3.SS6.SSS1)\)\. \(i\) The exact prefix identities hold to machine precision on every run, both in the prior\-retempered form \(Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)\) and the local\-update form \(Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)\), for every comparatorρα\\rho\_\{\\alpha\}in the one\-parameter family of \([166](https://arxiv.org/html/2607.08789#A3.E166)\) and not only atδi∗\\delta\_\{i^\{\*\}\}; the same residual scaling holds across the fullK∈\{8,16,32,64,128\}K\\in\\\{8,16,32,64,128\\\},T∈\{103,104,105\}T\\in\\\{10^\{3\},10^\{4\},10^\{5\}\\\}grid withlog10⁡max⁡\|ΔT\|=−9\.95\\log\_\{10\}\\max\|\\Delta\_\{T\}\|=\-9\.95, and the single\-step Bellman equalizer identity holds pointwise with a per\-step residual at most≈2×10−12\\approx 2\\times 10^\{\-12\}\(log10⁡max=−11\.67\\log\_\{10\}\\max=\-11\.67\) on the same grid\. \(ii\) The variance proxyWt​\(c\)=∑s≤tVarps​\(cs\)/2W\_\{t\}\(c\)=\\sum\_\{s\\leq t\}\\mathrm\{Var\}\_\{p\_\{s\}\}\(c\_\{s\}\)/2that drives the variance\-schedule variants is the leading\-order Taylor relaxation ofVt​\(c\)V\_\{t\}\(c\), with linear residual scaling inη\\etaand slopes of0\.99±0\.020\.99\\pm 0\.02on every family\. \(iii\) The lower side of the second\-order envelope is empirically tight on cycling\-adversarial paths: atT=8000T=8000the realizedPT/Γ​VT​\(c\)P\_\{T\}/\\sqrt\{\\Gamma V\_\{T\}\(c\)\}averages≈1\.33\\approx 1\.33against the asymptotic value2​C≈1\.412C\\approx 1\.41, with the gap closing asTTgrows \(the ratio rises1\.19→1\.26→1\.33→1\.361\.19\\to 1\.26\\to 1\.33\\to 1\.36acrossT∈\{2000,4000,8000,16000\}T\\in\\\{2000,4000,8000,16000\\\}\)\. \(iv\) The upper side is asymptotically tight on a single\-spike construction:PT/upperP\_\{T\}/\\textup\{upper\}rises from0\.050\.05atQ1≈0\.03Q\_\{1\}\\approx 0\.03to0\.950\.95atQ1≈499Q\_\{1\}\\approx 499, matching the closed\-form prediction; the maximum\-jump correctionQ∗T​\(c\)Q\_\{\*\}^\{T\}\(c\)in the upper bound is necessary, not an artefact of the proof\. \(v\) The matrix\-game side\-information recipe correctly attributes difficulty to forecast accuracy:VT​\(c\)V\_\{T\}\(c\)falls smoothly to zero as the forecast becomes perfect, while the original\-loss regret gap is correctly assigned to the predictable\-mismatch termMTM\_\{T\}\. \(vi\) The two schedules are competitive with state\-of\-the\-art adaptive online learning algorithms across predictable families, dominate local\-gap\-style baselines on cycling\-adversarial paths, and beat the best fixed expert on planted\-change\-point sequences without any explicit tracking machinery\. \(vii\) The fast\-rate behavior predicted by Theorem[4\.9](https://arxiv.org/html/2607.08789#S4.Thmtheorem9)and Corollary[4\.11](https://arxiv.org/html/2607.08789#S4.Thmtheorem11)is visible at finiteTT: when the comparator\-centered low\-noise condition holds, regret is constant inTTacrossT∈\[500,16,000\]T\\in\[500,16\{,\}000\]\(empirical log–log slope0\.0000\.000\), while the same algorithm reverts to the worst\-caseT\\sqrt\{T\}rate \(empirical slope0\.490\.49\) on a misspecified no\-gap family\. \(viii\) The Hoeffding range boundVT​\(c\)≤T/8V\_\{T\}\(c\)\\leq T/8is two to four orders of magnitude looser than the realized exactVT​\(c\)V\_\{T\}\(c\)on predictable families \(V4000≈0\.2V\_\{4000\}\\approx 0\.2vs the bound500500, slack ratio≥99\.92%\\geq 99\.92\\%\) and remains96%96\\%loose even on cycling\-adversarial paths; the framework’s path\-dependent budget is what makes the full second\-order story possible\.

##### Where the details live\.

Appendix[C](https://arxiv.org/html/2607.08789#A3)contains the full empirical battery: the four sequence families and schedule panel; the side\-information recipe; per\-family decomposition shares; variance\-proxy verification; envelope tightness on cycling\-adversarial paths and a single\-spike upper\-side construction; Hoeffding\-slack histograms; comparator\-by\-comparator identity check on both updates; head\-to\-head regret comparison against AdaHedge\[[20](https://arxiv.org/html/2607.08789#bib.bib20)\], NormalHedge\[[13](https://arxiv.org/html/2607.08789#bib.bib13)\], AdaNormalHedge\[[50](https://arxiv.org/html/2607.08789#bib.bib50)\], FTRL with1/21/2\-Tsallis entropy\[[75](https://arxiv.org/html/2607.08789#bib.bib75)\], and Squint\[[45](https://arxiv.org/html/2607.08789#bib.bib45)\]; stochastic\-luckiness fast\-rate verification; sleeping\-experts and shifting\-comparator validation; identity\-at\-scale and Bellman\-equalizer numerics on the\(K,T\)\(K,T\)grid; the benchmark\-trace protocol \(run on synthetic substitutes\) for horse\-racing, intraday electricity, and NYSE portfolio sequences; bandit decomposition; pressure\-target boosting on UCI letter / adult / covtype; structured matrix\-game self\-play \(Shapley RPS, Colonel\-Blotto\); and Thompson\-sampling/PPR\-martingale verification\.

## 8Related work and discussion

TableLABEL:tab:tradeoffsis meant as a reading guide for the surrounding literature\. Many papers in online learning appear to study different tradeoffs—worst\-case versus stochastic behavior, first\-order versus second\-order bounds, point experts versus mixtures, fixed horizons versus adaptive schedules, full versus partial feedback—but they often differ mainly in which part of the same information balance is emphasized or relaxed\. We organize the discussion below by those themes\.

### 8\.1Difficulty measures, second\-order regret, and schedule design

The first group of rows in TableLABEL:tab:tradeoffsconcerns how one measures the difficulty of a realized sequence\. The original weighted\-majority and Hedge analyses yield the familiarO~​\(T​log⁡K\)\\widetilde\{O\}\(\\sqrt\{T\\log K\}\)scale\[[48](https://arxiv.org/html/2607.08789#bib.bib48),[27](https://arxiv.org/html/2607.08789#bib.bib27),[12](https://arxiv.org/html/2607.08789#bib.bib12)\]\. A long subsequent line of work replaced the horizonTTby more sensitive first\- or second\-order quantities, including quadratic excess losses, quantile\-sensitive second\-order terms, mixability\-gap schedules, and parameter\-free potentials\[[31](https://arxiv.org/html/2607.08789#bib.bib31),[20](https://arxiv.org/html/2607.08789#bib.bib20),[45](https://arxiv.org/html/2607.08789#bib.bib45),[50](https://arxiv.org/html/2607.08789#bib.bib50),[56](https://arxiv.org/html/2607.08789#bib.bib56)\]\. Freund’s 2016 scaling\-time problem sharpened this agenda by asking specifically for regret guarantees driven by the variance of the realized loss under the learner’s own current distribution\[[25](https://arxiv.org/html/2607.08789#bib.bib25)\]; the recent NormalHedge result of\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]shows that a closely related curvature process can support simultaneous quantile adaptation\.

A useful way to place these works side by side is to distinguish several common “difficulty” scales\. Letet​\(i\):=ℓt​\(i\)−ℓt​\(k\)e\_\{t\}\(i\):=\\ell\_\{t\}\(i\)\-\\ell\_\{t\}\(k\)denote excess loss relative to a fixed comparator expertkk, and letμt:=⟨pt,ℓt⟩\\mu\_\{t\}:=\\langle p\_\{t\},\\ell\_\{t\}\\rangle\. For losses in\[0,1\]\[0,1\], the standard quantities satisfy

∑t=1TVari∼pt​\(ℓt​\(i\)\)⏟variance across actions≤∑t=1T∑i=1Kpt​\(i\)​et​\(i\)2⏟quadratic excess loss≤∑t=1T∑i=1Kpt​\(i\)​\|et​\(i\)\|⏟first\-order magnitude\\underbrace\{\\sum\_\{t=1\}^\{T\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(\\ell\_\{t\}\(i\)\)\}\_\{\\textup\{variance across actions\}\}\\leq\\underbrace\{\\sum\_\{t=1\}^\{T\}\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e\_\{t\}\(i\)^\{2\}\}\_\{\\textup\{quadratic excess loss\}\}\\leq\\underbrace\{\\sum\_\{t=1\}^\{T\}\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)\|e\_\{t\}\(i\)\|\}\_\{\\textup\{first\-order magnitude\}\}\(154\)Different algorithms and analyses are tuned to different points on this spectrum\[[12](https://arxiv.org/html/2607.08789#bib.bib12),[31](https://arxiv.org/html/2607.08789#bib.bib31),[45](https://arxiv.org/html/2607.08789#bib.bib45),[50](https://arxiv.org/html/2607.08789#bib.bib50),[30](https://arxiv.org/html/2607.08789#bib.bib30)\]\. AdaHedge and SafeBayes are especially relevant for the present paper because they tune the learning rate using the cumulative mixability gap itself\[[20](https://arxiv.org/html/2607.08789#bib.bib20),[37](https://arxiv.org/html/2607.08789#bib.bib37)\]\.

Our relation to this literature is twofold\. First, the quantity tracked here,Qt​\(c\)Q\_\{t\}\(c\), is not introduced as a proxy by an inequality; it is the exact finite\-temperature payment in the one\-step identity, defined after the composite\-loss reduction has already absorbed side information\. Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)shows that it is an average of tilted variances, so the familiar quadratic terms arise only after an additional relaxation\. Second, the schedule question splits naturally into two geometries\. For the prior\-retempered update, intrinsic\-time schedules play the role of second\-order, horizon\-free clocks\. For the local update, pressure\-target line searches stabilize a one\-step normalization condition instead of a cumulative variance budget\. This is why the paper can treat the rows of TableLABEL:tab:tradeoffsconcerning hard versus easy sequences, first versus second order, variance definitions, and horizon dependence inside one exact framework rather than by switching between unrelated proofs\.

The same perspective clarifies the bounded\-versus\-unbounded row of the table\. The exact identities require only that the relevant one\-step log\-normalizer be finite at the chosen temperature\. Boundedness enters only when one wants a simpler online upper bound onQt​\(c\)Q\_\{t\}\(c\), for instance through a Taylor or range inequality\. This is also why the robust\-score discussions in Sections[6\.4](https://arxiv.org/html/2607.08789#S6.SS4)and[A](https://arxiv.org/html/2607.08789#A1)fit naturally beside recent work on heavy\-tailed or unbounded losses\[[54](https://arxiv.org/html/2607.08789#bib.bib54)\]: the primitive object is still the log\-normalizer, not a bounded\-range assumption\.

##### Safe Bayes is adaptive Bayes seen on the wrong clock\.

The Safe\-Bayesian programme \(\[[37](https://arxiv.org/html/2607.08789#bib.bib37)\]and the misspecification analysis of\[[35](https://arxiv.org/html/2607.08789#bib.bib35)\]\) tempers the likelihood by an exponentη∈\(0,1\]\\eta\\in\(0,1\]to repair posterior consistency under model misspecification, choosingη∗\\eta^\{\*\}to track the KL\-optimal concentration rate\. In the present framework the same exponentηt\\eta\_\{t\}enters as the per\-round inverse\-step\-size, so Safe Bayes can be read as Bayes computed on a clock that ticks slower than the wall clock\. The two halves of the discipline align cleanly: the cumulative intrinsic\-time clockτT:=∑tηt2​Qt\\tau\_\{T\}:=\\sum\_\{t\}\\eta\_\{t\}^\{2\}Q\_\{t\}of §[3](https://arxiv.org/html/2607.08789#S3)is the integrated Safe\-Bayesian clock atηt≡η∗\\eta\_\{t\}\\equiv\\eta^\{\*\}, and the misspecification\-driven prescriptionη∗<1\\eta^\{\*\}<1is the wall\-clock face of a positive intrinsic\-time growth rate\. Conversely, our identities recover the Safe\-Bayesian guarantee at the choiceηt=η∗\\eta\_\{t\}=\\eta^\{\*\}as a corollary of the one\-step exact decomposition, but generalize to time\-varying schedules and to settings \(online, adversarial, partial feedback\) outside the i\.i\.d\. misspecification frame Safe Bayes was originally designed for\. The structural distinction is that Safe Bayes asks “which constantη\\etamakes Bayes calibrated under misspecification?” while the present framework asks “what is the exact information cost of any chosenηt\\eta\_\{t\}, calibrated or not?”—and the former is recovered as the calibration\-respecting subset of the latter\.

### 8\.2Comparator classes, structured priors, and parameter\-freeness

Another cluster of rows in TableLABEL:tab:tradeoffsconcerns the comparator\. Classical regret compares to the best single expert, but many later results compare to quantiles, mixtures, specialists, sleeping experts, or switching sequences\. Parameter\-free hedging\[[13](https://arxiv.org/html/2607.08789#bib.bib13)\], Squint\[[45](https://arxiv.org/html/2607.08789#bib.bib45)\], AdaNormalHedge\[[50](https://arxiv.org/html/2607.08789#bib.bib50)\], and the new NormalHedge scaling\-time result\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]are central examples in the quantile or prior\-mass direction\. Specialist and sleeping\-expert methods allow abstention or confidence weighting\[[29](https://arxiv.org/html/2607.08789#bib.bib29)\], and the Bayesian reinterpretation of\[[46](https://arxiv.org/html/2607.08789#bib.bib46)\]shows how naturally such structure fits with posterior averaging\. Tracking or switching regret compares to a sequence of experts with limited changes, in the tradition of fixed share\[[40](https://arxiv.org/html/2607.08789#bib.bib40)\]\. Internal and swap regret, especially important in game\-theoretic applications, are classically obtained by reductions from external regret\[[7](https://arxiv.org/html/2607.08789#bib.bib7)\]\.

The present paper’s viewpoint is that these are all changes of comparator class before they are changes of proof technique\. Once the comparator is encoded as a prior or a structured prior space, the same information balance applies\. This is explicit in the exact shifting\-comparator theorem of Section[4\.3](https://arxiv.org/html/2607.08789#S4.SS3), which writes dynamic regret exactly before any switch\-count relaxation, and in the sleeping/quantile discussions of Sections[4\.3](https://arxiv.org/html/2607.08789#S4.SS3)and[4\.4](https://arxiv.org/html/2607.08789#S4.SS4)\. Conceptually, structured priors turn many “new” comparator notions into ordinary PAC\-Bayesian regret on a richer space\.

This also reframes the parameter\-free versus adaptive row of TableLABEL:tab:tradeoffs\. In the present notation, a budgetΓ\\Gamma, a temperature schedule, or a pressure targetata\_\{t\}is not a hidden nuisance parameter; it is an explicit constraint or objective relative to which information is being measured\. One may still learn such quantities online\. Our dyadic controller in Section[4\.4](https://arxiv.org/html/2607.08789#S4.SS4)is one example\. Single\-copy simultaneous methods such as AdaNormalHedge, coin betting, and the NormalHedge scaling\-time construction remain important comparison points\[[50](https://arxiv.org/html/2607.08789#bib.bib50),[56](https://arxiv.org/html/2607.08789#bib.bib56),[30](https://arxiv.org/html/2607.08789#bib.bib30)\]\. What is different here is not the desire to avoid manual tuning, but the insistence on keeping the underlying information accounting exact\.

### 8\.3Predictable structure, partial feedback, and online optimization

A separate row of TableLABEL:tab:tradeoffsconcerns predictable structure\. Optimistic and variation\-based algorithms exploit the fact that part of the next loss vector is foreseeable from the past\[[39](https://arxiv.org/html/2607.08789#bib.bib39),[15](https://arxiv.org/html/2607.08789#bib.bib15),[70](https://arxiv.org/html/2607.08789#bib.bib70)\]\. The composite\-loss reduction in Section[2\.1](https://arxiv.org/html/2607.08789#S2.SS1)is a direct way of expressing that idea: predictable information is inserted through the side termutu\_\{t\}, and the exact online payment is then computed only on the residual sequencect=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}\. In other words, the paper does not place optimism beside the main theorem as an extra trick; it makes optimism one choice of what counts as already\-explained information\.

The same philosophical move underlies much of online convex optimization and adaptive first\-order methods\. Exponential weights is simultaneously an entropic mirror\-descent step and an FTRL update, and the OCO literature made that equivalence central\[[3](https://arxiv.org/html/2607.08789#bib.bib3),[76](https://arxiv.org/html/2607.08789#bib.bib76),[22](https://arxiv.org/html/2607.08789#bib.bib22)\]\. Adaptive Euclidean methods choose step sizes from geometry revealed by the realized gradients\. Our prior\-retempered schedules are the entropic analogue: they choose update strength from an exact cumulative quantity revealed by the path itself\. This is also why the same algebra survives the passage to continuous\-action OCO in Section[5\.6](https://arxiv.org/html/2607.08789#S5.SS6)\.

The full\-versus\-partial\-feedback row is similar\. Bandit and feedback\-graph algorithms introduce estimated losses, implicit exploration, and predictable bias corrections\[[5](https://arxiv.org/html/2607.08789#bib.bib5),[53](https://arxiv.org/html/2607.08789#bib.bib53),[2](https://arxiv.org/html/2607.08789#bib.bib2),[62](https://arxiv.org/html/2607.08789#bib.bib62)\]\. From the present viewpoint, these are front\-end changes to the score sequence rather than changes to the back\-end information identity\. Once one writes down the estimated composite losses, the same one\-step balance applies, with explicit martingale and bias terms tracking what the estimator added\. This is the reason the partial\-information section can parallel the full\-information one so closely\.

### 8\.4Local normalization, pressure targets, and boosting

The local pressure\-target update belongs to a different related\-work cluster from the retempered second\-order schedules\. Its closest relatives are one\-step coefficient choices in boosting and other multiplicative\-normalization procedures\. Already in the original game\-theoretic and boosting formulations of multiplicative weights\[[26](https://arxiv.org/html/2607.08789#bib.bib26),[27](https://arxiv.org/html/2607.08789#bib.bib27)\], the current round is summarized by a one\-step normalizer, and choosing the coefficient amounts to deciding how aggressively to trade margin gain against normalization\. The textbook AdaBoost coefficient is exactly the minimizer of that one\-step normalizer in the binary setting\[[27](https://arxiv.org/html/2607.08789#bib.bib27),[66](https://arxiv.org/html/2607.08789#bib.bib66)\]\. The drifting\-games analysis of\[[70](https://arxiv.org/html/2607.08789#bib.bib70)\]makes this normalization viewpoint especially explicit\.

The local results in Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)isolate the exact cumulative consequence of that viewpoint\. The schedule∑ipt​\(i\)​e−ηt​\(ct​\(i\)−at\)=1\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}\(c\_\{t\}\(i\)\-a\_\{t\}\)\}=1fixes a target level for the current normalized loss, and the cumulative identity records the resulting class\-conditioned terminal mass exactly\. This is why the natural applications of the local update are different from those of the retempered update\. The retempered schedule is the right object when one wants anytime second\-order control from a cumulative clock\. The local schedule is the right object when one wants to calibrate the present step itself, as in boosting or other procedures driven by one\-step margin or loss targets\.

This distinction is important conceptually\. Without it, the literature can look as if “learning\-rate tuning” were one topic\. In fact, the rows of TableLABEL:tab:tradeoffssplit into two different questions: cumulative intrinsic\-time equalization for the retempered update, and one\-step pressure calibration for the local update\. Keeping these apart helps explain why the two adaptive updates in this paper share a one\-step information cost but lead to different terminal objects\.

### 8\.5Individual\-sequence prediction, PAC\-Bayes language, and concentration

The paper is also close in spirit to several individual\-sequence traditions\. Prequential statistics evaluates a predictor by its sequential predictive performance on the realized path rather than by its fit to a data\-generating model\[[19](https://arxiv.org/html/2607.08789#bib.bib19)\]\. Game\-theoretic probability and defensive forecasting make the same move in a more explicitly adversarial language\[[74](https://arxiv.org/html/2607.08789#bib.bib74),[67](https://arxiv.org/html/2607.08789#bib.bib67)\]\. Predictive complexity and the aggregating algorithm interpret regret through code length and description complexity\[[73](https://arxiv.org/html/2607.08789#bib.bib73),[43](https://arxiv.org/html/2607.08789#bib.bib43)\]\. Universal coding and MDL supply the corresponding information\-theoretic background\[[61](https://arxiv.org/html/2607.08789#bib.bib61),[68](https://arxiv.org/html/2607.08789#bib.bib68),[52](https://arxiv.org/html/2607.08789#bib.bib52),[36](https://arxiv.org/html/2607.08789#bib.bib36)\]\. The early online\-learning minimax literature\[[11](https://arxiv.org/html/2607.08789#bib.bib11),[38](https://arxiv.org/html/2607.08789#bib.bib38),[9](https://arxiv.org/html/2607.08789#bib.bib9),[74](https://arxiv.org/html/2607.08789#bib.bib74),[23](https://arxiv.org/html/2607.08789#bib.bib23)\]can be read in the same spirit\.

Our use of PAC\-Bayes language is therefore intentionally pathwise\. The quantityKL​\(ρ∥π\)\\mathrm\{KL\}\(\\rho\\\|\\pi\)is a comparator description cost, not an i\.i\.d\. generalization penalty\. This is also why the row of TableLABEL:tab:tradeoffsabout algorithmic upper versus lower envelopes matters so much\. At the identity level there is no slack: the algorithm’s internal payment is decomposed exactly, and any later looseness comes only from a deliberate relaxation of exact objects such asQt​\(c\)Q\_\{t\}\(c\)orDTD\_\{T\}\.

The concentration results fit neatly into that picture\. The equivalence theorems of\[[60](https://arxiv.org/html/2607.08789#bib.bib60)\]show that deterministic regret inequalities and martingale tail bounds are often two views of the same statement\. Our sampled\-expert and confidence\-sequence theorems follow the same pattern\. The deterministic part of the accounting is kept exact first; probabilistic control is added only to the explicit martingale term afterwards\. This is why the high\-probability and anytime results in Section[3](https://arxiv.org/html/2607.08789#S3)feel like genuine extensions of the exact decomposition rather than separate analyses\.

### 8\.6Scaling time, lower bounds, and the scope of the reduction

The most direct external comparison is with the scaling\-time question of\[[25](https://arxiv.org/html/2607.08789#bib.bib25)\]and its recent NormalHedge resolution in\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]\. Our fixed\-budget theorem gives an exact PAC\-Bayes\-style counterpart: for any chosen budgetΓ\\Gamma, one run controls all comparators withKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, and the dyadic controller of Section[4\.4](https://arxiv.org/html/2607.08789#S4.SS4)converts those fixed\-budget guarantees into simultaneous quantile control with an explicit meta term\. This places the paper on the same conceptual map as the scaling\-time literature, but it is important to distinguish simultaneous validity from simultaneous adaptivity\. A single run with fixed budgetΓ\\Gammais valid for all simpler comparators, yet its learning rate is still anchored to that budget\. The strongest single\-copy simultaneous adaptation results remain a separate benchmark\[[50](https://arxiv.org/html/2607.08789#bib.bib50),[30](https://arxiv.org/html/2607.08789#bib.bib30)\]\.

The lower\-bound story is unchanged by the composite\-loss reduction\. Our setting strictly contains ordinary expert advice as the special caseut≡0u\_\{t\}\\equiv 0, so every impossibility result for experts is inherited verbatim\. In particular, any future attempt to strengthen the scaling\-time guarantees here must still confront the random\-walk lower bound emphasized in\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]:

Regϵ⁡\(T\)=Ω​\(\(∑t=1Tσt2\)​log⁡\(1/ϵ\)\)\\operatorname\{Reg\}\_\{\\epsilon\}\(T\)=\\Omega\\\!\\left\(\\sqrt\{\\left\(\\sum\_\{t=1\}^\{T\}\\sigma\_\{t\}^\{2\}\\right\)\\log\(1/\\epsilon\)\}\\right\)and, when the intrinsic\-time variable is the self\-variance under the learner’s own weights, the same discussion yields the sharper barrier

Regϵ⁡\(T\)=Ω​\(VT​log⁡\(1/ϵ\)\)\\operatorname\{Reg\}\_\{\\epsilon\}\(T\)=\\Omega\\\!\\left\(\\sqrt\{V\_\{T\}\\log\(1/\\epsilon\)\}\\right\)Mixed coincidence changes which loss sequence and which intrinsic\-time functional are natural to analyze; it does not make the hard instances disappear\.

Appendix I of\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]also clarifies how this interacts with the contextual\-bandit model\-selection impossibility theorem of\[[51](https://arxiv.org/html/2607.08789#bib.bib51)\]\. Their wrapper contradiction starts from a full\-information experts guarantee whose variance is measured under the learner’s own played distribution\. When importance weighting is wrapped around such a guarantee, the self\-variance case enjoys a two\-factorγ2\\gamma^\{2\}cancellation, and that is what drives the contradiction\. If the variance is measured under an auxiliary distribution instead, one of those factors disappears, the generic wrapper bound becomes larger, and the model\-selection lower bound survives\. This is a good way to interpret the present paper as well\. The fixed\-budget theorem is not single\-copy simultaneous, the dyadic controller pays an explicit meta term, and both are written after an exact reduction to the composite lossesct=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}\. The reduction changes the object being analyzed, but it does not circumvent the lower bounds\.

#### 8\.6\.1Comparator notions: quantiles, sleeping, and tracking

Regret itself has diversified in parallel\. Quantile or prior\-mass bounds compete with sets or mixtures of experts rather than just the single best expert; NormalHedge, Squint, AdaNormalHedge, and the recent scaling\-time refinement are central examples\[[13](https://arxiv.org/html/2607.08789#bib.bib13),[45](https://arxiv.org/html/2607.08789#bib.bib45),[50](https://arxiv.org/html/2607.08789#bib.bib50),[30](https://arxiv.org/html/2607.08789#bib.bib30)\]\. Specialists and sleeping experts allow abstention or confidences, and later Bayesian reinterpretations show how naturally those ideas fit with posterior averaging\[[29](https://arxiv.org/html/2607.08789#bib.bib29),[46](https://arxiv.org/html/2607.08789#bib.bib46)\]\. Tracking or switching regret compares to a sequence of experts with limited changes, classical fixed\-share territory\[[40](https://arxiv.org/html/2607.08789#bib.bib40)\]\. Our exact shifting\-comparator identity in Section[4\.3](https://arxiv.org/html/2607.08789#S4.SS3)is stated before any switch\-count relaxation: it first writes the dynamic regret exactly and only then bounds the path\-variation term\. Internal or swap regret is often obtained by wrapping an external\-regret learner through reductions such as\[[7](https://arxiv.org/html/2607.08789#bib.bib7)\]; conceptually, that too is a front\-end transformation of the comparator class followed by a back\-end regret theorem\.

From the mixed\-prior viewpoint, these comparator notions can also be encoded as structured prior spaces\. For example, ifΘS\\Theta\_\{S\}denotes the set of expert pathsθ=\(i1,…,iT\)\\theta=\(i\_\{1\},\\dots,i\_\{T\}\)with at mostSSswitches andα\\alphais a prior onΘS\\Theta\_\{S\}, then the same geometric\-mixture calculus aggregates over tracking paths rather than over single experts\. We do not pursue the efficient HMM\-style implementation here, but conceptually switching regret is just PAC\-Bayesian regret on a structured comparator space\.

#### 8\.6\.2Online learning and martingale concentration

There is also a parallel line of work showing that regret inequalities and concentration bounds are often different faces of the same underlying object\. The equivalence results in\[[60](https://arxiv.org/html/2607.08789#bib.bib60)\]connect deterministic pathwise regret inequalities, high\-probability tail bounds for martingales, and in\-expectation bounds\. The sampled\-expert identity of Theorem[3\.3](https://arxiv.org/html/2607.08789#S3.Thmtheorem3)and the confidence\-sequence corollary of Theorem[3\.6](https://arxiv.org/html/2607.08789#S3.Thmtheorem6)fit naturally into that picture\. Here the decomposition isolates a single martingale term and keeps the remaining cumulant/drift pieces deterministic conditional on the path\. The confidence\-sequence and LIL corollaries are therefore obtained by adding concentration to an already exact regret identity, rather than by re\-proving the regret bound in probabilistic form\.

#### 8\.6\.3Mirror descent, FTRL, OCO, and game\-theoretic prediction

Exponential weights is simultaneously an entropic mirror\-descent step and a follow\-the\-regularized\-leader \(FTRL\) update, and the online\-optimization literature made that equivalence central in both theory and practice\[[3](https://arxiv.org/html/2607.08789#bib.bib3),[76](https://arxiv.org/html/2607.08789#bib.bib76),[22](https://arxiv.org/html/2607.08789#bib.bib22)\]\. Adaptive variants\[[22](https://arxiv.org/html/2607.08789#bib.bib22)\]make the same philosophical move as our predictable\-rate schedules: the update magnitude is chosen from a cumulative geometry revealed by the realized data\. In the present paper that geometry is the exact finite\-temperature cumulant of the Bayes/exponential\-weights update\. We are therefore not proposing a new regularizer; rather, we isolate the exact information decomposition that any entropic exponential\-weights update already satisfies before one upper bounds it\. This is also why the same calculus survives the passage to continuous\-action OCO in Section[5\.6](https://arxiv.org/html/2607.08789#S5.SS6)\.

Historically, the same update also entered the subject through early game\-theoretic and boosting formulations\[[26](https://arxiv.org/html/2607.08789#bib.bib26),[28](https://arxiv.org/html/2607.08789#bib.bib28)\]\. These ideas are rooted in the broader tradition of individual\-sequence prediction, where algorithms are evaluated against the realized path rather than against any assumed data\-generating process\. The foundational work of\[[11](https://arxiv.org/html/2607.08789#bib.bib11)\]established sharp worst\-case bounds for prediction with expert advice, and\[[38](https://arxiv.org/html/2607.08789#bib.bib38)\]extended the individual\-sequence viewpoint to general loss functions with tight minimax characterizations\. This has been largely unified from an information\-theoretic perspective, connecting universal prediction to source coding and the minimum description length principle\[[52](https://arxiv.org/html/2607.08789#bib.bib52)\]\. A closely related probabilistic tradition is the prequential approach\[[19](https://arxiv.org/html/2607.08789#bib.bib19)\], which argues that statistical procedures should be evaluated by their sequential predictive performance on each individual realization, rather than by their behavior under a model\. A long tradition of work in this spirit\[[9](https://arxiv.org/html/2607.08789#bib.bib9),[74](https://arxiv.org/html/2607.08789#bib.bib74),[67](https://arxiv.org/html/2607.08789#bib.bib67)\]has analyzed prediction of individual sequences through a minimax lens, characterizing the best achievable regret without distributional assumptions\.

Another nearby tradition is game\-theoretic probability and defensive forecasting, which emphasizes sequential calibration, betting interpretations, and adversarial game structure rather than stochastic assumptions\[[23](https://arxiv.org/html/2607.08789#bib.bib23),[74](https://arxiv.org/html/2607.08789#bib.bib74),[67](https://arxiv.org/html/2607.08789#bib.bib67)\]\. The individual\-sequence interpretation through comparator description length is also close to the predictive\-complexity viewpoint developed around the aggregating algorithm\[[73](https://arxiv.org/html/2607.08789#bib.bib73),[43](https://arxiv.org/html/2607.08789#bib.bib43)\], and to the information\-theoretic tradition of universal coding, where the difficulty of an individual sequence is measured by its code length relative to a model class\[[61](https://arxiv.org/html/2607.08789#bib.bib61),[68](https://arxiv.org/html/2607.08789#bib.bib68),[36](https://arxiv.org/html/2607.08789#bib.bib36)\]\. Cover’s universal portfolio theory\[[17](https://arxiv.org/html/2607.08789#bib.bib17)\]applies the same philosophy to sequential investment, measuring individual\-sequence wealth rather than expected growth\. Our use of PAC\-Bayes language is different from these works, but the underlying aim is similar: keep the sequential game explicit in the absence of guiding structure of the sequence, and understand exactly how the learner’s update transfers information from one round to the next\.

#### 8\.6\.4What is solved, and what remains

The 2016 question in\[[25](https://arxiv.org/html/2607.08789#bib.bib25)\]asked for a second\-order regret theorem parameterized by the*variance across actions*and by the topε\\varepsilonquantile of experts\. In modern notation, the target regret bound considered there isRegε​\(T\)≲?VT​log⁡\(1/ε\)\\mathrm\{Reg\}\_\{\\varepsilon\}\(T\)\\stackrel\{\{\\scriptstyle?\}\}\{\{\\lesssim\}\}\\sqrt\{V\_\{T\}\\log\(1/\\varepsilon\)\}, simultaneously for allε\\varepsilon, whereVTV\_\{T\}is a suitable cumulative variance process of the instantaneous regrets under the learner’s current weights\.

As of this writing, that historical question should be regarded as substantially resolved rather than simply open\. From this paper’s perspective, we can exactly decompose the dynamics of Bayes\-compatible update rules, and the algorithms in this paper achieve the desired regret against appropriate comparator sets without any explicit dependence onTT\. It is evident that any future algorithms of this kind will also allocate information in precisely the manner this paper outlines\. So our results are fertile ground for designing new methods in this template\.

However, the broader question is a little more subtle, and we discuss it further because of its particular relevance in generalizations of this setting to partial\-information sequences\. Two recent works are especially pertinent: the NormalHedge scaling\-time theorem of\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]and the contextual\-bandit model\-selection analysis of\[[51](https://arxiv.org/html/2607.08789#bib.bib51)\]\. Both interact with the present paper in illuminating ways, and comparing approaches clarifies the distinctive aspects of each contribution\.

The scaling\-time result in\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]proves that a variant of NormalHedge achieves

Regε​\(T\)≤\(t0\+2​VT\)​\(log⁡\(t0\+2​VT\)\+2​log⁡\(1/ε\)\),t0=Θ​\(B2​log⁡N\)\\mathrm\{Reg\}\_\{\\varepsilon\}\(T\)\\leq\\sqrt\{\(t\_\{0\}\+2V\_\{T\}\)\\left\(\\log\(t\_\{0\}\+2V\_\{T\}\)\+2\\log\(1/\\varepsilon\)\\right\)\},\\qquad t\_\{0\}=\\Theta\(B^\{2\}\\log N\)with a matching lower bound of orderVT​log⁡\(1/ε\)\\sqrt\{V\_\{T\}\\log\(1/\\varepsilon\)\}\. Two features distinguish their result from ours: the additive offsett0t\_\{0\}contributes lower\-order logarithmic terms, andVTV\_\{T\}is computed under a curvature distributionqqinduced by the NormalHedge potential rather than under the play distributionpp\. In that sense, the quantile scaling\-time problem is solved up to the now\-standard offset and logarithmic bookkeeping\.

The present paper contributes a complementary, PAC\-Bayes\-flavored perspective on the same landscape\. By the paragraph on quantile comparators after Corollary[4\.1](https://arxiv.org/html/2607.08789#S4.Thmtheorem1), if the prior is uniform andρA\\rho\_\{A\}is uniform on a setAAof size\|A\|=ε​K\|A\|=\\varepsilon K, then Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)gives the exact identity

∑t=1T⟨pt,ct⟩−1\|A\|​∑i∈ACT​\(i\)=DT\+BT​\(ρA\)\+∑t=1Tηt​Qt​\(c\)\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},c\_\{t\}\\rangle\-\\frac\{1\}\{\|A\|\}\\sum\_\{i\\in A\}C\_\{T\}\(i\)=D\_\{T\}\+B\_\{T\}\(\\rho\_\{A\}\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(c\)\(155\)whereQt​\(c\)=ϕt​\(ηt\)/ηtQ\_\{t\}\(c\)=\\phi\_\{t\}\(\\eta\_\{t\}\)/\\eta\_\{t\}is the exact intrinsic\-time increment\. Running the second\-order schedule with complexity budgetΓ=log⁡\(1/ε\)\\Gamma=\\log\(1/\\varepsilon\)therefore yields

∑t=1T⟨pt,ct⟩−1\|A\|​∑i∈ACT​\(i\)≤\(1\+C2\)​log⁡\(1/ε\)\+Q∗T​\(c\)\+\(2​C\+C−1\)​VT​\(c\)​log⁡\(1/ε\)\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},c\_\{t\}\\rangle\-\\frac\{1\}\{\|A\|\}\\sum\_\{i\\in A\}C\_\{T\}\(i\)\\leq\(1\+C^\{2\}\)\\log\(1/\\varepsilon\)\+Q\_\{\*\}^\{T\}\(c\)\+\(2C\+C^\{\-1\}\)\\sqrt\{V\_\{T\}\(c\)\\log\(1/\\varepsilon\)\}
It is important to distinguish between simultaneous validity and simultaneous adaptivity\. Since the conditionKL\(ρ\|\|π\)≤ΓKL\(\\rho\|\|\\pi\)\\leq\\Gammatriggers this bound, a single run with budgetΓ\\Gammaprovides a regret bound that is simultaneously valid for all comparators simpler thanΓ\\Gamma\(e\.g\., all quantilesϵ≥e−Γ\\epsilon\\geq e^\{\-\\Gamma\}\)\. However, it is not simultaneously adaptive: the algorithm’s learning rateηt\\eta\_\{t\}is permanently anchored to the worst\-case budgetΓ\\Gamma, meaning it does not shrink its penalty if the actual target turns out to be much simpler\.

With that caveat, the fixed\-budget PAC\-Bayes analogue is therefore solved here for ordinary exponential weights and the exact cumulant intrinsic time, up to two edge terms intrinsic to the discrete path: the initialization costC2​ΓC^\{2\}\\Gammaand the largest single\-round jumpQ∗T​\(c\)Q\_\{\*\}^\{T\}\(c\)\. The discussion immediately after Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)explains that neither can be removed uniformly\.

Section[4\.4](https://arxiv.org/html/2607.08789#S4.SS4)adds a simple simultaneous quantile theorem on top of the fixed\-budget result: a logarithmic\-budget controller over dyadic values ofΓ\\Gammacombines the fixed\-budget copies into a single run whose regret is simultaneously controlled for allε\\varepsilon, with one extra meta\-regret term of the same intrinsic\-time form\. This places the exact cumulant theorem on the same conceptual map as the modern scaling\-time literature\.

##### Where the two perspectives meet\.

The key conceptual transfer between these approaches runs through the notion of intrinsic time\. In\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\], intrinsic time is measured through a curvature distribution tied to the NormalHedge potential\. In the present paper, it is measured through the exact cumulantQt​\(c\)Q\_\{t\}\(c\)of the Bayes\-rule update itself\. Both capture the same intuition—that the algorithm should track the difficulty revealed by the realized sequence—but through different geometric lenses\. The NormalHedge approach achieves simultaneous adaptation across all quantilesε\\varepsilonin a single copy; our approach keeps the identity exact and handles side information through the composite\-loss reduction\.

The natural remaining question is whether one can obtain a genuine single\-copy bound of orderVT♯​\(ρ\)​\(KL​\(ρ∥π\)\+1\)\\sqrt\{V\_\{T\}^\{\\sharp\}\(\\rho\)\(\\mathrm\{KL\}\(\\rho\\\|\\pi\)\+1\)\}with only mild lower\-order terms\. HereVT♯​\(ρ\)V\_\{T\}^\{\\sharp\}\(\\rho\)could be based on the learner distributionpp, on a curvature distributionqq, or on some other intrinsic second\-order process\. The recent NormalHedge theorem suggests that the curvature viewpoint may be the right one for simultaneous adaptation\. At the same time, the lower\-bound discussion in\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]and the contextual\-bandit impossibility result of\[[51](https://arxiv.org/html/2607.08789#bib.bib51)\]indicate that one should be cautious about insisting on the classicalpp\-variance*verbatim*together with full parameter\-freeness\.

Importantly, the mixed\-coincidence reduction is orthogonal to that difficulty\. Whatever the right future scaling\-time theorem turns out to be, it can be run on the composite lossesct=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}identified by Section[2\.1](https://arxiv.org/html/2607.08789#S2.SS1); the exact mismatch decomposition of Section[4\.1](https://arxiv.org/html/2607.08789#S4.SS1)then transfers the result back to the original losses\. Side priors are not the obstruction\. The hard part is simultaneous adaptation to both second\-order complexity and comparator complexity inside a single copy of the algorithm\.

##### Appendix I of\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]: wrapper variance and the self\-variance cancellation\.

A useful refinement of the lower\-bound picture comes from Appendix I of\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\], which sharpens exactly what is ruled out by the contextual\-bandit model\-selection impossibility theorem of\[[51](https://arxiv.org/html/2607.08789#bib.bib51)\]\. Their wrapper argument begins from a full\-information experts guarantee with variance measured under the learner’s own played distribution\. To avoid conflict with our notationutu\_\{t\}for side penalties, writeνt\\nu\_\{t\}for the auxiliary variance\-measuring distribution used in that discussion; the self\-variance case isνt=qt\\nu\_\{t\}=q\_\{t\}\. When the contextual\-bandit wrapper feeds scaled importance\-weighted losses to the expert algorithm, the crucial estimate in the self\-variance case is

𝔼​\[∑t=1TVari∼qt⁡\(ℓ^t,i\)\]=O​\(γ2​T\)\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=1\}^\{T\}\\operatorname\{Var\}\_\{i\\sim q\_\{t\}\}\(\\widehat\{\\ell\}\_\{t,i\}\)\\right\]=O\(\\gamma^\{2\}T\)Because the wrapper itself incurs a leading1/γ1/\\gammanormalization, this is exactly the cancellation that makes the contradiction with the model\-selection lower bound go through\. One gets a contextual\-bandit upper bound whoselog⁡k\\sqrt\{\\log k\}term carries no compensating negative power ofγ\\gamma, and that upper bound can be driven below the model\-selection lower bound by a suitable choice ofγ\\gammaandkk\.

Appendix I also explains why that contradiction is more limited than may initially be apparent\. First, if the full\-information guarantee is allowed to measure variance under an auxiliary distributionνt\\nu\_\{t\}rather than under the played distribution itself, the best generic wrapper estimate loses one power ofγ\\gamma:

𝔼​\[∑t=1TVari∼νt⁡\(ℓ^t,i\)\]=O​\(γ​T\)rather thanO​\(γ2​T\)\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=1\}^\{T\}\\operatorname\{Var\}\_\{i\\sim\\nu\_\{t\}\}\(\\widehat\{\\ell\}\_\{t,i\}\)\\right\]=O\(\\gamma T\)\\qquad\\text\{rather than\}\\qquad O\(\\gamma^\{2\}T\)This is not a cosmetic change\. In the self\-variance case the same distribution both triggers the rare importance\-weighted coordinate and averages its variance, so the wrapper gets two small factors: oneγ\\gammafrom the event that the informative arm is played, and a secondγ\\gammafrom the variance being measured under the same played distribution\. When variance is measured under an auxiliaryνt\\nu\_\{t\}, only the first factor survives;νt\\nu\_\{t\}may still put order\-one mass on the inflated coordinate, so the per\-round variance is onlyO​\(γ\)O\(\\gamma\)rather thanO​\(γ2\)O\(\\gamma^\{2\}\)\. After the outer1/γ1/\\gammarescaling in the wrapper, the contextual\-bandit upper bound becomes of orderγ​T\+T/γ\\gamma T\+\\sqrt\{T/\\gamma\}rather thanγ​T\+T\\gamma T\+\\sqrt\{T\}, and that larger term makes the model\-selection lower bound hold by default\.

Second,\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]shows that even in the self\-variance case a bound of the formRegϵ⁡\(T\)=O~​\(\(log⁡N\+VT\)​log⁡\(1/ϵ\)\)\\operatorname\{Reg\}\_\{\\epsilon\}\(T\)=\\widetilde\{O\}\\\!\\left\(\\sqrt\{\(\\log N\+V\_\{T\}\)\\log\(1/\\epsilon\)\}\\right\)escapes the wrapper lower bound: the extralog⁡N\\log Nterm is already large enough that the model\-selection lower bound is dominated by the wrapper upper bound in every parameter regime\. Our fixed\-budget theorem and the outer simultaneous controller of Section[4\.4](https://arxiv.org/html/2607.08789#S4.SS4)are both consistent with this picture\. The former is not simultaneous; the latter pays an explicit meta term; and the theorem is stated after an exact reduction to the composite lossesct=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}, with complexity term equal to the exact cumulant intrinsic timeVT​\(c\)V\_\{T\}\(c\)plus the explicit mismatch correction∑t\(⟨u,ut⟩−⟨pt,ut⟩\)\\sum\_\{t\}\\left\(\\langle u,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\\right\)\. Settingut≡0u\_\{t\}\\equiv 0recovers the usual experts problem, while for general side factors one must still pay for prior mismatch separately\. The lesson is structural\. What transfers black\-box through mixed coincidence is the passage from several priors to one composite loss\. What does*not*transfer black\-box is any sharp quantile theorem whose proof depends on the geometry of a particular potential and on the precise choice of variance\-measuring distribution\.

### 8\.7Discussion and open directions

Exponential weights can be viewed not only as a single algorithm, but as an instance of a general identity about geometric mixtures and entropic duality\. The mixed\-coincidence perspective determines exactly how several priors or side factors become composite losses\. The two branches of the variable\-temperature update then compose the shared one\-step information balance in structurally parallel ways: the retempered chain produces a global cumulant decomposition, the local chain produces an exact cumulative normalization identity\. That these two different\-looking objects share the same front end, the same per\-round cumulant functional as their unit of account, and the same terminal comparator\-information structure is the main unifying point of the paper\.

The cumulant theorem of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)isolates three pieces of the retempered regret: the exact one\-round centered cumulants, the temperature\-change drift, and the terminal PAC\-Bayes complexity term\. The pressure\-target identities of Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)isolate a parallel three\-piece structure for the local recursion: the exact one\-round cumulant cost, the cumulative normalization \(in place of drift\), and the terminal partition \(in place of retempered free energy\)\. Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)shows that the intrinsic\-time increment itself is an exact average of tilted variances in both cases, so both branches record the curvature revealed by the realized path\. The familiar square\-root intrinsic\-time bound is then a corollary of the retempered chain, while the exact event\-restricted partition identities are a corollary of the local chain\.

The luckiness results sharpen that picture in the same\-scale stochastic regime\. For point comparators they give a short route to constant expected regret; for posterior comparators they isolate the exact additional stochastic condition needed by the comparator\-centered argument\. The sampled\-expert theorem shows that the same exact chain persists pathwise once the martingale term is written explicitly\.

The analysis is not specific to Hedge in the narrow sense\. The Bayes\-rule template of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)only assumes that expertwise evidence is accumulated multiplicatively and then updated at a predictable temperature\. That viewpoint makes the paper simultaneously tighter and more general than variance\-first Taylor\-expansion analyses: tighter because it keeps the exact cumulant costs visible, and more general because the same proof layer can be reused across a whole family of predictable schedules and Bayes\-rule update designs\.

The later sections demonstrate that the same algebra is not confined to one theorem family\. After the composite\-loss reduction is in place, forgetting\-kernel mixtures, pooled opponent models, composite\-loss Exp3, contextual\-bandit wrappers, continuous\-action OCO, boosting, and direct multi\-prior PAC\-Bayes penalties all become different instances of the same identity, and the machinery around it\. For ordinary exponential weights with the exact cumulant intrinsic time, the fixed\-budget PAC\-Bayes counterpart of the scaling\-time question is therefore solved here up to two unavoidable edge terms \(Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)\), and the dyadic controller of Section[4\.4](https://arxiv.org/html/2607.08789#S4.SS4)converts that fixed\-budget guarantee into simultaneous quantile control with one explicit meta term\. As discussed in Section[8\.6](https://arxiv.org/html/2607.08789#S8.SS6), this is*simultaneously valid*but not*single\-copy simultaneously adaptive*; the strongest single\-copy results remain a separate benchmark\[[50](https://arxiv.org/html/2607.08789#bib.bib50),[30](https://arxiv.org/html/2607.08789#bib.bib30)\]\. The composite\-loss view, the pressure\-target schedule, and the multi\-prior side\-factor reduction nonetheless open versatile avenues to designing further algorithms\[[39](https://arxiv.org/html/2607.08789#bib.bib39),[15](https://arxiv.org/html/2607.08789#bib.bib15),[70](https://arxiv.org/html/2607.08789#bib.bib70),[58](https://arxiv.org/html/2607.08789#bib.bib58),[62](https://arxiv.org/html/2607.08789#bib.bib62),[30](https://arxiv.org/html/2607.08789#bib.bib30)\], and the same exact accounting governs whatever new schedule, prior, or feedback model is plugged into the template\.

##### Open directions\.

The exact decomposition framework naturally suggests several avenues for further work\.

Variance processes defined with respect to bespoke distributions \(as in\[[30](https://arxiv.org/html/2607.08789#bib.bib30)\]\) may behave in complex ways after the composite\-loss translation, opening interesting algorithmic possibilities while still respecting the lower bounds discussed in Section[8\.6](https://arxiv.org/html/2607.08789#S8.SS6)\. A second direction concerns the side\-prior weightsαt,w\\alpha\_\{t,w\}themselves: in many settings these should be learned online, suggesting a two\-level architecture in which one procedure updates the action distribution and another updates the prior mixture\. The composite\-loss view also suggests multi\-prior versions of contextual\-bandit and model\-selection reductions, in the spirit of the open problem in\[[24](https://arxiv.org/html/2607.08789#bib.bib24)\]\.

On the exploration side, Section[6\.6](https://arxiv.org/html/2607.08789#S6.SS6)hints at replacing single\-posterior information gains by multiway coincidence or testing\-exponent scores, connecting to information\-directed sampling and mirror\-descent stability\[[47](https://arxiv.org/html/2607.08789#bib.bib47),[64](https://arxiv.org/html/2607.08789#bib.bib64)\]\. In repeated games, multiscale opponent\-model pooling is a natural response to drift and periodicity\. In such situations, our path\-specific information theory implies dynamic\-regret guarantees in terms of model\-tracking penalties, allowing for flexible adaptive learning rate schedules\. Finally, the continuum geometric\-pool viewpoint turns mixing over scales, temperatures, or regularization strengths into a first\-class variational object, potentially yielding principled adaptive schedules for these parameters, and supplanting hand\-designed grids\. A particularly interesting open question in this direction is whether scaling\-time adaptation extends to unbounded\-loss settings such as calibrated\-expert regimes\[[54](https://arxiv.org/html/2607.08789#bib.bib54)\]\.

### 8\.8Speculative connections

The following observations are raised here as short proposals rather than finished results; each merits a future treatment\.

##### Tilted\-variance representation as a path\-adapted Edgeworth refinement\.

Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)writesQt​\(c\)Q\_\{t\}\(c\)as an explicit integral ofss\-tilted variances of the current score vector\. This is the exact finite\-temperature refinement of the leading variance term in an Edgeworth/cumulant expansion, with the tilting parameters∈\[0,1\]s\\in\[0,1\]interpolating between the played distributionptp\_\{t\}and the one\-step Gibbs update\. Plotting the tilted\-variance integrand on real data would give a path\-adapted diagnostic: flat integrands mean the second\-order proxyWt​\(c\)W\_\{t\}\(c\)is already accurate, while sharply\-curved ones indicate genuine finite\-temperature effects\. Formalizing this as a controlled surrogate\-tightness bound looks feasible and would sharpen theWtW\_\{t\}vs\.VtV\_\{t\}distinction in Algorithm[1](https://arxiv.org/html/2607.08789#alg1)\.

##### Pressure\-target line search as a free\-energy thermostat\.

The unit\-potential rule∑ipt​\(i\)​e−ηt​ct​\(i\)=1\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}=1coincides with the canonical ensemble equation at inverse temperatureηt\\eta\_\{t\}with zero free energy\. This identifies the pressure\-target schedule with a*thermostat*: each round is calibrated so the instantaneous free energy equals the targetata\_\{t\}\. Importing tools from non\-equilibrium statistical mechanics \(Jarzynski equality, Crooks fluctuation theorem\) to the cumulative normalization identity∏tZt=e−∑tηt​at\\prod\_\{t\}Z\_\{t\}=e^\{\-\\sum\_\{t\}\\eta\_\{t\}a\_\{t\}\}could yield exact exchange\-fluctuation relations for the local update, withata\_\{t\}playing the role of a controlled reservoir temperature\.

##### Regret spectrumDq​\(T\)D\_\{q\}\(T\)as a finite\-time Rényi fractal\.

The spectrumDq​\(T\)=−Hq​\(pT\+1\)/log⁡TD\_\{q\}\(T\)=\-H\_\{q\}\(p\_\{T\+1\}\)/\\log Tintroduced at the end of Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)is formally the Rényi dimension profile of the terminal weights at log\-scaleTT\. This connects adaptive Bayes updates to the multifractal\-spectrum literature: a nondegenerate curvature ofq↦Dq​\(T\)q\\mapsto D\_\{q\}\(T\)certifies a multiscale hierarchy of competitive experts, while a flat spectrum means the terminal posterior concentrates on essentially one scale\. Relating the shape of this spectrum to the cumulative intrinsic timeVTV\_\{T\}or to the terminal free\-energyAT​\(ηT\)A\_\{T\}\(\\eta\_\{T\}\)is a clean follow\-up question\.

##### Two\-level composite\-loss architecture for online meta\-learning\.

Combining the multi\-prior side\-factor reduction \(Section[2\.1](https://arxiv.org/html/2607.08789#S2.SS1)\) with the dyadic budget controller \(Section[4\.4](https://arxiv.org/html/2607.08789#S4.SS4)\) suggests a natural two\-level architecture: an inner algorithm runs retempered Hedge on composite lossesct=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}where the side factorst=∏wπt,wαt,ws\_\{t\}=\\prod\_\{w\}\\pi\_\{t,w\}^\{\\alpha\_\{t,w\}\}pools several candidate priors, and an outer controller updates the pooling weightsαt\\alpha\_\{t\}by another instance of the same exact identity\. Because both levels inherit exact information accounting, the end\-to\-end regret bound is a sum of two explicit intrinsic\-time terms plus an explicit cross\-entropy mismatch, without any unrealized constants\.

##### Pressure\-target \+ feedback graph = graph\-aware step\-size calibration\.

Proposition[6\.10](https://arxiv.org/html/2607.08789#S6.Thmtheorem10)treats feedback\-graph topology by changing only the estimator, retaining the retempered second\-order schedule\. The pressure\-target schedule of Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)is unexplored in the feedback\-graph setting, where a local line\-search against the realized observation ratiosot​\(a\)o\_\{t\}\(a\)would naturally concentrate the update on frequently\-observed arms and relax it on rarely\-observed ones\. Whether this produces aα​\(Gt\)​T\\sqrt\{\\alpha\(G\_\{t\}\)T\}\-type bound through the exact cumulative normalization route \(rather than through the standard independence\-number argument\) is an open question\.

##### Pathwise regret as a variational dual\.

Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)writes the pathwise regret as a sum of per\-round equalizer values, where each round’s payoff is the support function of a KL ball\. This matches the strong\-duality pattern of variational\-form two\-player games: the retempered chain is the*primal*accounting of information spent on each round, and a corresponding min\-max problem on KL\-ball strategies is the*dual*whose value equals that per\-round payment\. Making this duality rigorous \(in the stochastic\-interface sense\) could clarify why Gibbs strategies are equalizers and why the intrinsic\-time increment coincides with the CGF payment in both framings\.

## Appendix AProbabilistic experts and logarithmic pooling

The mixed coincidence identity also applies to a different simplex: instead of weighting actions, one can weight probabilistic experts\. This appendix records the resulting exact decomposition for logarithmic opinion pools\.

LetYYbe an outcome space and suppose that each experte∈\[N\]e\\in\[N\]reports a predictive distributionπt,e∈Δ​\(Y\)\\pi\_\{t,e\}\\in\\Delta\(Y\)on roundtt\. Given weightsαt∈Δ​\(\[N\]\)\\alpha\_\{t\}\\in\\Delta\(\[N\]\), define the pooled forecast

pt​\(y\)=∏e=1Nπt,e​\(y\)αt,e∑z∈Y∏e=1Nπt,e​\(z\)αt,e\.p\_\{t\}\(y\)=\\frac\{\\prod\_\{e=1\}^\{N\}\\pi\_\{t,e\}\(y\)^\{\\alpha\_\{t,e\}\}\}\{\\sum\_\{z\\in Y\}\\prod\_\{e=1\}^\{N\}\\pi\_\{t,e\}\(z\)^\{\\alpha\_\{t,e\}\}\}\.\(156\)This is the classical logarithmic opinion pool\[[33](https://arxiv.org/html/2607.08789#bib.bib33),[41](https://arxiv.org/html/2607.08789#bib.bib41)\], which is a direct instance of the mixed coincidence identity\.

###### Proposition A\.1\(Exact pooled log\-loss decomposition\)\.

For any realized outcomeyt∈Yy\_\{t\}\\in Y,

−log⁡pt​\(yt\)=∑e=1Nαt,e​\(−log⁡πt,e​\(yt\)\)−𝒞αt​\(πt,1:N\),\-\\log p\_\{t\}\(y\_\{t\}\)=\\sum\_\{e=1\}^\{N\}\\alpha\_\{t,e\}\\left\(\-\\log\\pi\_\{t,e\}\(y\_\{t\}\)\\right\)\-\\mathcal\{C\}\_\{\\alpha\_\{t\}\}\(\\pi\_\{t,1:N\}\),\(157\)where𝒞αt​\(πt,1:N\):=−log​∑y∈Y∏e=1Nπt,e​\(y\)αt,e\\mathcal\{C\}\_\{\\alpha\_\{t\}\}\(\\pi\_\{t,1:N\}\):=\-\\log\\sum\_\{y\\in Y\}\\prod\_\{e=1\}^\{N\}\\pi\_\{t,e\}\(y\)^\{\\alpha\_\{t,e\}\}\. Ifαt∈Δ​\(\[N\]\)\\alpha\_\{t\}\\in\\Delta\(\[N\]\), then𝒞αt​\(πt,1:N\)≥0\\mathcal\{C\}\_\{\\alpha\_\{t\}\}\(\\pi\_\{t,1:N\}\)\\geq 0\.

Thus the learner’s log loss is the weighted expert log loss minus a nonnegative coincidence discount\. The usual weighted\-expert benchmark is therefore missing a geometric\-agreement bonus\.

### Online learning of pooling weights

Define the convex loss on the simplex bygt​\(α\):=−∑e=1Nαe​log⁡πt,e​\(yt\)\+log⁡\(∑y∈Y∏e=1Nπt,e​\(y\)αe\)g\_\{t\}\(\\alpha\):=\-\\sum\_\{e=1\}^\{N\}\\alpha\_\{e\}\\log\\pi\_\{t,e\}\(y\_\{t\}\)\+\\log\\\!\\left\(\\sum\_\{y\\in Y\}\\prod\_\{e=1\}^\{N\}\\pi\_\{t,e\}\(y\)^\{\\alpha\_\{e\}\}\\right\)\. By construction,gt​\(αt\)=−log⁡pt​\(yt\)g\_\{t\}\(\\alpha\_\{t\}\)=\-\\log p\_\{t\}\(y\_\{t\}\)\. Therefore any online convex optimization guarantee for the sequence\{gt\}\\\{g\_\{t\}\\\}yields an online logarithmic\-pooling guarantee\.

###### Theorem A\.2\(Generic meta\-regret bound for online logarithmic pooling\)\.

Letαt∈Δ​\(\[N\]\)\\alpha\_\{t\}\\in\\Delta\(\[N\]\)be produced by any online convex optimization method satisfying

∑t=1Tgt​\(αt\)≤minα∈Δ​\(\[N\]\)​∑t=1Tgt​\(α\)\+RT\\sum\_\{t=1\}^\{T\}g\_\{t\}\(\\alpha\_\{t\}\)\\leq\\min\_\{\\alpha\\in\\Delta\(\[N\]\)\}\\sum\_\{t=1\}^\{T\}g\_\{t\}\(\\alpha\)\+R\_\{T\}Then the cumulative pooled log loss obeys

∑t=1T−log⁡pt​\(yt\)≤minα∈Δ​\(\[N\]\)⁡\{∑t=1T∑e=1Nαe​\(−log⁡πt,e​\(yt\)\)−∑t=1T𝒞α​\(πt,1:N\)\}\+RT\.\\sum\_\{t=1\}^\{T\}\-\\log p\_\{t\}\(y\_\{t\}\)\\leq\\min\_\{\\alpha\\in\\Delta\(\[N\]\)\}\\left\\\{\\sum\_\{t=1\}^\{T\}\\sum\_\{e=1\}^\{N\}\\alpha\_\{e\}\\left\(\-\\log\\pi\_\{t,e\}\(y\_\{t\}\)\\right\)\-\\sum\_\{t=1\}^\{T\}\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{t,1:N\}\)\\right\\\}\+R\_\{T\}\.\(158\)

The benchmark in \([158](https://arxiv.org/html/2607.08789#A1.E158)\) is stronger than the usual “best convex combination of expert log losses” because of the explicit coincidence bonus\. The theorem is agnostic about howRTR\_\{T\}is obtained\. If the expert forecasts are clipped away from zero, entropic mirror descent gives the usualO​\(T​log⁡N\)O\(\\sqrt\{T\\log N\}\)\-type control\. Without clipping, log loss is unbounded, and the calibrated\-expert setting studied in\[[54](https://arxiv.org/html/2607.08789#bib.bib54)\]becomes especially relevant\.

##### A conceptual bridge to Sections[2\.1](https://arxiv.org/html/2607.08789#S2.SS1)and[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)\.

Section[2\.1](https://arxiv.org/html/2607.08789#S2.SS1)converts side priors into composite losses for adaptive Hedge over actions, and the two updates of Sections[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)and[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)compose the resulting information flow cumulatively\. The present appendix instead treats experts themselves as priors over outcomes\. In both cases the same quantity appears: a coincidence discount obtained by geometrically pooling several measures\. The difference is only which simplex carries the optimization\.

## Appendix BProofs

We collect the proofs in one appendix, grouped by the section of the main text where each result is stated\. Each proof is labeled by the theorem, lemma, proposition, or corollary it establishes\.

### B\.1Proofs for Section[2](https://arxiv.org/html/2607.08789#S2)\(core identities\)

#### Proof of Theorem[2\.1](https://arxiv.org/html/2607.08789#S2.Thmtheorem1)

###### Proof\.

By definition,log⁡pα⋆​\(x\)=∑w=1Wαw​log⁡μw​\(x\)−log⁡Z​\(α\)\\log p^\{\\star\}\_\{\\alpha\}\(x\)=\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\log\\mu\_\{w\}\(x\)\-\\log Z\(\\alpha\)\. Therefore, for anyq∈Δ​\(𝒳\)q\\in\\Delta\(\\mathcal\{X\}\),

KL​\(q∥pα⋆\)\\displaystyle\\mathrm\{KL\}\(q\\\|p^\{\\star\}\_\{\\alpha\}\)=∑x∈𝒳q​\(x\)​log⁡q​\(x\)pα⋆​\(x\)\\displaystyle=\\sum\_\{x\\in\\mathcal\{X\}\}q\(x\)\\log\\frac\{q\(x\)\}\{p^\{\\star\}\_\{\\alpha\}\(x\)\}=∑x∈𝒳q​\(x\)​log⁡q​\(x\)−∑w=1Wαw​∑x∈𝒳q​\(x\)​log⁡μw​\(x\)\+log⁡Z​\(α\)\\displaystyle=\\sum\_\{x\\in\\mathcal\{X\}\}q\(x\)\\log q\(x\)\-\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\sum\_\{x\\in\\mathcal\{X\}\}q\(x\)\\log\\mu\_\{w\}\(x\)\+\\log Z\(\\alpha\)=−H​\(q\)\+∑w=1Wαw​H​\(q,μw\)\+log⁡Z​\(α\)\\displaystyle=\-\\mathrm\{H\}\(q\)\+\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{H\}\(q,\\mu\_\{w\}\)\+\\log Z\(\\alpha\)Rearranging proves \([3](https://arxiv.org/html/2607.08789#S2.E3)\)\.

Now assume that eachμw\\mu\_\{w\}is a probability distribution and thatα∈Δ​\(\[W\]\)\\alpha\\in\\Delta\(\[W\]\)\. Because∑wαw=1\\sum\_\{w\}\\alpha\_\{w\}=1,∑w=1Wαw​KL​\(q∥μw\)=∑w=1Wαw​H​\(q,μw\)−H​\(q\)\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{KL\}\(q\\\|\\mu\_\{w\}\)=\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{H\}\(q,\\mu\_\{w\}\)\-\\mathrm\{H\}\(q\)\. Combining this with \([3](https://arxiv.org/html/2607.08789#S2.E3)\) yields∑w=1Wαw​KL​\(q∥μw\)=−log⁡Z​\(α\)\+KL​\(q∥pα⋆\)\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{KL\}\(q\\\|\\mu\_\{w\}\)=\-\\log Z\(\\alpha\)\+\\mathrm\{KL\}\(q\\\|p^\{\\star\}\_\{\\alpha\}\)\. Since the relative entropy is nonnegative and vanishes only atq=pα⋆q=p^\{\\star\}\_\{\\alpha\}, the minimum is−log⁡Z​\(α\)\-\\log Z\(\\alpha\)and the minimizer is unique\. ∎

#### Proof of Theorem[2\.2](https://arxiv.org/html/2607.08789#S2.Thmtheorem2)

###### Proof\.

From \([4](https://arxiv.org/html/2607.08789#S2.E4)\),qt\+1​\(i\)=qt​\(i\)​e−ηt​ℓt​\(i\)​st​\(i\)Ztq\_\{t\+1\}\(i\)=\\frac\{q\_\{t\}\(i\)e^\{\-\\eta\_\{t\}\\ell\_\{t\}\(i\)\}s\_\{t\}\(i\)\}\{Z\_\{t\}\}\. Hence

KL​\(ρ∥qt\+1\)\\displaystyle\\mathrm\{KL\}\(\\rho\\\|q\_\{t\+1\}\)=∑i=1Kρ​\(i\)​log⁡ρ​\(i\)qt\+1​\(i\)\\displaystyle=\\sum\_\{i=1\}^\{K\}\\rho\(i\)\\log\\frac\{\\rho\(i\)\}\{q\_\{t\+1\}\(i\)\}=∑i=1Kρ​\(i\)​log⁡ρ​\(i\)​Ztqt​\(i\)​e−ηt​ℓt​\(i\)​st​\(i\)\\displaystyle=\\sum\_\{i=1\}^\{K\}\\rho\(i\)\\log\\frac\{\\rho\(i\)Z\_\{t\}\}\{q\_\{t\}\(i\)e^\{\-\\eta\_\{t\}\\ell\_\{t\}\(i\)\}s\_\{t\}\(i\)\}=KL​\(ρ∥qt\)\+log⁡Zt\+ηt​⟨ρ,ℓt⟩−∑i=1Kρ​\(i\)​log⁡st​\(i\)\\displaystyle=\\mathrm\{KL\}\(\\rho\\\|q\_\{t\}\)\+\\log Z\_\{t\}\+\\eta\_\{t\}\\langle\\rho,\\ell\_\{t\}\\rangle\-\\sum\_\{i=1\}^\{K\}\\rho\(i\)\\log s\_\{t\}\(i\)Sinceut​\(i\)=−ηt−1​log⁡st​\(i\)u\_\{t\}\(i\)=\-\\eta\_\{t\}^\{\-1\}\\log s\_\{t\}\(i\)andct​\(i\)=ℓt​\(i\)\+ut​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\+u\_\{t\}\(i\), this is equivalent to \([5](https://arxiv.org/html/2607.08789#S2.E5)\)\.

If \([6](https://arxiv.org/html/2607.08789#S2.E6)\) holds, then

−∑i=1Kρ​\(i\)​log⁡st​\(i\)=−∑i=1Kρ​\(i\)​∑w=1Jtαt,w​log⁡μt,w​\(i\)=∑w=1Jtαt,w​H​\(ρ,μt,w\)\-\\sum\_\{i=1\}^\{K\}\\rho\(i\)\\log s\_\{t\}\(i\)=\-\\sum\_\{i=1\}^\{K\}\\rho\(i\)\\sum\_\{w=1\}^\{J\_\{t\}\}\\alpha\_\{t,w\}\\log\\mu\_\{t,w\}\(i\)=\\sum\_\{w=1\}^\{J\_\{t\}\}\\alpha\_\{t,w\}\\mathrm\{H\}\(\\rho,\\mu\_\{t,w\}\)which yields \([7](https://arxiv.org/html/2607.08789#S2.E7)\)\. ∎

#### Proof of Corollaries[2\.4](https://arxiv.org/html/2607.08789#S2.Thmtheorem4),[4\.2](https://arxiv.org/html/2607.08789#S4.Thmtheorem2), and[2\.5](https://arxiv.org/html/2607.08789#S2.Thmtheorem5)

###### Proof\.

Whenηt≡η\\eta\_\{t\}\\equiv\\eta, summing \([5](https://arxiv.org/html/2607.08789#S2.E5)\) overt=1,…,Tt=1,\\dots,Tgives

∑t=1Tmt\+η−1​KL​\(ρ∥qT\+1\)=η−1​KL​\(ρ∥q1\)\+∑t=1T⟨ρ,ct⟩\\sum\_\{t=1\}^\{T\}m\_\{t\}\+\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|q\_\{T\+1\}\)=\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|q\_\{1\}\)\+\\sum\_\{t=1\}^\{T\}\\langle\\rho,c\_\{t\}\\ranglewhich is exactly \([9](https://arxiv.org/html/2607.08789#S2.E9)\)\. The displayed equivalent form in the statement is just∑t⟨ρ,ct⟩=⟨ρ,LT⟩\+∑t⟨ρ,ut⟩\\sum\_\{t\}\\langle\\rho,c\_\{t\}\\rangle=\\langle\\rho,L\_\{T\}\\rangle\+\\sum\_\{t\}\\langle\\rho,u\_\{t\}\\rangle\.

For \([66](https://arxiv.org/html/2607.08789#S4.E66)\), letIt∼qtI\_\{t\}\\sim q\_\{t\}and letXt:=ct​\(It\)X\_\{t\}:=c\_\{t\}\(I\_\{t\}\)\. ThenXt∈\[at,bt\]X\_\{t\}\\in\[a\_\{t\},b\_\{t\}\]almost surely andmt=−η−1​log⁡𝔼​\[e−η​Xt\]m\_\{t\}=\-\\eta^\{\-1\}\\log\\mathbb\{E\}\[e^\{\-\\eta X\_\{t\}\}\]\. the standard bounded\-range exponential\-moment bound implies

−η−1​log⁡𝔼​\[e−η​Xt\]≥𝔼​\[Xt\]−η8​\(bt−at\)2=⟨qt,ct⟩−η8​\(bt−at\)2\-\\eta^\{\-1\}\\log\\mathbb\{E\}\[e^\{\-\\eta X\_\{t\}\}\]\\geq\\mathbb\{E\}\[X\_\{t\}\]\-\\frac\{\\eta\}\{8\}\(b\_\{t\}\-a\_\{t\}\)^\{2\}=\\langle q\_\{t\},c\_\{t\}\\rangle\-\\frac\{\\eta\}\{8\}\(b\_\{t\}\-a\_\{t\}\)^\{2\}Combining this with \([5](https://arxiv.org/html/2607.08789#S2.E5)\) \(with constantη\\eta\) gives

⟨qt,ct⟩−⟨ρ,ct⟩≤η−1​\(KL​\(ρ∥qt\)−KL​\(ρ∥qt\+1\)\)\+η8​\(bt−at\)2\\langle q\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle\\leq\\eta^\{\-1\}\\left\(\\mathrm\{KL\}\(\\rho\\\|q\_\{t\}\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{t\+1\}\)\\right\)\+\\frac\{\\eta\}\{8\}\(b\_\{t\}\-a\_\{t\}\)^\{2\}Summing overttproves \([66](https://arxiv.org/html/2607.08789#S4.E66)\)\. Finally,⟨qt,ℓt⟩−⟨ρ,ℓt⟩=⟨qt,ct⟩−⟨ρ,ct⟩\+⟨ρ,ut⟩−⟨qt,ut⟩\\langle q\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,\\ell\_\{t\}\\rangle=\\langle q\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle\+\\langle\\rho,u\_\{t\}\\rangle\-\\langle q\_\{t\},u\_\{t\}\\rangle, and summing yields \([67](https://arxiv.org/html/2607.08789#S4.E67)\)\. The identity \([10](https://arxiv.org/html/2607.08789#S2.E10)\) expandsut​\(i\)=−η−1​∑wαt,w​log⁡μt,w​\(i\)u\_\{t\}\(i\)=\-\\eta^\{\-1\}\\sum\_\{w\}\\alpha\_\{t,w\}\\log\\mu\_\{t,w\}\(i\)\. ∎

#### Proof of Lemma[2\.6](https://arxiv.org/html/2607.08789#S2.Thmtheorem6)

###### Proof\.

By definition ofqX⋆q\_\{X\}^\{\\star\},log⁡qX⋆​\(i\)π​\(i\)=X​\(i\)−log⁡\(∑j=1Kπ​\(j\)​eX​\(j\)\)\\log\\frac\{q\_\{X\}^\{\\star\}\(i\)\}\{\\pi\(i\)\}=X\(i\)\-\\log\\\!\\left\(\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{X\(j\)\}\\right\)\. Multiplying byρ​\(i\)\\rho\(i\), summing overii, and rearranging givesKL​\(ρ∥qX⋆\)=KL​\(ρ∥π\)−⟨ρ,X⟩\+log⁡\(∑j=1Kπ​\(j\)​eX​\(j\)\)\\mathrm\{KL\}\(\\rho\\\|q\_\{X\}^\{\\star\}\)=\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\langle\\rho,X\\rangle\+\\log\\\!\\left\(\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{X\(j\)\}\\right\), which is exactly \([12](https://arxiv.org/html/2607.08789#S2.E12)\)\. SubstitutingX​\(i\)=−η​x​\(i\)X\(i\)=\-\\eta x\(i\)gives \([13](https://arxiv.org/html/2607.08789#S2.E13)\) withqη,x⋆​\(i\)∝π​\(i\)​e−η​x​\(i\)q\_\{\\eta,x\}^\{\\star\}\(i\)\\propto\\pi\(i\)e^\{\-\\eta x\(i\)\}\. ∎

#### Proof of Lemma[2\.7](https://arxiv.org/html/2607.08789#S2.Thmtheorem7)

###### Proof\.

Fixttandη\>0\\eta\>0, and defineqη​\(i\):=π​\(i\)​e−η​Ct​\(i\)∑j=1Kπ​\(j\)​e−η​Ct​\(j\)q\_\{\\eta\}\(i\):=\\frac\{\\pi\(i\)e^\{\-\\eta C\_\{t\}\(i\)\}\}\{\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{\-\\eta C\_\{t\}\(j\)\}\}\. Then for anyq∈Δ​\(\[K\]\)q\\in\\Delta\(\[K\]\),

KL​\(q∥qη\)\\displaystyle\\mathrm\{KL\}\(q\\\|q\_\{\\eta\}\)=∑i=1Kq​\(i\)​log⁡q​\(i\)qη​\(i\)\\displaystyle=\\sum\_\{i=1\}^\{K\}q\(i\)\\log\\frac\{q\(i\)\}\{q\_\{\\eta\}\(i\)\}=KL​\(q∥π\)\+η​⟨q,Ct⟩\+log⁡\(∑j=1Kπ​\(j\)​e−η​Ct​\(j\)\)\\displaystyle=\\mathrm\{KL\}\(q\\\|\\pi\)\+\\eta\\langle q,C\_\{t\}\\rangle\+\\log\\left\(\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{\-\\eta C\_\{t\}\(j\)\}\\right\)=KL​\(q∥π\)\+η​⟨q,Ct⟩−η​At​\(η\)\\displaystyle=\\mathrm\{KL\}\(q\\\|\\pi\)\+\\eta\\langle q,C\_\{t\}\\rangle\-\\eta A\_\{t\}\(\\eta\)Rearranging givesAt​\(η\)\+η−1​KL​\(q∥qη\)=⟨q,Ct⟩\+η−1​KL​\(q∥π\)A\_\{t\}\(\\eta\)\+\\eta^\{\-1\}\\mathrm\{KL\}\(q\\\|q\_\{\\eta\}\)=\\langle q,C\_\{t\}\\rangle\+\\eta^\{\-1\}\\mathrm\{KL\}\(q\\\|\\pi\)\. Since the left\-hand KL term is nonnegative and vanishes atq=qηq=q\_\{\\eta\}, this proves \([15](https://arxiv.org/html/2607.08789#S2.E15)\)\.

The mapη↦At​\(η\)\\eta\\mapsto A\_\{t\}\(\\eta\)is nonincreasing because for each fixedqqthe expression⟨q,Ct⟩\+η−1​KL​\(q∥π\)\\langle q,C\_\{t\}\\rangle\+\\eta^\{\-1\}\\mathrm\{KL\}\(q\\\|\\pi\)is nonincreasing inη\\eta, andAt​\(η\)A\_\{t\}\(\\eta\)is the pointwise minimum of those functions\. ∎

#### Proof of Theorem[2\.8](https://arxiv.org/html/2607.08789#S2.Thmtheorem8)

###### Proof\.

LetHT:=∑t=1T⟨pt,ct⟩,MT:=∑t=1TmtH\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},c\_\{t\}\\rangle,\\qquad M\_\{T\}:=\\sum\_\{t=1\}^\{T\}m\_\{t\}\. ThenRTc​\(ρ\)=HT−⟨ρ,CT⟩=\(HT−MT\)\+\(MT−⟨ρ,CT⟩\)R\_\{T\}^\{c\}\(\\rho\)=H\_\{T\}\-\\langle\\rho,C\_\{T\}\\rangle=\(H\_\{T\}\-M\_\{T\}\)\+\(M\_\{T\}\-\\langle\\rho,C\_\{T\}\\rangle\)\.

For the first term, letμt:=⟨pt,ct⟩\\mu\_\{t\}:=\\langle p\_\{t\},c\_\{t\}\\rangle\. By definition ofψt\\psi\_\{t\},

ψt​\(−ηt\)=log⁡𝔼i∼pt​exp⁡\(−ηt​\(ct​\(i\)−μt\)\)=ηt​\(μt−mt\)=ηt​δt​\(c\)\\psi\_\{t\}\(\-\\eta\_\{t\}\)=\\log\\mathbb\{E\}\_\{i\\sim p\_\{t\}\}\\exp\\\!\\left\(\-\\eta\_\{t\}\(c\_\{t\}\(i\)\-\\mu\_\{t\}\)\\right\)=\\eta\_\{t\}\(\\mu\_\{t\}\-m\_\{t\}\)=\\eta\_\{t\}\\delta\_\{t\}\(c\)Therefore

HT−MT=∑t=1Tδt​\(c\)=∑t=1Tϕt​\(ηt\)\.H\_\{T\}\-M\_\{T\}=\\sum\_\{t=1\}^\{T\}\\delta\_\{t\}\(c\)=\\sum\_\{t=1\}^\{T\}\\phi\_\{t\}\(\\eta\_\{t\}\)\.\(159\)
For the second term, \([11](https://arxiv.org/html/2607.08789#S2.E11)\) gives

mt=−ηt−1​log⁡∑i=1Kπ​\(i\)​e−ηt​Ct​\(i\)∑j=1Kπ​\(j\)​e−ηt​Ct−1​\(j\)=At​\(ηt\)−At−1​\(ηt\)m\_\{t\}=\-\\eta\_\{t\}^\{\-1\}\\log\\frac\{\\sum\_\{i=1\}^\{K\}\\pi\(i\)e^\{\-\\eta\_\{t\}C\_\{t\}\(i\)\}\}\{\\sum\_\{j=1\}^\{K\}\\pi\(j\)e^\{\-\\eta\_\{t\}C\_\{t\-1\}\(j\)\}\}=A\_\{t\}\(\\eta\_\{t\}\)\-A\_\{t\-1\}\(\\eta\_\{t\}\)Summing overttand regrouping,MT=AT​\(ηT\)\+∑t=1T−1\(At​\(ηt\)−At​\(ηt\+1\)\)M\_\{T\}=A\_\{T\}\(\\eta\_\{T\}\)\+\\sum\_\{t=1\}^\{T\-1\}\\left\(A\_\{t\}\(\\eta\_\{t\}\)\-A\_\{t\}\(\\eta\_\{t\+1\}\)\\right\)\. Applying \([14](https://arxiv.org/html/2607.08789#S2.E14)\) witht=Tt=T,η=ηT\\eta=\\eta\_\{T\}, andq=ρq=\\rhoyields

AT​\(ηT\)−⟨ρ,CT⟩=KL​\(ρ∥π\)−KL​\(ρ∥qT,ηT\)ηT\.A\_\{T\}\(\\eta\_\{T\}\)\-\\langle\\rho,C\_\{T\}\\rangle=\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{T,\\eta\_\{T\}\}\)\}\{\\eta\_\{T\}\}\.\(160\)Substituting \([159](https://arxiv.org/html/2607.08789#A2.E159)\)–\([160](https://arxiv.org/html/2607.08789#A2.E160)\) intoRTc​\(ρ\)=\(HT−MT\)\+\(MT−⟨ρ,CT⟩\)R\_\{T\}^\{c\}\(\\rho\)=\(H\_\{T\}\-M\_\{T\}\)\+\(M\_\{T\}\-\\langle\\rho,C\_\{T\}\\rangle\)proves \([16](https://arxiv.org/html/2607.08789#S2.E16)\)\.

If\(ηt\)\(\\eta\_\{t\}\)is nonincreasing, thenηt≥ηt\+1\\eta\_\{t\}\\geq\\eta\_\{t\+1\}for everytt\. SinceAt​\(η\)A\_\{t\}\(\\eta\)is nonincreasing inη\\etaby Lemma[2\.7](https://arxiv.org/html/2607.08789#S2.Thmtheorem7),At​\(ηt\)−At​\(ηt\+1\)≤0A\_\{t\}\(\\eta\_\{t\}\)\-A\_\{t\}\(\\eta\_\{t\+1\}\)\\leq 0, and summing proves the result\. ∎

#### Proof of Corollary[2\.9](https://arxiv.org/html/2607.08789#S2.Thmtheorem9)

###### Proof\.

Ifηt≡η\\eta\_\{t\}\\equiv\\eta, the temperature\-change drift in \([16](https://arxiv.org/html/2607.08789#S2.E16)\) vanishes\. MoreoverqT,η=pT\+1q\_\{T,\\eta\}=p\_\{T\+1\}by definition\. Substituting these facts into Theorem[2\.8](https://arxiv.org/html/2607.08789#S2.Thmtheorem8)gives \([17](https://arxiv.org/html/2607.08789#S2.E17)\)\. ∎

#### Proof of Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)

###### Proof\.

Theorem[2\.8](https://arxiv.org/html/2607.08789#S2.Thmtheorem8)givesRTc​\(ρ\)=∑t=1Tϕt​\(ηt\)\+DT\+BT​\(ρ\)R\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\phi\_\{t\}\(\\eta\_\{t\}\)\+D\_\{T\}\+B\_\{T\}\(\\rho\)\. By the definition in \([18](https://arxiv.org/html/2607.08789#S2.E18)\),ϕt​\(ηt\)=ηt​Qt​\(c\)\\phi\_\{t\}\(\\eta\_\{t\}\)=\\eta\_\{t\}Q\_\{t\}\(c\), so \([19](https://arxiv.org/html/2607.08789#S2.E19)\) follows immediately\. Sinceϕt​\(ηt\)=δt​\(c\)=⟨pt,ct⟩−mt\\phi\_\{t\}\(\\eta\_\{t\}\)=\\delta\_\{t\}\(c\)=\\langle p\_\{t\},c\_\{t\}\\rangle\-m\_\{t\}, Jensen’s inequality givesmt≤⟨pt,ct⟩m\_\{t\}\\leq\\langle p\_\{t\},c\_\{t\}\\rangleand thereforeϕt​\(ηt\)≥0\\phi\_\{t\}\(\\eta\_\{t\}\)\\geq 0, soQt​\(c\)≥0Q\_\{t\}\(c\)\\geq 0\. Ifηt≡η\\eta\_\{t\}\\equiv\\eta, thenDT=0D\_\{T\}=0, and \([20](https://arxiv.org/html/2607.08789#S2.E20)\) follows from the same identity\. ∎

#### Proof of Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)

###### Proof\.

Fixttandη\>0\\eta\>0, and writeX​\(i\):=ct​\(i\)−μtX\(i\):=c\_\{t\}\(i\)\-\\mu\_\{t\}withμt:=⟨pt,ct⟩\\mu\_\{t\}:=\\langle p\_\{t\},c\_\{t\}\\rangle\. Thenψt​\(λ\)=log​∑i=1Kpt​\(i\)​eλ​X​\(i\)\\psi\_\{t\}\(\\lambda\)=\\log\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\\lambda X\(i\)\}\. Ifνλ​\(i\):=pt​\(i\)​eλ​X​\(i\)∑j=1Kpt​\(j\)​eλ​X​\(j\)\\nu\_\{\\lambda\}\(i\):=\\frac\{p\_\{t\}\(i\)e^\{\\lambda X\(i\)\}\}\{\\sum\_\{j=1\}^\{K\}p\_\{t\}\(j\)e^\{\\lambda X\(j\)\}\}, then differentiation of the log\-partition function givesψt′​\(λ\)=𝔼νλ​\[X\],ψt′′​\(λ\)=Varνλ​\(X\)=Varνλ​\(ct\)\\psi\_\{t\}^\{\\prime\}\(\\lambda\)=\\mathbb\{E\}\_\{\\nu\_\{\\lambda\}\}\[X\],\\qquad\\psi\_\{t\}^\{\\prime\\prime\}\(\\lambda\)=\\mathrm\{Var\}\_\{\\nu\_\{\\lambda\}\}\(X\)=\\mathrm\{Var\}\_\{\\nu\_\{\\lambda\}\}\(c\_\{t\}\)\. Sinceψt​\(0\)=0\\psi\_\{t\}\(0\)=0andψt′​\(0\)=𝔼pt​\[X\]=0\\psi\_\{t\}^\{\\prime\}\(0\)=\\mathbb\{E\}\_\{p\_\{t\}\}\[X\]=0, the integral remainder formula yieldsψt​\(−η\)=η2​∫01\(1−s\)​ψt′′​\(−s​η\)​𝑑s\\psi\_\{t\}\(\-\\eta\)=\\eta^\{2\}\\int\_\{0\}^\{1\}\(1\-s\)\\psi\_\{t\}^\{\\prime\\prime\}\(\-s\\eta\)\\,ds\. Butν−s​η=pt,s\(η\)\\nu\_\{\-s\\eta\}=p\_\{t,s\}^\{\(\\eta\)\}by definition, soψt​\(−η\)=η2​∫01\(1−s\)​Vari∼pt,s\(η\)​\(ct​\(i\)\)​𝑑s\\psi\_\{t\}\(\-\\eta\)=\\eta^\{2\}\\int\_\{0\}^\{1\}\(1\-s\)\\mathrm\{Var\}\_\{i\\sim p\_\{t,s\}^\{\(\\eta\)\}\}\(c\_\{t\}\(i\)\)\\,ds\. Dividing byη\\etagives the formula forϕt​\(η\)\\phi\_\{t\}\(\\eta\), and dividing byηt2\\eta\_\{t\}^\{2\}gives the formula forQt​\(c\)Q\_\{t\}\(c\)\.

The integrand is nonnegative, soQt​\(c\)≥0Q\_\{t\}\(c\)\\geq 0\. IfQt​\(c\)=0Q\_\{t\}\(c\)=0, then the continuous nonnegative functions↦Vari∼pt,s\(ηt\)​\(ct​\(i\)\)s\\mapsto\\mathrm\{Var\}\_\{i\\sim p\_\{t,s\}^\{\(\\eta\_\{t\}\)\}\}\(c\_\{t\}\(i\)\)has zero weighted integral on\[0,1\]\[0,1\], so it vanishes for everys∈\[0,1\]s\\in\[0,1\]\. In particular,Vari∼pt​\(ct​\(i\)\)=Vari∼pt,0\(ηt\)​\(ct​\(i\)\)=0\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(c\_\{t\}\(i\)\)=\\mathrm\{Var\}\_\{i\\sim p\_\{t,0\}^\{\(\\eta\_\{t\}\)\}\}\(c\_\{t\}\(i\)\)=0, soct​\(i\)c\_\{t\}\(i\)isptp\_\{t\}\-almost surely constant\. The converse implication is immediate\. ∎

###### Proof of Corollary[2\.3](https://arxiv.org/html/2607.08789#S2.Thmtheorem3)\.\.

Subtract⟨qt,ct⟩\\langle q\_\{t\},c\_\{t\}\\ranglefrom both sides of \([5](https://arxiv.org/html/2607.08789#S2.E5)\) and usemt=−ηt−1​log⁡Ztm\_\{t\}=\-\\eta\_\{t\}^\{\-1\}\\log Z\_\{t\}\. The original\-loss form follows fromct=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}\. Whenηt≡η\\eta\_\{t\}\\equiv\\eta, sum \([8](https://arxiv.org/html/2607.08789#S2.E8)\) overtt\. ∎

### B\.2Proofs for Section[3](https://arxiv.org/html/2607.08789#S3)\(schedules and intrinsic time\)

#### Proof of Proposition[3\.1](https://arxiv.org/html/2607.08789#S3.Thmtheorem1)

###### Proof\.

The standard bounded\-range exponential\-moment bound\[[10](https://arxiv.org/html/2607.08789#bib.bib10), Lemma 2\.2\]states that for any random variableYYsupported on an interval of lengthLLwith𝔼​\[Y\]=0\\mathbb\{E\}\[Y\]=0and anyη∈ℝ\\eta\\in\\mathbb\{R\},

log⁡𝔼​\[eη​Y\]≤η2​L2/8\\log\\mathbb\{E\}\[e^\{\\eta Y\}\]\\leq\\eta^\{2\}L^\{2\}/8Apply this toY=⟨pt,ct⟩−ct​\(It\)Y=\\langle p\_\{t\},c\_\{t\}\\rangle\-c\_\{t\}\(I\_\{t\}\)withIt∼ptI\_\{t\}\\sim p\_\{t\}, which is supported on the interval\[⟨pt,ct⟩−bt,⟨pt,ct⟩−at\]\[\\langle p\_\{t\},c\_\{t\}\\rangle\-b\_\{t\},\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\]of lengthbt−atb\_\{t\}\-a\_\{t\}and has mean zero\. Then

ψt​\(−ηt\)=log⁡𝔼​\[e−ηt​\(ct​\(It\)−⟨pt,ct⟩\)\]=log⁡𝔼​\[eηt​Y\]≤ηt2​\(bt−at\)2/8\\psi\_\{t\}\(\-\\eta\_\{t\}\)=\\log\\mathbb\{E\}\[e^\{\-\\eta\_\{t\}\(c\_\{t\}\(I\_\{t\}\)\-\\langle p\_\{t\},c\_\{t\}\\rangle\)\}\]=\\log\\mathbb\{E\}\[e^\{\\eta\_\{t\}Y\}\]\\leq\\eta\_\{t\}^\{2\}\(b\_\{t\}\-a\_\{t\}\)^\{2\}/8soQt​\(c\)=ηt−2​ψt​\(−ηt\)≤\(bt−at\)2/8Q\_\{t\}\(c\)=\\eta\_\{t\}^\{\-2\}\\psi\_\{t\}\(\-\\eta\_\{t\}\)\\leq\(b\_\{t\}\-a\_\{t\}\)^\{2\}/8\. ∎

#### Proof of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)

###### Proof\.

Write

qt:=Qt​\(c\),St:=∑s=1tqs,q∗:=Q∗T​\(c\)=max1≤t≤T⁡qt,GT:=∑t=1Tηt​qtq\_\{t\}:=Q\_\{t\}\(c\),\\qquad S\_\{t\}:=\\sum\_\{s=1\}^\{t\}q\_\{s\},\\qquad q\_\{\*\}:=Q\_\{\*\}^\{T\}\(c\)=\\max\_\{1\\leq t\\leq T\}q\_\{t\},\\qquad G\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}q\_\{t\}ThenSt=Vt​\(c\)S\_\{t\}=V\_\{t\}\(c\)andST=VT​\(c\)S\_\{T\}=V\_\{T\}\(c\)\. Also,

ηt=f​\(St−1\),f​\(x\):=\{1,if​x≤C2​Γ,C​Γ/x,if​x\>C2​Γ,\\eta\_\{t\}=f\(S\_\{t\-1\}\),\\qquad f\(x\):=\\begin\{cases\}1,&\\text\{if \}x\\leq C^\{2\}\\Gamma,\\\\\[4\.30554pt\] C\\sqrt\{\\Gamma/x\},&\\text\{if \}x\>C^\{2\}\\Gamma,\\end\{cases\}whereffis nonincreasing on\[0,∞\)\[0,\\infty\)\.

For the lower bound, monotonicity offfgivesqt​f​\(St−1\)=∫St−1Stf​\(St−1\)​𝑑x≥∫St−1Stf​\(x\)​𝑑xq\_\{t\}f\(S\_\{t\-1\}\)=\\int\_\{S\_\{t\-1\}\}^\{S\_\{t\}\}f\(S\_\{t\-1\}\)\\,dx\\geq\\int\_\{S\_\{t\-1\}\}^\{S\_\{t\}\}f\(x\)\\,dx\. Summing overttyieldsGT≥∫0STf​\(x\)​𝑑xG\_\{T\}\\geq\\int\_\{0\}^\{S\_\{T\}\}f\(x\)\\,dx\. IfST≤C2​ΓS\_\{T\}\\leq C^\{2\}\\Gamma, then∫0STf​\(x\)​𝑑x=ST\\int\_\{0\}^\{S\_\{T\}\}f\(x\)\\,dx=S\_\{T\}, andST≥2​C​Γ​ST−C2​ΓS\_\{T\}\\geq 2C\\sqrt\{\\Gamma S\_\{T\}\}\-C^\{2\}\\Gammaby\(ST−C​Γ\)2≥0\(\\sqrt\{S\_\{T\}\}\-C\\sqrt\{\\Gamma\}\)^\{2\}\\geq 0\. IfST\>C2​ΓS\_\{T\}\>C^\{2\}\\Gamma, then

∫0STf​\(x\)​𝑑x=C2​Γ\+C​Γ​∫C2​ΓSTx−1/2​𝑑x=2​C​Γ​ST−C2​Γ\\int\_\{0\}^\{S\_\{T\}\}f\(x\)\\,dx=C^\{2\}\\Gamma\+C\\sqrt\{\\Gamma\}\\int\_\{C^\{2\}\\Gamma\}^\{S\_\{T\}\}x^\{\-1/2\}\\,dx=2C\\sqrt\{\\Gamma S\_\{T\}\}\-C^\{2\}\\GammaSo in all casesGT≥2​C​Γ​ST−C2​ΓG\_\{T\}\\geq 2C\\sqrt\{\\Gamma S\_\{T\}\}\-C^\{2\}\\Gamma\.

For the upper bound, use again thatffis nonincreasing\. For everyx∈\[St−1,St\]x\\in\[S\_\{t\-1\},S\_\{t\}\]we havex−qt≤St−1x\-q\_\{t\}\\leq S\_\{t\-1\}, sof​\(St−1\)≤f​\(\(x−qt\)\+\)≤f​\(\(x−q∗\)\+\)f\(S\_\{t\-1\}\)\\leq f\\left\(\(x\-q\_\{t\}\)\_\{\+\}\\right\)\\leq f\\left\(\(x\-q\_\{\*\}\)\_\{\+\}\\right\)\. Therefore

qt​f​\(St−1\)=∫St−1Stf​\(St−1\)​𝑑x≤∫St−1Stf​\(\(x−q∗\)\+\)​𝑑xq\_\{t\}f\(S\_\{t\-1\}\)=\\int\_\{S\_\{t\-1\}\}^\{S\_\{t\}\}f\(S\_\{t\-1\}\)\\,dx\\leq\\int\_\{S\_\{t\-1\}\}^\{S\_\{t\}\}f\\left\(\(x\-q\_\{\*\}\)\_\{\+\}\\right\)\\,dxSumming overttgives

GT≤∫0STf​\(\(x−q∗\)\+\)​𝑑x≤q∗​f​\(0\)\+∫0STf​\(x\)​𝑑x=q∗\+∫0STf​\(x\)​𝑑xG\_\{T\}\\leq\\int\_\{0\}^\{S\_\{T\}\}f\\left\(\(x\-q\_\{\*\}\)\_\{\+\}\\right\)\\,dx\\leq q\_\{\*\}f\(0\)\+\\int\_\{0\}^\{S\_\{T\}\}f\(x\)\\,dx=q\_\{\*\}\+\\int\_\{0\}^\{S\_\{T\}\}f\(x\)\\,dxSince∫0STf​\(x\)​𝑑x≤C2​Γ\+2​C​Γ​ST\\int\_\{0\}^\{S\_\{T\}\}f\(x\)\\,dx\\leq C^\{2\}\\Gamma\+2C\\sqrt\{\\Gamma S\_\{T\}\}, we obtainGT≤C2​Γ\+q∗\+2​C​Γ​STG\_\{T\}\\leq C^\{2\}\\Gamma\+q\_\{\*\}\+2C\\sqrt\{\\Gamma S\_\{T\}\}, which proves \([26](https://arxiv.org/html/2607.08789#S3.E26)\)\.

Equation \([27](https://arxiv.org/html/2607.08789#S3.E27)\) follows by addingDT\+BT​\(ρ\)D\_\{T\}\+B\_\{T\}\(\\rho\)to \([26](https://arxiv.org/html/2607.08789#S3.E26)\) and using \([19](https://arxiv.org/html/2607.08789#S2.E19)\)\. IfKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, thenBT​\(ρ\)≤Γ​ηT−1B\_\{T\}\(\\rho\)\\leq\\Gamma\\eta\_\{T\}^\{\-1\}\. Moreover,

ηT−1=\{1,if​VT−1​\(c\)=0,max⁡\{1,C−1​VT−1​\(c\)Γ\},if​VT−1​\(c\)\>0,\\eta\_\{T\}^\{\-1\}=\\begin\{cases\}1,&\\text\{if \}V\_\{T\-1\}\(c\)=0,\\\\\[4\.30554pt\] \\max\\\!\\left\\\{1,\\,C^\{\-1\}\\sqrt\{\\frac\{V\_\{T\-1\}\(c\)\}\{\\Gamma\}\}\\right\\\},&\\text\{if \}V\_\{T\-1\}\(c\)\>0,\\end\{cases\}soΓ​ηT−1≤Γ\+C−1​Γ​VT​\(c\)\\Gamma\\eta\_\{T\}^\{\-1\}\\leq\\Gamma\+C^\{\-1\}\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\. Combining this with the upper half of \([27](https://arxiv.org/html/2607.08789#S3.E27)\) andDT≤0D\_\{T\}\\leq 0gives \([28](https://arxiv.org/html/2607.08789#S3.E28)\)\. The caseC=1/2C=1/\\sqrt\{2\}is \([29](https://arxiv.org/html/2607.08789#S3.E29)\), whileC=1C=1is the corresponding unoptimized specialization of \([28](https://arxiv.org/html/2607.08789#S3.E28)\)\. ∎

#### Proof of Theorem[3\.3](https://arxiv.org/html/2607.08789#S3.Thmtheorem3)and Corollary[3\.4](https://arxiv.org/html/2607.08789#S3.Thmtheorem4)

###### Proof\.

By definition,

R^Tc​\(ρ\)=∑t=1T\(ct​\(It\)−⟨pt,ct⟩\)\+∑t=1T⟨pt,ct⟩−⟨ρ,CT⟩=MTsam\+RTc​\(ρ\)\\widehat\{R\}\_\{T\}^\{c\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\left\(c\_\{t\}\(I\_\{t\}\)\-\\langle p\_\{t\},c\_\{t\}\\rangle\\right\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,C\_\{T\}\\rangle=M\_\{T\}^\{\\mathrm\{sam\}\}\+R\_\{T\}^\{c\}\(\\rho\)Substituting \([19](https://arxiv.org/html/2607.08789#S2.E19)\) proves \([30](https://arxiv.org/html/2607.08789#S3.E30)\)\. The two\-sided estimate \([31](https://arxiv.org/html/2607.08789#S3.E31)\) is then immediate from Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)\. IfKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, the upper bound \([32](https://arxiv.org/html/2607.08789#S3.E32)\) follows from \([28](https://arxiv.org/html/2607.08789#S3.E28)\)\. Finally,

∑t=1Tℓt​\(It\)−⟨ρ,LT⟩=∑t=1Tct​\(It\)−⟨ρ,CT⟩\+∑t=1T\(⟨ρ,ut⟩−ut​\(It\)\)\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}\(I\_\{t\}\)\-\\langle\\rho,L\_\{T\}\\rangle=\\sum\_\{t=1\}^\{T\}c\_\{t\}\(I\_\{t\}\)\-\\langle\\rho,C\_\{T\}\\rangle\+\\sum\_\{t=1\}^\{T\}\\left\(\\langle\\rho,u\_\{t\}\\rangle\-u\_\{t\}\(I\_\{t\}\)\\right\)which is exactly Theorem[3\.3](https://arxiv.org/html/2607.08789#S3.Thmtheorem3)for the original losses\. ∎

### B\.3Proofs for Section[4](https://arxiv.org/html/2607.08789#S4)\(side information, comparators, and luckiness\)

#### Proof of Corollaries[4\.1](https://arxiv.org/html/2607.08789#S4.Thmtheorem1),[4\.3](https://arxiv.org/html/2607.08789#S4.Thmtheorem3), and[4\.4](https://arxiv.org/html/2607.08789#S4.Thmtheorem4)

###### Proof\.

Fromct=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}we have, for each roundtt,⟨pt,ℓt⟩−⟨ρ,ℓt⟩=⟨pt,ct⟩−⟨ρ,ct⟩\+⟨ρ,ut⟩−⟨pt,ut⟩\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,\\ell\_\{t\}\\rangle=\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle\+\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle\. Summing overttand applying \([19](https://arxiv.org/html/2607.08789#S2.E19)\) gives \([65](https://arxiv.org/html/2607.08789#S4.E65)\)\. If \([68](https://arxiv.org/html/2607.08789#S4.E68)\) holds, thenut​\(i\)=−ηt−1​∑w=1Jtαt,w​log⁡μt,w​\(i\)u\_\{t\}\(i\)=\-\\eta\_\{t\}^\{\-1\}\\sum\_\{w=1\}^\{J\_\{t\}\}\\alpha\_\{t,w\}\\log\\mu\_\{t,w\}\(i\), so

⟨ρ,ut⟩−⟨pt,ut⟩\\displaystyle\\langle\\rho,u\_\{t\}\\rangle\-\\langle p\_\{t\},u\_\{t\}\\rangle=−ηt−1​∑w=1Jtαt,w​∑i=1K\(ρ​\(i\)−pt​\(i\)\)​log⁡μt,w​\(i\)\\displaystyle=\-\\eta\_\{t\}^\{\-1\}\\sum\_\{w=1\}^\{J\_\{t\}\}\\alpha\_\{t,w\}\\sum\_\{i=1\}^\{K\}\\left\(\\rho\(i\)\-p\_\{t\}\(i\)\\right\)\\log\\mu\_\{t,w\}\(i\)=∑w=1Jtαt,wηt​\(H​\(ρ,μt,w\)−H​\(pt,μt,w\)\)\\displaystyle=\\sum\_\{w=1\}^\{J\_\{t\}\}\\frac\{\\alpha\_\{t,w\}\}\{\\eta\_\{t\}\}\\left\(\\mathrm\{H\}\(\\rho,\\mu\_\{t,w\}\)\-\\mathrm\{H\}\(p\_\{t\},\\mu\_\{t,w\}\)\\right\)Summing overttproves \([69](https://arxiv.org/html/2607.08789#S4.E69)\)\. The two\-sided statement \([70](https://arxiv.org/html/2607.08789#S4.E70)\) follows by substituting \([27](https://arxiv.org/html/2607.08789#S3.E27)\) into \([65](https://arxiv.org/html/2607.08789#S4.E65)\), and \([71](https://arxiv.org/html/2607.08789#S4.E71)\) follows from \([28](https://arxiv.org/html/2607.08789#S3.E28)\)\. ∎

#### Proof of Proposition[4\.8](https://arxiv.org/html/2607.08789#S4.Thmtheorem8)

###### Proof\.

Fixtt, letIt∼ptI\_\{t\}\\sim p\_\{t\}, and writeXt:=ct​\(It\)X\_\{t\}:=c\_\{t\}\(I\_\{t\}\)andyt:=⟨ρ,ct⟩y\_\{t\}:=\\langle\\rho,c\_\{t\}\\rangle\. Since bothXtX\_\{t\}andyty\_\{t\}lie in\[0,1\]\[0,1\], the differenceZt:=Xt−ytZ\_\{t\}:=X\_\{t\}\-y\_\{t\}belongs to\[−1,1\]\[\-1,1\]\. Moreover,mt=−ηt−1​log⁡𝔼​\[e−ηt​Xt\]=yt−ηt−1​log⁡𝔼​\[e−ηt​Zt\]m\_\{t\}=\-\\eta\_\{t\}^\{\-1\}\\log\\mathbb\{E\}\[e^\{\-\\eta\_\{t\}X\_\{t\}\}\]=y\_\{t\}\-\\eta\_\{t\}^\{\-1\}\\log\\mathbb\{E\}\[e^\{\-\\eta\_\{t\}Z\_\{t\}\}\]\. Becauseηt≤1\\eta\_\{t\}\\leq 1andZt≥−1Z\_\{t\}\\geq\-1, we have−ηt​Zt≤1\-\\eta\_\{t\}Z\_\{t\}\\leq 1\. The inequalityex≤1\+x\+\(e−2\)​x2e^\{x\}\\leq 1\+x\+\(e\-2\)x^\{2\}forx≤1x\\leq 1therefore gives

𝔼​\[e−ηt​Zt\]\\displaystyle\\mathbb\{E\}\[e^\{\-\\eta\_\{t\}Z\_\{t\}\}\]≤1−ηt​𝔼​\[Zt\]\+\(e−2\)​ηt2​𝔼​\[Zt2\]\\displaystyle\\leq 1\-\\eta\_\{t\}\\mathbb\{E\}\[Z\_\{t\}\]\+\(e\-2\)\\eta\_\{t\}^\{2\}\\mathbb\{E\}\[Z\_\{t\}^\{2\}\]≤exp⁡\(−ηt​𝔼​\[Zt\]\+\(e−2\)​ηt2​𝔼​\[Zt2\]\)\\displaystyle\\leq\\exp\\left\(\-\\eta\_\{t\}\\mathbb\{E\}\[Z\_\{t\}\]\+\(e\-2\)\\eta\_\{t\}^\{2\}\\mathbb\{E\}\[Z\_\{t\}^\{2\}\]\\right\)Substituting into the expression formtm\_\{t\}yieldsmt≥yt\+𝔼​\[Zt\]−\(e−2\)​ηt​𝔼​\[Zt2\]=⟨pt,ct⟩−\(e−2\)​ηt​Ψt​\(ρ\)m\_\{t\}\\geq y\_\{t\}\+\\mathbb\{E\}\[Z\_\{t\}\]\-\(e\-2\)\\eta\_\{t\}\\mathbb\{E\}\[Z\_\{t\}^\{2\}\]=\\langle p\_\{t\},c\_\{t\}\\rangle\-\(e\-2\)\\eta\_\{t\}\\Psi\_\{t\}\(\\rho\)\. Hence

δt​\(c\)=⟨pt,ct⟩−mt≤\(e−2\)​ηt​Ψt​\(ρ\),Qt​\(c\)=δt​\(c\)ηt≤\(e−2\)​Ψt​\(ρ\)\\delta\_\{t\}\(c\)=\\langle p\_\{t\},c\_\{t\}\\rangle\-m\_\{t\}\\leq\(e\-2\)\\eta\_\{t\}\\Psi\_\{t\}\(\\rho\),\\qquad Q\_\{t\}\(c\)=\\frac\{\\delta\_\{t\}\(c\)\}\{\\eta\_\{t\}\}\\leq\(e\-2\)\\Psi\_\{t\}\(\\rho\)which is exactly \([76](https://arxiv.org/html/2607.08789#S4.E76)\)\.

Now assume in addition that\(ηt\)\(\\eta\_\{t\}\)is nonincreasing\. Theorem[2\.8](https://arxiv.org/html/2607.08789#S2.Thmtheorem8)and∑t=1T−1\(At​\(ηt\)−At​\(ηt\+1\)\)≤0\\sum\_\{t=1\}^\{T\-1\}\\left\(A\_\{t\}\(\\eta\_\{t\}\)\-A\_\{t\}\(\\eta\_\{t\+1\}\)\\right\)\\leq 0giveRTc​\(ρ\)≤∑t=1Tϕt​\(ηt\)\+KL​\(ρ∥π\)ηTR\_\{T\}^\{c\}\(\\rho\)\\leq\\sum\_\{t=1\}^\{T\}\\phi\_\{t\}\(\\eta\_\{t\}\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\_\{T\}\}\. Usingϕt​\(ηt\)=ηt​Qt​\(c\)\\phi\_\{t\}\(\\eta\_\{t\}\)=\\eta\_\{t\}Q\_\{t\}\(c\)and the bound just proved,ϕt​\(ηt\)≤\(e−2\)​ηt​Ψt​\(ρ\)for every​t\\phi\_\{t\}\(\\eta\_\{t\}\)\\leq\(e\-2\)\\eta\_\{t\}\\Psi\_\{t\}\(\\rho\)\\qquad\\text\{for every \}t, and summing overttproves \([77](https://arxiv.org/html/2607.08789#S4.E77)\)\. ∎

#### Proof of Theorem[4\.9](https://arxiv.org/html/2607.08789#S4.Thmtheorem9)

###### Proof\.

Leta:=e−2a:=e\-2and writexT:=𝔼​\[RTc​\(ρ\)\]x\_\{T\}:=\\mathbb\{E\}\[R\_\{T\}^\{c\}\(\\rho\)\]\. Since the learning rate is fixed, Proposition[4\.8](https://arxiv.org/html/2607.08789#S4.Thmtheorem8)gives the pathwise boundRTc​\(ρ\)≤η−1​KL​\(ρ∥π\)\+a​η​∑t=1TΨt​\(ρ\)R\_\{T\}^\{c\}\(\\rho\)\\leq\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|\\pi\)\+a\\eta\\sum\_\{t=1\}^\{T\}\\Psi\_\{t\}\(\\rho\)\. Taking expectations yields

xT≤η−1​KL​\(ρ∥π\)\+a​η​∑t=1T𝔼​\[Ψt​\(ρ\)\]x\_\{T\}\\leq\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|\\pi\)\+a\\eta\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\[\\Psi\_\{t\}\(\\rho\)\]\(161\)We now control the second term by the regret itself\. Letℱt−1\\mathcal\{F\}\_\{t\-1\}be the sigma\-field generated by the firstt−1t\-1rounds\. Sinceptp\_\{t\}isℱt−1\\mathcal\{F\}\_\{t\-1\}\-measurable andctc\_\{t\}is independent ofℱt−1\\mathcal\{F\}\_\{t\-1\},

𝔼​\[Ψt​\(ρ\)∣ℱt−1\]\\displaystyle\\mathbb\{E\}\[\\Psi\_\{t\}\(\\rho\)\\mid\\mathcal\{F\}\_\{t\-1\}\]=∑i=1Kpt​\(i\)​𝔼​\[\(ct​\(i\)−⟨ρ,ct⟩\)2\]\\displaystyle=\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)\\,\\mathbb\{E\}\\bigl\[\(c\_\{t\}\(i\)\-\\langle\\rho,c\_\{t\}\\rangle\)^\{2\}\\bigr\]≤κρ​∑i=1Kpt​\(i\)​\(μ​\(i\)−⟨ρ,μ⟩\)by \([78](https://arxiv.org/html/2607.08789#S4.E78)\)\\displaystyle\\leq\\kappa\_\{\\rho\}\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)\\left\(\\mu\(i\)\-\\langle\\rho,\\mu\\rangle\\right\)\\qquad\\text\{by \\eqref\{eq:rho\-massart\}\}=κρ​𝔼​\[⟨pt,ct⟩−⟨ρ,ct⟩∣ℱt−1\]\\displaystyle=\\kappa\_\{\\rho\}\\,\\mathbb\{E\}\\bigl\[\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]Taking expectations and summing overtttherefore gives

∑t=1T𝔼​\[Ψt​\(ρ\)\]≤κρ​xT\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\[\\Psi\_\{t\}\(\\rho\)\]\\leq\\kappa\_\{\\rho\}x\_\{T\}\(162\)Substituting \([162](https://arxiv.org/html/2607.08789#A2.E162)\) into \([161](https://arxiv.org/html/2607.08789#A2.E161)\) yieldsxT≤η−1​KL​\(ρ∥π\)\+a​κρ​η​xTx\_\{T\}\\leq\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|\\pi\)\+a\\kappa\_\{\\rho\}\\eta\\,x\_\{T\}\. Sincea​κρ​η<1a\\kappa\_\{\\rho\}\\eta<1, rearranging proves \([79](https://arxiv.org/html/2607.08789#S4.E79)\)\.

For the explicit choiceη:=min⁡\{1,12​\(e−2\)​κρ\}\\eta:=\\min\\\!\\left\\\{1,\\frac\{1\}\{2\(e\-2\)\\kappa\_\{\\rho\}\}\\right\\\}, we havea​κρ​η≤1/2a\\kappa\_\{\\rho\}\\eta\\leq 1/2, so𝔼​\[RTc​\(ρ\)\]≤2​K​L​\(ρ∥π\)η\\mathbb\{E\}\[R\_\{T\}^\{c\}\(\\rho\)\]\\leq\\frac\{2\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\}\. Ifκρ≤1/\(2​a\)\\kappa\_\{\\rho\}\\leq 1/\(2a\)thenη=1\\eta=1and the last display is at most2​K​L​\(ρ∥π\)2\\mathrm\{KL\}\(\\rho\\\|\\pi\)\. Ifκρ\>1/\(2​a\)\\kappa\_\{\\rho\}\>1/\(2a\)thenη=1/\(2​a​κρ\)\\eta=1/\(2a\\kappa\_\{\\rho\}\)and the last display becomes4​a​κρ​KL​\(ρ∥π\)4a\\kappa\_\{\\rho\}\\mathrm\{KL\}\(\\rho\\\|\\pi\)\. In either case,𝔼​\[RTc​\(ρ\)\]≤2​\(1\+2​a​κρ\)​KL​\(ρ∥π\)\\mathbb\{E\}\[R\_\{T\}^\{c\}\(\\rho\)\]\\leq 2\\left\(1\+2a\\kappa\_\{\\rho\}\\right\)\\mathrm\{KL\}\(\\rho\\\|\\pi\), which is \([80](https://arxiv.org/html/2607.08789#S4.E80)\)\. ∎

#### Proof of Corollary[4\.10](https://arxiv.org/html/2607.08789#S4.Thmtheorem10)

###### Proof\.

Part \(i\) is exactly Theorem[4\.9](https://arxiv.org/html/2607.08789#S4.Thmtheorem9)specialized toρ=δk∗\\rho=\\delta\_\{k^\{\*\}\}, because then \([78](https://arxiv.org/html/2607.08789#S4.E78)\) becomes \([81](https://arxiv.org/html/2607.08789#S4.E81)\) andKL​\(δk∗∥π\)=log⁡\(1/π​\(k∗\)\)\\mathrm\{KL\}\(\\delta\_\{k^\{\*\}\}\\\|\\pi\)=\\log\(1/\\pi\(k^\{\*\}\)\)\.

For part \(ii\), leti≠k∗i\\neq k^\{\*\}\. Sincect​\(i\),ct​\(k∗\)∈\[0,1\]c\_\{t\}\(i\),c\_\{t\}\(k^\{\*\}\)\\in\[0,1\], we have\|ct​\(i\)−ct​\(k∗\)\|≤1\|c\_\{t\}\(i\)\-c\_\{t\}\(k^\{\*\}\)\|\\leq 1and therefore\(ct​\(i\)−ct​\(k∗\)\)2≤1\(c\_\{t\}\(i\)\-c\_\{t\}\(k^\{\*\}\)\)^\{2\}\\leq 1almost surely\. Therefore,𝔼​\[\(ct​\(i\)−ct​\(k∗\)\)2\]≤1≤μ​\(i\)−μ​\(k∗\)dmin\\mathbb\{E\}\\bigl\[\(c\_\{t\}\(i\)\-c\_\{t\}\(k^\{\*\}\)\)^\{2\}\\bigr\]\\leq 1\\leq\\frac\{\\mu\(i\)\-\\mu\(k^\{\*\}\)\}\{d\_\{\\min\}\}\. Fori=k∗i=k^\{\*\}the inequality is trivial because the left\-hand side is0\. Thus \([81](https://arxiv.org/html/2607.08789#S4.E81)\) holds withc∗=1/dminc\_\{\*\}=1/d\_\{\\min\}, and part \(i\) yields the claimed bound\. ∎

#### Proof of Corollary[4\.11](https://arxiv.org/html/2607.08789#S4.Thmtheorem11)

###### Proof\.

Leta:=e−2a:=e\-2and write

xT:=𝔼​\[RTc​\(ρ\)\],VT:=VT​\(c\)=∑t=1TQt​\(c\),Q∗:=Q∗T​\(c\)x\_\{T\}:=\\mathbb\{E\}\\bigl\[R\_\{T\}^\{c\}\(\\rho\)\\bigr\],\\qquad V\_\{T\}:=V\_\{T\}\(c\)=\\sum\_\{t=1\}^\{T\}Q\_\{t\}\(c\),\\qquad Q\_\{\*\}:=Q\_\{\*\}^\{T\}\(c\)FixC\>0C\>0and run the schedule \([25](https://arxiv.org/html/2607.08789#S3.E25)\)\. By \([28](https://arxiv.org/html/2607.08789#S3.E28)\),RTc​\(ρ\)≤\(1\+C2\)​Γ\+Q∗\+\(2​C\+C−1\)​Γ​VTR\_\{T\}^\{c\}\(\\rho\)\\leq\(1\+C^\{2\}\)\\Gamma\+Q\_\{\*\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma V\_\{T\}\}\. Taking expectations and using Jensen’s inequality for the concave square root gives

xT≤\(1\+C2\)​Γ\+𝔼​\[Q∗\]\+\(2​C\+C−1\)​Γ​𝔼​\[VT\]x\_\{T\}\\leq\(1\+C^\{2\}\)\\Gamma\+\\mathbb\{E\}\[Q\_\{\*\}\]\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma\\,\\mathbb\{E\}\[V\_\{T\}\]\}\(163\)Next, Proposition[4\.8](https://arxiv.org/html/2607.08789#S4.Thmtheorem8)impliesQt​\(c\)≤a​Ψt​\(ρ\)pathwise for every​tQ\_\{t\}\(c\)\\leq a\\,\\Psi\_\{t\}\(\\rho\)\\qquad\\text\{pathwise for every \}t\. Hence, conditioning onℱt−1\\mathcal\{F\}\_\{t\-1\}and using \([78](https://arxiv.org/html/2607.08789#S4.E78)\) exactly as in the proof of Theorem[4\.9](https://arxiv.org/html/2607.08789#S4.Thmtheorem9),

𝔼​\[Qt​\(c\)∣ℱt−1\]≤a​𝔼​\[Ψt​\(ρ\)∣ℱt−1\]≤a​κρ​𝔼​\[⟨pt,ct⟩−⟨ρ,ct⟩∣ℱt−1\]\\mathbb\{E\}\[Q\_\{t\}\(c\)\\mid\\mathcal\{F\}\_\{t\-1\}\]\\leq a\\,\\mathbb\{E\}\[\\Psi\_\{t\}\(\\rho\)\\mid\\mathcal\{F\}\_\{t\-1\}\]\\leq a\\kappa\_\{\\rho\}\\,\\mathbb\{E\}\\bigl\[\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]Summing overttgives

𝔼​\[VT\]≤a​κρ​xT\\mathbb\{E\}\[V\_\{T\}\]\\leq a\\kappa\_\{\\rho\}x\_\{T\}\(164\)Combining \([163](https://arxiv.org/html/2607.08789#A2.E163)\) and \([164](https://arxiv.org/html/2607.08789#A2.E164)\) yieldsxT≤\(1\+C2\)​Γ\+𝔼​\[Q∗\]\+\(2​C\+C−1\)​a​κρ​Γ​xTx\_\{T\}\\leq\(1\+C^\{2\}\)\\Gamma\+\\mathbb\{E\}\[Q\_\{\*\}\]\+\(2C\+C^\{\-1\}\)\\sqrt\{a\\kappa\_\{\\rho\}\\Gamma x\_\{T\}\}\. Applying2​u​v≤u2\+v22uv\\leq u^\{2\}\+v^\{2\}withu=xTu=\\sqrt\{x\_\{T\}\}andv=\(2​C\+C−1\)​a​κρ​Γv=\(2C\+C^\{\-1\}\)\\sqrt\{a\\kappa\_\{\\rho\}\\Gamma\}, we obtainxT≤\(1\+C2\)​Γ\+𝔼​\[Q∗\]\+xT2\+\(2​C\+C−1\)22​a​κρ​Γx\_\{T\}\\leq\(1\+C^\{2\}\)\\Gamma\+\\mathbb\{E\}\[Q\_\{\*\}\]\+\\frac\{x\_\{T\}\}\{2\}\+\\frac\{\(2C\+C^\{\-1\}\)^\{2\}\}\{2\}a\\kappa\_\{\\rho\}\\Gamma\. Rearranging provesxT≤2​\(1\+C2\)​Γ\+2​𝔼​\[Q∗\]\+\(2​C\+C−1\)2​a​κρ​Γx\_\{T\}\\leq 2\(1\+C^\{2\}\)\\Gamma\+2\\,\\mathbb\{E\}\[Q\_\{\*\}\]\+\(2C\+C^\{\-1\}\)^\{2\}a\\kappa\_\{\\rho\}\\Gamma, which is \([82](https://arxiv.org/html/2607.08789#S4.E82)\)\. The final displayed bound in Corollary[4\.11](https://arxiv.org/html/2607.08789#S4.Thmtheorem11)then follows from Proposition[3\.1](https://arxiv.org/html/2607.08789#S3.Thmtheorem1)and the choiceC=1/2C=1/\\sqrt\{2\}\. ∎

###### Proof of Corollary[4\.4](https://arxiv.org/html/2607.08789#S4.Thmtheorem4)\.\.

Apply Corollary[4\.1](https://arxiv.org/html/2607.08789#S4.Thmtheorem1)withut=−mtu\_\{t\}=\-m\_\{t\}\. This gives

∑t=1T⟨pt,ℓt⟩−⟨ρ,LT⟩=DT\+BT​\(ρ\)\+∑t=1Tηt​Qt​\(ℓ−m\)\+∑t=1T⟨pt−ρ,mt⟩\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}\\rangle=D\_\{T\}\+B\_\{T\}\(\\rho\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(\\ell\-m\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-\\rho,m\_\{t\}\\ranglewhich is the first display\. The residual\-range bound follows from Proposition[3\.1](https://arxiv.org/html/2607.08789#S3.Thmtheorem1)applied toct=ℓt−mtc\_\{t\}=\\ell\_\{t\}\-m\_\{t\}\. The fixed\-rate statement then usesDT=0D\_\{T\}=0andBT​\(ρ\)≤KL​\(ρ∥π\)/ηB\_\{T\}\(\\rho\)\\leq\\mathrm\{KL\}\(\\rho\\\|\\pi\)/\\eta, while the second\-order\-schedule statement is exactly Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)applied to the same residual sequence\. ∎

### B\.4Proofs for Section[5](https://arxiv.org/html/2607.08789#S5)\(full\-information extensions\)

#### Proof of Proposition[A\.1](https://arxiv.org/html/2607.08789#A1.Thmtheorem1)

###### Proof\.

From \([156](https://arxiv.org/html/2607.08789#A1.E156)\),

log⁡pt​\(yt\)=∑e=1Nαt,e​log⁡πt,e​\(yt\)−log​∑y∈Y∏e=1Nπt,e​\(y\)αt,e\\log p\_\{t\}\(y\_\{t\}\)=\\sum\_\{e=1\}^\{N\}\\alpha\_\{t,e\}\\log\\pi\_\{t,e\}\(y\_\{t\}\)\-\\log\\sum\_\{y\\in Y\}\\prod\_\{e=1\}^\{N\}\\pi\_\{t,e\}\(y\)^\{\\alpha\_\{t,e\}\}and rearranging gives \([157](https://arxiv.org/html/2607.08789#A1.E157)\)\. Ifαt∈Δ​\(\[N\]\)\\alpha\_\{t\}\\in\\Delta\(\[N\]\), then by weighted AM\-GM,∏e=1Nπt,e​\(y\)αt,e≤∑e=1Nαt,e​πt,e​\(y\)\\prod\_\{e=1\}^\{N\}\\pi\_\{t,e\}\(y\)^\{\\alpha\_\{t,e\}\}\\leq\\sum\_\{e=1\}^\{N\}\\alpha\_\{t,e\}\\pi\_\{t,e\}\(y\)for everyyy\. Summing overyyyields

∑y∈Y∏e=1Nπt,e​\(y\)αt,e≤∑e=1Nαt,e​∑y∈Yπt,e​\(y\)=1\\sum\_\{y\\in Y\}\\prod\_\{e=1\}^\{N\}\\pi\_\{t,e\}\(y\)^\{\\alpha\_\{t,e\}\}\\leq\\sum\_\{e=1\}^\{N\}\\alpha\_\{t,e\}\\sum\_\{y\\in Y\}\\pi\_\{t,e\}\(y\)=1so𝒞αt​\(πt,1:N\)≥0\\mathcal\{C\}\_\{\\alpha\_\{t\}\}\(\\pi\_\{t,1:N\}\)\\geq 0\. ∎

#### Proof of Theorem[A\.2](https://arxiv.org/html/2607.08789#A1.Thmtheorem2)

###### Proof\.

By definition,gt​\(αt\)=−log⁡pt​\(yt\)g\_\{t\}\(\\alpha\_\{t\}\)=\-\\log p\_\{t\}\(y\_\{t\}\)\. The online convex optimization guarantee gives

∑t=1T−log⁡pt​\(yt\)=∑t=1Tgt​\(αt\)≤minα∈Δ​\(\[N\]\)​∑t=1Tgt​\(α\)\+RT\\sum\_\{t=1\}^\{T\}\-\\log p\_\{t\}\(y\_\{t\}\)=\\sum\_\{t=1\}^\{T\}g\_\{t\}\(\\alpha\_\{t\}\)\\leq\\min\_\{\\alpha\\in\\Delta\(\[N\]\)\}\\sum\_\{t=1\}^\{T\}g\_\{t\}\(\\alpha\)\+R\_\{T\}Expandinggt​\(α\)g\_\{t\}\(\\alpha\)using Proposition[A\.1](https://arxiv.org/html/2607.08789#A1.Thmtheorem1)yieldsgt​\(α\)=∑e=1Nαe​\(−log⁡πt,e​\(yt\)\)−𝒞α​\(πt,1:N\)g\_\{t\}\(\\alpha\)=\\sum\_\{e=1\}^\{N\}\\alpha\_\{e\}\\left\(\-\\log\\pi\_\{t,e\}\(y\_\{t\}\)\\right\)\-\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{t,1:N\}\), which is exactly \([158](https://arxiv.org/html/2607.08789#A1.E158)\)\. ∎

#### Proof of Proposition[5\.6](https://arxiv.org/html/2607.08789#S5.Thmtheorem6)

###### Proof\.

Using \([90](https://arxiv.org/html/2607.08789#S5.E90)\),

∏w=1Wπt,w​\(i\)αt,w∝exp⁡\(−η​∑w=1Wαt,w​\(M​q~t,w\)i\)\\prod\_\{w=1\}^\{W\}\\pi\_\{t,w\}\(i\)^\{\\alpha\_\{t,w\}\}\\propto\\exp\\\!\\left\(\-\\eta\\sum\_\{w=1\}^\{W\}\\alpha\_\{t,w\}\(M\\widetilde\{q\}\_\{t,w\}\)\_\{i\}\\right\)which is \([91](https://arxiv.org/html/2607.08789#S5.E91)\)\. Ifλt=∑wαt,w\>0\\lambda\_\{t\}=\\sum\_\{w\}\\alpha\_\{t,w\}\>0andq¯t=λt−1​∑wαt,w​q~t,w\\bar\{q\}\_\{t\}=\\lambda\_\{t\}^\{\-1\}\\sum\_\{w\}\\alpha\_\{t,w\}\\widetilde\{q\}\_\{t,w\}, then linearity of matrix multiplication gives∑w=1Wαt,w​\(M​q~t,w\)i=λt​\(M​q¯t\)i\\sum\_\{w=1\}^\{W\}\\alpha\_\{t,w\}\(M\\widetilde\{q\}\_\{t,w\}\)\_\{i\}=\\lambda\_\{t\}\(M\\bar\{q\}\_\{t\}\)\_\{i\}, which yields \([92](https://arxiv.org/html/2607.08789#S5.E92)\)\. ∎

#### Proof of Proposition[5\.7](https://arxiv.org/html/2607.08789#S5.Thmtheorem7)

###### Proof\.

Definef​\(p,α\):=∑w=1Wαw​KL​\(p∥πw\),p∈Δ​\(\[K\]\),α∈Δ​\(\[W\]\)f\(p,\\alpha\):=\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{KL\}\(p\\\|\\pi\_\{w\}\),\\qquad p\\in\\Delta\(\[K\]\),\\ \\alpha\\in\\Delta\(\[W\]\)\. Because eachπw\\pi\_\{w\}has strictly positive coordinates,ffis finite and continuous, convex inpp, and linear inα\\alpha\. Therefore the minimax theorem in\[[69](https://arxiv.org/html/2607.08789#bib.bib69)\]applies:

maxα∈Δ​\(\[W\]\)⁡minp∈Δ​\(\[K\]\)⁡f​\(p,α\)=minp∈Δ​\(\[K\]\)⁡maxα∈Δ​\(\[W\]\)⁡f​\(p,α\)\\max\_\{\\alpha\\in\\Delta\(\[W\]\)\}\\min\_\{p\\in\\Delta\(\[K\]\)\}f\(p,\\alpha\)=\\min\_\{p\\in\\Delta\(\[K\]\)\}\\max\_\{\\alpha\\in\\Delta\(\[W\]\)\}f\(p,\\alpha\)For fixedα\\alpha, Theorem[2\.1](https://arxiv.org/html/2607.08789#S2.Thmtheorem1)givesminp∈Δ​\(\[K\]\)⁡f​\(p,α\)=𝒞α​\(π1:W\)\\min\_\{p\\in\\Delta\(\[K\]\)\}f\(p,\\alpha\)=\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\), with minimizerpα⋆p^\{\\star\}\_\{\\alpha\}\. For fixedpp, linearity inα\\alphagivesmaxα∈Δ​\(\[W\]\)⁡f​\(p,α\)=maxw∈\[W\]⁡KL​\(p∥πw\)\\max\_\{\\alpha\\in\\Delta\(\[W\]\)\}f\(p,\\alpha\)=\\max\_\{w\\in\[W\]\}\\mathrm\{KL\}\(p\\\|\\pi\_\{w\}\)\. Combining the last three displays proves \([93](https://arxiv.org/html/2607.08789#S5.E93)\)\. Ifα⋆\\alpha^\{\\star\}maximizes𝒞α​\(π1:W\)\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\), thenpα⋆⋆p^\{\\star\}\_\{\\alpha^\{\\star\}\}minimizesf​\(⋅,α⋆\)f\(\\cdot,\\alpha^\{\\star\}\)and attains the minimax value, so it is a robust center of geometric\-mixture form\. ∎

###### Proof of Corollary[5\.1](https://arxiv.org/html/2607.08789#S5.Thmtheorem1)\.\.

Corollary[2\.9](https://arxiv.org/html/2607.08789#S2.Thmtheorem9)gives

RTc​\(ρ\)=η​∑t=1TQt​\(c\)\+KL​\(ρ∥π\)−KL​\(ρ∥pT\+1\)ηR\_\{T\}^\{c\}\(\\rho\)=\\eta\\sum\_\{t=1\}^\{T\}Q\_\{t\}\(c\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\}\{\\eta\}By Proposition[3\.1](https://arxiv.org/html/2607.08789#S3.Thmtheorem1),Qt​\(c\)≤\(bt−at\)2/8Q\_\{t\}\(c\)\\leq\(b\_\{t\}\-a\_\{t\}\)^\{2\}/8on every round\. Adding and subtractingη​ST/8\\eta S\_\{T\}/8gives \([83](https://arxiv.org/html/2607.08789#S5.E83)\), and \([84](https://arxiv.org/html/2607.08789#S5.E84)\) is immediate\. ∎

###### Proof of Corollary[5\.2](https://arxiv.org/html/2607.08789#S5.Thmtheorem2)\.\.

SubstituteηΓ\\eta\_\{\\Gamma\}into \([83](https://arxiv.org/html/2607.08789#S5.E83)\)\. SinceΓ/ηΓ\+ηΓ​ST/8=Γ​ST/2\\Gamma/\\eta\_\{\\Gamma\}\+\\eta\_\{\\Gamma\}S\_\{T\}/8=\\sqrt\{\\Gamma S\_\{T\}/2\}andKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma, the displayed inequality follows\. ∎

###### Proof of Proposition[5\.3](https://arxiv.org/html/2607.08789#S5.Thmtheorem3)\.\.

Multiply \([8](https://arxiv.org/html/2607.08789#S2.E8)\) byηt\\eta\_\{t\}and sum overtt\. ∎

###### Proof of Proposition[5\.4](https://arxiv.org/html/2607.08789#S5.Thmtheorem4)\.\.

Expanding the two weighted divergences gives

Dσ​\(ρ∥pt\)−Dσ​\(ρ∥pt\+1\)\+Dσ​\(pt∥pt\+1\)=∑i=1Kσi​\(pt​\(i\)−ρ​\(i\)\)​log⁡pt​\(i\)pt\+1​\(i\)D\_\{\\sigma\}\(\\rho\\\|p\_\{t\}\)\-D\_\{\\sigma\}\(\\rho\\\|p\_\{t\+1\}\)\+D\_\{\\sigma\}\(p\_\{t\}\\\|p\_\{t\+1\}\)=\\sum\_\{i=1\}^\{K\}\\sigma\_\{i\}\\bigl\(p\_\{t\}\(i\)\-\\rho\(i\)\\bigr\)\\log\\frac\{p\_\{t\}\(i\)\}\{p\_\{t\+1\}\(i\)\}From the update,σi​log⁡pt​\(i\)pt\+1​\(i\)=ηt​ℓt​\(i\)\+λt\\sigma\_\{i\}\\log\\frac\{p\_\{t\}\(i\)\}\{p\_\{t\+1\}\(i\)\}=\\eta\_\{t\}\\ell\_\{t\}\(i\)\+\\lambda\_\{t\}for everyii\. Theλt\\lambda\_\{t\}contribution vanishes after summing because bothptp\_\{t\}andρ\\rhohave unit mass, leaving

Dσ​\(ρ∥pt\)−Dσ​\(ρ∥pt\+1\)\+Dσ​\(pt∥pt\+1\)=ηt​\(⟨pt,ℓt⟩−⟨ρ,ℓt⟩\)D\_\{\\sigma\}\(\\rho\\\|p\_\{t\}\)\-D\_\{\\sigma\}\(\\rho\\\|p\_\{t\+1\}\)\+D\_\{\\sigma\}\(p\_\{t\}\\\|p\_\{t\+1\}\)=\\eta\_\{t\}\\bigl\(\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,\\ell\_\{t\}\\rangle\\bigr\)Dividing byηt\\eta\_\{t\}proves \([88](https://arxiv.org/html/2607.08789#S5.E88)\)\. ∎

###### Proof of Corollary[5\.16](https://arxiv.org/html/2607.08789#S5.Thmtheorem16)\.\.

The training\-error claim is the special caseθ=0\\theta=0of the margin\-tail bound, because every mistake has nonpositive margin\. For the tail bound, use𝟏​\{MT​\(i\)≤θ​AT\}≤eθ​AT−MT​\(i\)\\mathbf\{1\}\\\{M\_\{T\}\(i\)\\leq\\theta A\_\{T\}\\\}\\leq e^\{\\theta A\_\{T\}\-M\_\{T\}\(i\)\}pointwise and average overii\. This givesℰT​\(θ\)≤1N​∑i=1Neθ​AT−MT​\(i\)=eθ​AT​ℒTexp\\mathcal\{E\}\_\{T\}\(\\theta\)\\leq\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}e^\{\\theta A\_\{T\}\-M\_\{T\}\(i\)\}=e^\{\\theta A\_\{T\}\}\\mathcal\{L\}\_\{T\}^\{\\exp\}, which is \([113](https://arxiv.org/html/2607.08789#S5.E113)\)\. Substituting the variational identity \([110](https://arxiv.org/html/2607.08789#S5.E110)\) yields \([114](https://arxiv.org/html/2607.08789#S5.E114)\)\. ∎

###### Proof of Corollary[5\.18](https://arxiv.org/html/2607.08789#S5.Thmtheorem18)\.\.

DifferentiateZt​\(α\)Z\_\{t\}\(\\alpha\)and set the derivative to zero to obtain \([121](https://arxiv.org/html/2607.08789#S5.E121)\)\. Substituting that value intoZt​\(α\)Z\_\{t\}\(\\alpha\)gives \([122](https://arxiv.org/html/2607.08789#S5.E122)\)\. The loss identity \([123](https://arxiv.org/html/2607.08789#S5.E123)\) is then just the product formulaℒTexp=∏tZt​\(αt⋆\)\\mathcal\{L\}\_\{T\}^\{\\exp\}=\\prod\_\{t\}Z\_\{t\}\(\\alpha\_\{t\}^\{\\star\}\), while \([124](https://arxiv.org/html/2607.08789#S5.E124)\) and \([125](https://arxiv.org/html/2607.08789#S5.E125)\) follow from Corollary[5\.16](https://arxiv.org/html/2607.08789#S5.Thmtheorem16)\. Finally,−12​log⁡\(1−μt2\)≥μt2/2≥2​γ2\-\\frac\{1\}\{2\}\\log\(1\-\\mu\_\{t\}^\{2\}\)\\geq\\mu\_\{t\}^\{2\}/2\\geq 2\\gamma^\{2\}, which proves \([126](https://arxiv.org/html/2607.08789#S5.E126)\)\.∎

### B\.5Proofs for Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)\(local updates and pressure\)

#### Proof of Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8)

###### Proof\.

Setf​\(η\):=∑i=1Kpt​\(i\)​e−η​\(ct​\(i\)−at\)f\(\\eta\):=\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\-\\eta\(c\_\{t\}\(i\)\-a\_\{t\}\)\}\. ThenffisC∞C^\{\\infty\}onℝ\\mathbb\{R\}, strictly convex becausectc\_\{t\}is nonconstant, and satisfiesf​\(0\)=1f\(0\)=1andf′​\(0\)=−\(⟨pt,ct⟩−at\)<0f^\{\\prime\}\(0\)=\-\(\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\)<0by the hypothesisat<⟨pt,ct⟩a\_\{t\}<\\langle p\_\{t\},c\_\{t\}\\rangle\. Sincemini⁡\(ct​\(i\)−at\)<0\\min\_\{i\}\(c\_\{t\}\(i\)\-a\_\{t\}\)<0by the hypothesisat\>mini⁡ct​\(i\)a\_\{t\}\>\\min\_\{i\}c\_\{t\}\(i\), the exponent−η​\(ct​\(imin\)−at\)→\+∞\-\\eta\(c\_\{t\}\(i\_\{\\min\}\)\-a\_\{t\}\)\\to\+\\inftyasη→∞\\eta\\to\\infty, sof​\(η\)→∞f\(\\eta\)\\to\\infty\. A strictly convexC∞C^\{\\infty\}function withf​\(0\)=1f\(0\)=1,f′​\(0\)<0f^\{\\prime\}\(0\)<0, andlimη→∞f​\(η\)=\+∞\\lim\_\{\\eta\\to\\infty\}f\(\\eta\)=\+\\inftyhas a unique positive minimizerη⋆\\eta^\{\\star\}and a unique positive rootηt\\eta\_\{t\}on\(η⋆,∞\)\(\\eta^\{\\star\},\\infty\)at whichf​\(ηt\)=1f\(\\eta\_\{t\}\)=1\. Thisηt\\eta\_\{t\}is exactly the solution of \([40](https://arxiv.org/html/2607.08789#S3.E40)\), and dividing its logarithm by−ηt\-\\eta\_\{t\}gives \([39](https://arxiv.org/html/2607.08789#S3.E39)\)\.

Ifctc\_\{t\}andata\_\{t\}are simultaneously scaled byb\>0b\>0, thenfb​\(η\)=∑ipt​\(i\)​e−η​\(b​ct​\(i\)−b​at\)=f​\(b​η\)f\_\{b\}\(\\eta\)=\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta\(bc\_\{t\}\(i\)\-ba\_\{t\}\)\}=f\(b\\eta\), so the unique positive root scales asηt/b\\eta\_\{t\}/b\.

Finally, when \([40](https://arxiv.org/html/2607.08789#S3.E40)\) holds, the update based on the shifted scoresct−at​𝟏c\_\{t\}\-a\_\{t\}\\mathbf\{1\}ispt\+1​\(i\)=pt​\(i\)​e−ηt​\(ct​\(i\)−at\)p\_\{t\+1\}\(i\)=p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}\(c\_\{t\}\(i\)\-a\_\{t\}\)\}\. Because \([40](https://arxiv.org/html/2607.08789#S3.E40)\) implieseηt​at=1/∑jpt​\(j\)​e−ηt​ct​\(j\)e^\{\\eta\_\{t\}a\_\{t\}\}=1/\\sum\_\{j\}p\_\{t\}\(j\)e^\{\-\\eta\_\{t\}c\_\{t\}\(j\)\}, this is the same normalized update aspt\+1​\(i\)∝pt​\(i\)​e−ηt​ct​\(i\)p\_\{t\+1\}\(i\)\\propto p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}\. Thereforelog⁡pt​\(i\)pt\+1​\(i\)=ηt​\(ct​\(i\)−at\)\\log\\frac\{p\_\{t\}\(i\)\}\{p\_\{t\+1\}\(i\)\}=\\eta\_\{t\}\(c\_\{t\}\(i\)\-a\_\{t\}\), and multiplying byρ​\(i\)\\rho\(i\)and summing overiiproves \([41](https://arxiv.org/html/2607.08789#S3.E41)\)\. ∎

#### Proof of Corollary[3\.9](https://arxiv.org/html/2607.08789#S3.Thmtheorem9)

###### Proof\.

Immediate specialization of Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8)toat=0a\_\{t\}=0: the feasibility conditionmini⁡ct​\(i\)<0<⟨pt,ct⟩\\min\_\{i\}c\_\{t\}\(i\)<0<\\langle p\_\{t\},c\_\{t\}\\rangleis the hypothesis, and \([40](https://arxiv.org/html/2607.08789#S3.E40)\) then reads∑ipt​\(i\)​e−ηt​ct​\(i\)=1\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}=1\. ∎

#### Proof of Proposition[3\.11](https://arxiv.org/html/2607.08789#S3.Thmtheorem11)

###### Proof\.

At a common temperatureη\\eta, the local recursionpt\+1\(η\)​\(i\)∝pt\(η\)​\(i\)​e−η​ct​\(i\)p\_\{t\+1\}^\{\(\\eta\)\}\(i\)\\propto p\_\{t\}^\{\(\\eta\)\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}with initial distributionπ\\piunrolls to

pt\(η\)​\(i\)=π​\(i\)​e−η​∑s=1t−1cs​\(i\)∑jπ​\(j\)​e−η​∑s=1t−1cs​\(j\)=π​\(i\)​e−η​Ct−1​\(i\)∑jπ​\(j\)​e−η​Ct−1​\(j\)p\_\{t\}^\{\(\\eta\)\}\(i\)=\\frac\{\\pi\(i\)e^\{\-\\eta\\sum\_\{s=1\}^\{t\-1\}c\_\{s\}\(i\)\}\}\{\\sum\_\{j\}\\pi\(j\)e^\{\-\\eta\\sum\_\{s=1\}^\{t\-1\}c\_\{s\}\(j\)\}\}=\\frac\{\\pi\(i\)e^\{\-\\eta C\_\{t\-1\}\(i\)\}\}\{\\sum\_\{j\}\\pi\(j\)e^\{\-\\eta C\_\{t\-1\}\(j\)\}\}which is identical to the prior\-retempered update \([11](https://arxiv.org/html/2607.08789#S2.E11)\) at the same fixedη\\eta\. Equation \([46](https://arxiv.org/html/2607.08789#S3.E46)\) is then exactly Corollary[2\.9](https://arxiv.org/html/2607.08789#S2.Thmtheorem9)withψt\\psi\_\{t\}replaced byψt\(η\)\\psi\_\{t\}^\{\(\\eta\)\}\(i\.e\., evaluated onpt\(η\)p\_\{t\}^\{\(\\eta\)\}\)\. ∎

#### Proof of Corollary[3\.12](https://arxiv.org/html/2607.08789#S3.Thmtheorem12)

###### Proof\.

Start from \([46](https://arxiv.org/html/2607.08789#S3.E46)\), add and subtract∑t=1Tvt​\(η\)\\sum\_\{t=1\}^\{T\}v\_\{t\}\(\\eta\):

RTc,\(η\)​\(ρ\)=∑t=1Tvt​\(η\)\+∑t=1T\(ψt\(η\)​\(−η\)η−vt​\(η\)\)\+KL​\(ρ∥π\)η−KL​\(ρ∥pT\+1\(η\)\)ηR\_\{T\}^\{c,\(\\eta\)\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}v\_\{t\}\(\\eta\)\+\\sum\_\{t=1\}^\{T\}\\left\(\\frac\{\\psi\_\{t\}^\{\(\\eta\)\}\(\-\\eta\)\}\{\\eta\}\-v\_\{t\}\(\\eta\)\\right\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|\\pi\)\}\{\\eta\}\-\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{\(\\eta\)\}\)\}\{\\eta\}Rearranging and using the hypothesisvt​\(η\)≥ψt\(η\)​\(−η\)/ηv\_\{t\}\(\\eta\)\\geq\\psi\_\{t\}^\{\(\\eta\)\}\(\-\\eta\)/\\etagives \([47](https://arxiv.org/html/2607.08789#S3.E47)\) with nonnegative slackΔTloc​\(ρ,η\)\\Delta\_\{T\}^\{\\mathrm\{loc\}\}\(\\rho,\\eta\)as displayed\. ∎

#### Proof of Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13)

###### Proof\.

Because \([40](https://arxiv.org/html/2607.08789#S3.E40)\) holds at every round,Zt:=∑ipt​\(i\)​e−ηt​ct​\(i\)=e−ηt​atZ\_\{t\}:=\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}=e^\{\-\\eta\_\{t\}a\_\{t\}\}\. Unrolling the recursive updatepT\+1​\(i\)=p1​\(i\)​∏t=1Te−ηt​ct​\(i\)/Ztp\_\{T\+1\}\(i\)=p\_\{1\}\(i\)\\prod\_\{t=1\}^\{T\}e^\{\-\\eta\_\{t\}c\_\{t\}\(i\)\}/Z\_\{t\}gives

pT\+1​\(i\)=p1​\(i\)​exp⁡\(−∑t=1Tηt​ct​\(i\)\+∑t=1Tηt​at\)=p1​\(i\)​e−GT​\(i\)p\_\{T\+1\}\(i\)=p\_\{1\}\(i\)\\exp\\\!\\left\(\-\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}c\_\{t\}\(i\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}a\_\{t\}\\right\)=p\_\{1\}\(i\)e^\{\-G\_\{T\}\(i\)\}which is the first part of \([48](https://arxiv.org/html/2607.08789#S3.E48)\); the second part∑ip1​\(i\)​e−GT​\(i\)=∑ipT\+1​\(i\)=1\\sum\_\{i\}p\_\{1\}\(i\)e^\{\-G\_\{T\}\(i\)\}=\\sum\_\{i\}p\_\{T\+1\}\(i\)=1follows from the fact thatpT\+1∈Δ​\(\[K\]\)p\_\{T\+1\}\\in\\Delta\(\[K\]\)\. Taking logarithms ofpT\+1​\(i\)/p1​\(i\)=e−GT​\(i\)p\_\{T\+1\}\(i\)/p\_\{1\}\(i\)=e^\{\-G\_\{T\}\(i\)\}, multiplying byρ​\(i\)\\rho\(i\), and summing overiiyields

−KL​\(ρ∥p1\)\+KL​\(ρ∥pT\+1\)=−∑iρ​\(i\)​GT​\(i\)=−∑t=1Tηt​\(⟨ρ,ct⟩−at\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{1\}\)\+\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)=\-\\sum\_\{i\}\\rho\(i\)G\_\{T\}\(i\)=\-\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(\\langle\\rho,c\_\{t\}\\rangle\-a\_\{t\}\\bigr\)which is \([49](https://arxiv.org/html/2607.08789#S3.E49)\) after reversing signs\. For \([50](https://arxiv.org/html/2607.08789#S3.E50)\), note thatat=mt​\(ηt\)=μt−ηt−1​ψt​\(−ηt\)a\_\{t\}=m\_\{t\}\(\\eta\_\{t\}\)=\\mu\_\{t\}\-\\eta\_\{t\}^\{\-1\}\\psi\_\{t\}\(\-\\eta\_\{t\}\), soηt​\(μt−at\)=ψt​\(−ηt\)=ηt2​Qtloc​\(c\)\\eta\_\{t\}\(\\mu\_\{t\}\-a\_\{t\}\)=\\psi\_\{t\}\(\-\\eta\_\{t\}\)=\\eta\_\{t\}^\{2\}Q\_\{t\}^\{\\mathrm\{loc\}\}\(c\)\. Summing and combining with \([49](https://arxiv.org/html/2607.08789#S3.E49)\) proves the claim\. ∎

#### Proof of Proposition[3\.14](https://arxiv.org/html/2607.08789#S3.Thmtheorem14)

###### Proof\.

For everyη\>0\\eta\>0and everyii,

pt​\(i\)1−η​rt​\(i\)η=pt​\(i\)1−η​\(pt​\(i\)​e−ct​\(i\)Zt​\(1\)\)η=pt​\(i\)​e−η​ct​\(i\)Zt​\(1\)ηp\_\{t\}\(i\)^\{1\-\\eta\}r\_\{t\}\(i\)^\{\\eta\}=p\_\{t\}\(i\)^\{1\-\\eta\}\\left\(\\frac\{p\_\{t\}\(i\)e^\{\-c\_\{t\}\(i\)\}\}\{Z\_\{t\}\(1\)\}\\right\)^\{\\eta\}=\\frac\{p\_\{t\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}\}\{Z\_\{t\}\(1\)^\{\\eta\}\}Summing overiigives𝒞t​\(η\)=Zt​\(η\)/Zt​\(1\)η\\mathcal\{C\}\_\{t\}\(\\eta\)=Z\_\{t\}\(\\eta\)/Z\_\{t\}\(1\)^\{\\eta\}, which is \([51](https://arxiv.org/html/2607.08789#S3.E51)\)\. Taking logarithms giveslog⁡Zt​\(η\)=η​log⁡Zt​\(1\)\+log⁡𝒞t​\(η\)\\log Z\_\{t\}\(\\eta\)=\\eta\\log Z\_\{t\}\(1\)\+\\log\\mathcal\{C\}\_\{t\}\(\\eta\), hence

δtloc​\(η\)=μt\+η−1​log⁡Zt​\(η\)=μt\+log⁡Zt​\(1\)\+η−1​log⁡𝒞t​\(η\)\\delta\_\{t\}^\{\\mathrm\{loc\}\}\(\\eta\)=\\mu\_\{t\}\+\\eta^\{\-1\}\\log Z\_\{t\}\(\\eta\)=\\mu\_\{t\}\+\\log Z\_\{t\}\(1\)\+\\eta^\{\-1\}\\log\\mathcal\{C\}\_\{t\}\(\\eta\)In particularδtloc​\(1\)=μt\+log⁡Zt​\(1\)\\delta\_\{t\}^\{\\mathrm\{loc\}\}\(1\)=\\mu\_\{t\}\+\\log Z\_\{t\}\(1\)\. Subtracting the last two displays gives the first equality in \([52](https://arxiv.org/html/2607.08789#S3.E52)\)\. For the second equality,log⁡𝒞t​\(η\)=\(η−1\)​Dη​\(rt∥pt\)\\log\\mathcal\{C\}\_\{t\}\(\\eta\)=\(\\eta\-1\)D\_\{\\eta\}\(r\_\{t\}\\\|p\_\{t\}\)follows directly from the definition of the order\-η\\etaRényi divergence, and dividing byη\\etagivesη−1​log⁡𝒞t​\(η\)=−\(1−η\)​Dη​\(rt∥pt\)/η\\eta^\{\-1\}\\log\\mathcal\{C\}\_\{t\}\(\\eta\)=\-\(1\-\\eta\)D\_\{\\eta\}\(r\_\{t\}\\\|p\_\{t\}\)/\\eta\. For0<η<10<\\eta<1, the Rényi divergenceDη​\(rt∥pt\)≥0D\_\{\\eta\}\(r\_\{t\}\\\|p\_\{t\}\)\\geq 0, solog⁡𝒞t​\(η\)≤0\\log\\mathcal\{C\}\_\{t\}\(\\eta\)\\leq 0, i\.e\.,𝒞t​\(η\)∈\(0,1\]\\mathcal\{C\}\_\{t\}\(\\eta\)\\in\(0,1\]\. ∎

#### Proof of Proposition[3\.15](https://arxiv.org/html/2607.08789#S3.Thmtheorem15)

###### Proof\.

From Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13),pT\+1​\(E\)=∑i∈Ep1​\(i\)​e−GT​\(i\)=𝒵T​\(E\)p\_\{T\+1\}\(E\)=\\sum\_\{i\\in E\}p\_\{1\}\(i\)e^\{\-G\_\{T\}\(i\)\}=\\mathcal\{Z\}\_\{T\}\(E\)\. For anyρ∈Δ​\(E\)\\rho\\in\\Delta\(E\), the identity

KL​\(ρ∥pT\+1E\)=∑i∈Eρ​\(i\)​log⁡ρ​\(i\)​pT\+1​\(E\)pT\+1​\(i\)=KL​\(ρ∥pT\+1\)\+log⁡pT\+1​\(E\)\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{E\}\)=\\sum\_\{i\\in E\}\\rho\(i\)\\log\\frac\{\\rho\(i\)p\_\{T\+1\}\(E\)\}\{p\_\{T\+1\}\(i\)\}=\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\+\\log p\_\{T\+1\}\(E\)is immediate frompT\+1E​\(i\)=pT\+1​\(i\)/pT\+1​\(E\)p\_\{T\+1\}^\{E\}\(i\)=p\_\{T\+1\}\(i\)/p\_\{T\+1\}\(E\)fori∈Ei\\in E\. Substituting \([49](https://arxiv.org/html/2607.08789#S3.E49)\) andlog⁡pT\+1​\(E\)=log⁡𝒵T​\(E\)\\log p\_\{T\+1\}\(E\)=\\log\\mathcal\{Z\}\_\{T\}\(E\)gives \([53](https://arxiv.org/html/2607.08789#S3.E53)\)\. Subtracting \([53](https://arxiv.org/html/2607.08789#S3.E53)\) from the identity∑tηt​\(μt−at\)=∑tηt2​Qtloc​\(c\)\\sum\_\{t\}\\eta\_\{t\}\(\\mu\_\{t\}\-a\_\{t\}\)=\\sum\_\{t\}\\eta\_\{t\}^\{2\}Q\_\{t\}^\{\\mathrm\{loc\}\}\(c\)\(which follows fromat=mt​\(ηt\)a\_\{t\}=m\_\{t\}\(\\eta\_\{t\}\)\) yields \([54](https://arxiv.org/html/2607.08789#S3.E54)\)\. Equations \([55](https://arxiv.org/html/2607.08789#S3.E55)\) and \([56](https://arxiv.org/html/2607.08789#S3.E56)\) follow by takingminρ∈Δ​\(E\)\\min\_\{\\rho\\in\\Delta\(E\)\}andsupρ∈Δ​\(E\)\\sup\_\{\\rho\\in\\Delta\(E\)\}of \([53](https://arxiv.org/html/2607.08789#S3.E53)\) and \([54](https://arxiv.org/html/2607.08789#S3.E54)\) respectively\. Nonnegativity ofKL​\(ρ∥pT\+1E\)\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{E\}\), vanishing atρ=pT\+1E\\rho=p\_\{T\+1\}^\{E\}, identifies the extremizer in both cases\. ∎

#### Proof of Corollary[3\.16](https://arxiv.org/html/2607.08789#S3.Thmtheorem16)and Corollary[3\.17](https://arxiv.org/html/2607.08789#S3.Thmtheorem17)

###### Proof\.

For Corollary[3\.16](https://arxiv.org/html/2607.08789#S3.Thmtheorem16): apply Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8)to the centered scoresct=−gtc\_\{t\}=\-g\_\{t\}and targetat=−bta\_\{t\}=\-b\_\{t\}\. The feasibility conditionmini⁡ct​\(i\)<at<⟨pt,ct⟩\\min\_\{i\}c\_\{t\}\(i\)<a\_\{t\}<\\langle p\_\{t\},c\_\{t\}\\rangletranslates to0<bt<maxi⁡gt​\(i\)0<b\_\{t\}<\\max\_\{i\}g\_\{t\}\(i\)\(using⟨pt,gt⟩=0\\langle p\_\{t\},g\_\{t\}\\rangle=0\), which is the hypothesis\. The pressure\-target root equation becomes∑ipt​\(i\)​eηt​gt​\(i\)=eηt​bt\\sum\_\{i\}p\_\{t\}\(i\)e^\{\\eta\_\{t\}g\_\{t\}\(i\)\}=e^\{\\eta\_\{t\}b\_\{t\}\}, i\.e\.,log​∑ipt​\(i\)​eηt​gt​\(i\)=ηt​bt\\log\\sum\_\{i\}p\_\{t\}\(i\)e^\{\\eta\_\{t\}g\_\{t\}\(i\)\}=\\eta\_\{t\}b\_\{t\}, which is \([57](https://arxiv.org/html/2607.08789#S3.E57)\)\. Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13)then givespT\+1​\(i\)=p1​\(i\)​exp⁡\(∑tηt​\(gt​\(i\)−bt\)\)p\_\{T\+1\}\(i\)=p\_\{1\}\(i\)\\exp\(\\sum\_\{t\}\\eta\_\{t\}\(g\_\{t\}\(i\)\-b\_\{t\}\)\)with∑ip1​\(i\)​exp⁡\(∑tηt​\(gt​\(i\)−bt\)\)=1\\sum\_\{i\}p\_\{1\}\(i\)\\exp\(\\sum\_\{t\}\\eta\_\{t\}\(g\_\{t\}\(i\)\-b\_\{t\}\)\)=1, which is \([58](https://arxiv.org/html/2607.08789#S3.E58)\)\. Equation \([49](https://arxiv.org/html/2607.08789#S3.E49)\) \(with the same substitutions\) is \([59](https://arxiv.org/html/2607.08789#S3.E59)\)\. The upper bound \([60](https://arxiv.org/html/2607.08789#S3.E60)\) follows from nonnegativity ofKL​\(ρ∥pT\+1\)\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\.

For Corollary[3\.17](https://arxiv.org/html/2607.08789#S3.Thmtheorem17):ηt​bt=log​∑ipt​\(i\)​eηt​gt​\(i\)=κt​ηt2\\eta\_\{t\}b\_\{t\}=\\log\\sum\_\{i\}p\_\{t\}\(i\)e^\{\\eta\_\{t\}g\_\{t\}\(i\)\}=\\kappa\_\{t\}\\eta\_\{t\}^\{2\}by definition ofκt\\kappa\_\{t\}, so \([57](https://arxiv.org/html/2607.08789#S3.E57)\) is equivalent to \([61](https://arxiv.org/html/2607.08789#S3.E61)\)\. Substitutingbt=κt​ηtb\_\{t\}=\\kappa\_\{t\}\\eta\_\{t\}into \([59](https://arxiv.org/html/2607.08789#S3.E59)\) gives \([62](https://arxiv.org/html/2607.08789#S3.E62)\)\. Ifκ¯t≥κt\\bar\{\\kappa\}\_\{t\}\\geq\\kappa\_\{t\}, then∑tκt​ηt2≤∑tκ¯t​ηt2\\sum\_\{t\}\\kappa\_\{t\}\\eta\_\{t\}^\{2\}\\leq\\sum\_\{t\}\\bar\{\\kappa\}\_\{t\}\\eta\_\{t\}^\{2\}, and dropping the nonnegative termKL​\(ρ∥pT\+1\)\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)from \([62](https://arxiv.org/html/2607.08789#S3.E62)\) gives \([63](https://arxiv.org/html/2607.08789#S3.E63)\)\. ∎

#### Proof of Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)

###### Proof\.

\([41](https://arxiv.org/html/2607.08789#S3.E41)\) rearranged gives

⟨pt,ct⟩−⟨ρ,ct⟩=\(⟨pt,ct⟩−at\)\+\(at−⟨ρ,ct⟩\)=\(⟨pt,ct⟩−at\)\+KL​\(ρ∥pt\)−KL​\(ρ∥pt\+1\)ηt\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho,c\_\{t\}\\rangle=\\bigl\(\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\\bigr\)\+\\bigl\(a\_\{t\}\-\\langle\\rho,c\_\{t\}\\rangle\\bigr\)=\\bigl\(\\langle p\_\{t\},c\_\{t\}\\rangle\-a\_\{t\}\\bigr\)\+\\frac\{\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\-\\mathrm\{KL\}\(\\rho\\\|p\_\{t\+1\}\)\}\{\\eta\_\{t\}\}which is \([42](https://arxiv.org/html/2607.08789#S3.E42)\)\. Summing overttyields \([43](https://arxiv.org/html/2607.08789#S3.E43)\)\. Applying the Abel transform∑t=1Tat​\(Kt−Kt\+1\)=a1​K1−aT​KT\+1\+∑t=2TKt​\(at−at−1\)\\sum\_\{t=1\}^\{T\}a\_\{t\}\(K\_\{t\}\-K\_\{t\+1\}\)=a\_\{1\}K\_\{1\}\-a\_\{T\}K\_\{T\+1\}\+\\sum\_\{t=2\}^\{T\}K\_\{t\}\(a\_\{t\}\-a\_\{t\-1\}\)withat=1/ηta\_\{t\}=1/\\eta\_\{t\}andKt=KL​\(ρ∥pt\)K\_\{t\}=\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)gives \([44](https://arxiv.org/html/2607.08789#S3.E44)\)\. If\(ηt\)\(\\eta\_\{t\}\)is nondecreasing, then1/ηt−1/ηt−1≤01/\\eta\_\{t\}\-1/\\eta\_\{t\-1\}\\leq 0andKL​\(ρ∥pt\)≥0\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\\geq 0, so the last two terms in \([44](https://arxiv.org/html/2607.08789#S3.E44)\) are nonpositive, and dropping both \(as well as the nonnegativeKL​\(ρ∥pT\+1\)/ηT\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)/\\eta\_\{T\}\) proves \([45](https://arxiv.org/html/2607.08789#S3.E45)\)\. ∎

###### Proof of Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)\.\.

Equation \([42](https://arxiv.org/html/2607.08789#S3.E42)\) is just \([41](https://arxiv.org/html/2607.08789#S3.E41)\) rearranged after adding and subtracting⟨pt,ct⟩\\langle p\_\{t\},c\_\{t\}\\rangle\. Summing overttgives \([43](https://arxiv.org/html/2607.08789#S3.E43)\)\. Applying the Abel transform identity∑t=1Tat​\(Kt−Kt\+1\)=a1​K1−aT​KT\+1\+∑t=2TKt​\(at−at−1\)\\sum\_\{t=1\}^\{T\}a\_\{t\}\\left\(K\_\{t\}\-K\_\{t\+1\}\\right\)=a\_\{1\}K\_\{1\}\-a\_\{T\}K\_\{T\+1\}\+\\sum\_\{t=2\}^\{T\}K\_\{t\}\(a\_\{t\}\-a\_\{t\-1\}\)withat=1/ηta\_\{t\}=1/\\eta\_\{t\}andKt=KL​\(ρ∥pt\)K\_\{t\}=\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)yields \([44](https://arxiv.org/html/2607.08789#S3.E44)\)\. If\(ηt\)\(\\eta\_\{t\}\)is nondecreasing, thenat−at−1≤0a\_\{t\}\-a\_\{t\-1\}\\leq 0andKL​\(ρ∥pt\)≥0\\mathrm\{KL\}\(\\rho\\\|p\_\{t\}\)\\geq 0, so the last two terms in \([44](https://arxiv.org/html/2607.08789#S3.E44)\) are nonpositive\. ∎

###### Proof of Proposition[3\.11](https://arxiv.org/html/2607.08789#S3.Thmtheorem11)\.\.

For a common temperatureη\\eta, the local recursion and the prior\-retempered update \([11](https://arxiv.org/html/2607.08789#S2.E11)\) coincide:

pt\(η\)​\(i\)=π​\(i\)​e−η​Ct−1​\(i\)∑jπ​\(j\)​e−η​Ct−1​\(j\)p\_\{t\}^\{\(\\eta\)\}\(i\)=\\frac\{\\pi\(i\)e^\{\-\\eta C\_\{t\-1\}\(i\)\}\}\{\\sum\_\{j\}\\pi\(j\)e^\{\-\\eta C\_\{t\-1\}\(j\)\}\}Equation \([46](https://arxiv.org/html/2607.08789#S3.E46)\) is therefore exactly the fixed\-rate identity \([17](https://arxiv.org/html/2607.08789#S2.E17)\)\. ∎

###### Proof of Corollary[3\.12](https://arxiv.org/html/2607.08789#S3.Thmtheorem12)\.\.

Add and subtract∑t=1Tvt​\(η\)\\sum\_\{t=1\}^\{T\}v\_\{t\}\(\\eta\)in \([46](https://arxiv.org/html/2607.08789#S3.E46)\)\. ∎

###### Proof of Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13)\.\.

From the recursive update and the identityZt=e−ηt​atZ\_\{t\}=e^\{\-\\eta\_\{t\}a\_\{t\}\},

pT\+1​\(i\)=p1​\(i\)​exp⁡\(−∑t=1Tηt​ct​\(i\)\)∏t=1TZt=p1​\(i\)​exp⁡\(−∑t=1Tηt​\(ct​\(i\)−at\)\)p\_\{T\+1\}\(i\)=\\frac\{p\_\{1\}\(i\)\\exp\\\!\\left\(\-\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}c\_\{t\}\(i\)\\right\)\}\{\\prod\_\{t=1\}^\{T\}Z\_\{t\}\}=p\_\{1\}\(i\)\\exp\\\!\\left\(\-\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\bigl\(c\_\{t\}\(i\)\-a\_\{t\}\\bigr\)\\right\)which proves \([48](https://arxiv.org/html/2607.08789#S3.E48)\)\. Taking logarithms, multiplying byρ​\(i\)\\rho\(i\), and summing overiiyields \([49](https://arxiv.org/html/2607.08789#S3.E49)\)\. Sinceat=mt​\(ηt\)=μt−ηt−1​ψt​\(−ηt\)a\_\{t\}=m\_\{t\}\(\\eta\_\{t\}\)=\\mu\_\{t\}\-\\eta\_\{t\}^\{\-1\}\\psi\_\{t\}\(\-\\eta\_\{t\}\), rearranging gives \([50](https://arxiv.org/html/2607.08789#S3.E50)\)\. ∎

###### Proof of Proposition[3\.14](https://arxiv.org/html/2607.08789#S3.Thmtheorem14)\.\.

By the definition ofrtr\_\{t\},

pt​\(i\)1−η​rt​\(i\)η=pt​\(i\)​e−η​ct​\(i\)Zt​\(1\)ηp\_\{t\}\(i\)^\{1\-\\eta\}r\_\{t\}\(i\)^\{\\eta\}=\\frac\{p\_\{t\}\(i\)e^\{\-\\eta c\_\{t\}\(i\)\}\}\{Z\_\{t\}\(1\)^\{\\eta\}\}Summing overiiyields \([51](https://arxiv.org/html/2607.08789#S3.E51)\)\. Taking logarithms and addingμt\\mu\_\{t\}gives the first identity in \([52](https://arxiv.org/html/2607.08789#S3.E52)\)\. Sincelog⁡𝒞t​\(η\)=\(η−1\)​Dη​\(rt∥pt\)\\log\\mathcal\{C\}\_\{t\}\(\\eta\)=\(\\eta\-1\)D\_\{\\eta\}\(r\_\{t\}\\\|p\_\{t\}\), the second follows immediately\. For0<η<10<\\eta<1, nonnegativity of Rényi divergence gives𝒞t​\(η\)≤1\\mathcal\{C\}\_\{t\}\(\\eta\)\\leq 1\. ∎

###### Proof of Proposition[3\.15](https://arxiv.org/html/2607.08789#S3.Thmtheorem15)\.\.

Since𝒵T​\(E\)=pT\+1​\(E\)\\mathcal\{Z\}\_\{T\}\(E\)=p\_\{T\+1\}\(E\)by Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13), for everyρ∈Δ​\(E\)\\rho\\in\\Delta\(E\)one has

KL​\(ρ∥pT\+1E\)=∑i∈Eρ​\(i\)​log⁡ρ​\(i\)pT\+1​\(i\)\+log⁡pT\+1​\(E\)=KL​\(ρ∥pT\+1\)\+log⁡𝒵T​\(E\)\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}^\{E\}\)=\\sum\_\{i\\in E\}\\rho\(i\)\\log\\frac\{\\rho\(i\)\}\{p\_\{T\+1\}\(i\)\}\+\\log p\_\{T\+1\}\(E\)=\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)\+\\log\\mathcal\{Z\}\_\{T\}\(E\)Substituting \([49](https://arxiv.org/html/2607.08789#S3.E49)\) gives \([53](https://arxiv.org/html/2607.08789#S3.E53)\)\. Subtracting \([53](https://arxiv.org/html/2607.08789#S3.E53)\) from∑t=1Tηt​\(μt−at\)=∑t=1Tηt2​Qtloc​\(c\)\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\(\\mu\_\{t\}\-a\_\{t\}\)=\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}^\{2\}Q\_\{t\}^\{\\mathrm\{loc\}\}\(c\)yields \([54](https://arxiv.org/html/2607.08789#S3.E54)\)\. Minimizing \([53](https://arxiv.org/html/2607.08789#S3.E53)\) and maximizing \([54](https://arxiv.org/html/2607.08789#S3.E54)\) overρ∈Δ​\(E\)\\rho\\in\\Delta\(E\)prove \([55](https://arxiv.org/html/2607.08789#S3.E55)\) and \([56](https://arxiv.org/html/2607.08789#S3.E56)\); both extrema are attained atρ=pT\+1E\\rho=p\_\{T\+1\}^\{E\}\. ∎

###### Proof of Corollary[3\.16](https://arxiv.org/html/2607.08789#S3.Thmtheorem16)\.\.

Ifgt≡0g\_\{t\}\\equiv 0, the statements are immediate\. Otherwise apply Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8)and Proposition[3\.13](https://arxiv.org/html/2607.08789#S3.Thmtheorem13)withct=−gtc\_\{t\}=\-g\_\{t\}andat=−bta\_\{t\}=\-b\_\{t\}\. The last claim is immediate from the definition ofgtg\_\{t\}\. ∎

###### Proof of Corollary[3\.17](https://arxiv.org/html/2607.08789#S3.Thmtheorem17)\.\.

Equation \([61](https://arxiv.org/html/2607.08789#S3.E61)\) is just \([57](https://arxiv.org/html/2607.08789#S3.E57)\) rewritten asηt​bt=κt​ηt2\\eta\_\{t\}b\_\{t\}=\\kappa\_\{t\}\\eta\_\{t\}^\{2\}\. Likewise, \([62](https://arxiv.org/html/2607.08789#S3.E62)\) is exactly \([59](https://arxiv.org/html/2607.08789#S3.E59)\) withbt=κt​ηtb\_\{t\}=\\kappa\_\{t\}\\eta\_\{t\}\. Finally, ifκ¯t≥κt\\bar\{\\kappa\}\_\{t\}\\geq\\kappa\_\{t\}, then \([62](https://arxiv.org/html/2607.08789#S3.E62)\) and the nonnegativity ofKL​\(ρ∥pT\+1\)\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)imply \([63](https://arxiv.org/html/2607.08789#S3.E63)\)\. ∎

### B\.6Proofs for Section[6](https://arxiv.org/html/2607.08789#S6)\(partial information with contextual bandits\)

#### Proof of Theorem[6\.4](https://arxiv.org/html/2607.08789#S6.Thmtheorem4)and Corollary[6\.5](https://arxiv.org/html/2607.08789#S6.Thmtheorem5)

###### Proof\.

Condition on an arbitrary realization of the estimated lossesc^1,…,c^T\\hat\{c\}\_\{1\},\\dots,\\hat\{c\}\_\{T\}\. The update \([134](https://arxiv.org/html/2607.08789#S6.E134)\) is exactly the variable\-temperature Bayes recursion of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2), now run on the realized sequencec^t\\hat\{c\}\_\{t\}\. Applying Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)withctc\_\{t\}replaced byc^t\\hat\{c\}\_\{t\}gives \([135](https://arxiv.org/html/2607.08789#S6.E135)\); the fixed\-rate statement is the corresponding fixed\-rate specialization\. Corollary[6\.5](https://arxiv.org/html/2607.08789#S6.Thmtheorem5)follows in the same way from Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2), again applied pathwise to the realized estimated losses\. ∎

#### Proof of Theorem[6\.6](https://arxiv.org/html/2607.08789#S6.Thmtheorem6)

###### Proof\.

By definition,

R~Tc​\(u\)=∑t=1T\(ct​\(At\)−⟨u,ct⟩\)=MTplay\+∑t=1T⟨μt−u,ct⟩\\widetilde\{R\}\_\{T\}^\{c\}\(u\)=\\sum\_\{t=1\}^\{T\}\\left\(c\_\{t\}\(A\_\{t\}\)\-\\langle u,c\_\{t\}\\rangle\\right\)=M\_\{T\}^\{\\mathrm\{play\}\}\+\\sum\_\{t=1\}^\{T\}\\langle\\mu\_\{t\}\-u,c\_\{t\}\\rangleSplit the last term as

∑t=1T⟨μt−u,ct⟩=ΞT\+∑t=1T⟨pt−u,ct⟩=ΞT−MTest​\(u\)\+BiasT​\(u\)\+∑t=1T⟨pt−u,c^t⟩\\sum\_\{t=1\}^\{T\}\\langle\\mu\_\{t\}\-u,c\_\{t\}\\rangle=\\Xi\_\{T\}\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,c\_\{t\}\\rangle=\\Xi\_\{T\}\-M\_\{T\}^\{\\mathrm\{est\}\}\(u\)\+\\mathrm\{Bias\}\_\{T\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}\\rangleApplying \([135](https://arxiv.org/html/2607.08789#S6.E135)\) to the realized sequencec^t\\hat\{c\}\_\{t\}gives \([137](https://arxiv.org/html/2607.08789#S6.E137)\), and substituting \([136](https://arxiv.org/html/2607.08789#S6.E136)\) yields \([138](https://arxiv.org/html/2607.08789#S6.E138)\)\. Equation \([139](https://arxiv.org/html/2607.08789#S6.E139)\) is immediate fromct=ℓt\+utc\_\{t\}=\\ell\_\{t\}\+u\_\{t\}\.

For the martingale statements, sinceAt∼μtA\_\{t\}\\sim\\mu\_\{t\}conditionally on the past,𝔼​\[ct​\(At\)−⟨μt,ct⟩∣ℱt−1\]=0\\mathbb\{E\}\[c\_\{t\}\(A\_\{t\}\)\-\\langle\\mu\_\{t\},c\_\{t\}\\rangle\\mid\\mathcal\{F\}\_\{t\-1\}\]=0, soMTplayM\_\{T\}^\{\\mathrm\{play\}\}is a martingale\. Also,

𝔼​\[⟨pt−u,c^t−c¯t⟩∣ℱt−1\]=⟨pt−u,𝔼​\[c^t∣ℱt−1\]−c¯t⟩=0\\mathbb\{E\}\[\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}\-\\bar\{c\}\_\{t\}\\rangle\\mid\\mathcal\{F\}\_\{t\-1\}\]=\\langle p\_\{t\}\-u,\\mathbb\{E\}\[\\hat\{c\}\_\{t\}\\mid\\mathcal\{F\}\_\{t\-1\}\]\-\\bar\{c\}\_\{t\}\\rangle=0soMTest​\(u\)M\_\{T\}^\{\\mathrm\{est\}\}\(u\)is a martingale\. Taking expectations in \([137](https://arxiv.org/html/2607.08789#S6.E137)\) gives the final display\. ∎

#### Proof of Corollary[6\.7](https://arxiv.org/html/2607.08789#S6.Thmtheorem7)and Propositions[6\.8](https://arxiv.org/html/2607.08789#S6.Thmtheorem8)–[6\.9](https://arxiv.org/html/2607.08789#S6.Thmtheorem9)

###### Proof\.

For Corollary[6\.7](https://arxiv.org/html/2607.08789#S6.Thmtheorem7), conditional onℱt−1\\mathcal\{F\}\_\{t\-1\}and for every armaa,𝔼​\[c^t​\(a\)∣ℱt−1\]=ut​\(a\)\+ℓt​\(a\)​μt​\(a\)μt​\(a\)=ct​\(a\)\\mathbb\{E\}\[\\hat\{c\}\_\{t\}\(a\)\\mid\\mathcal\{F\}\_\{t\-1\}\]=u\_\{t\}\(a\)\+\\frac\{\\ell\_\{t\}\(a\)\\mu\_\{t\}\(a\)\}\{\\mu\_\{t\}\(a\)\}=c\_\{t\}\(a\)\. Thusc¯t=ct\\bar\{c\}\_\{t\}=c\_\{t\}, soBiasT​\(u\)=0\\mathrm\{Bias\}\_\{T\}\(u\)=0, and Theorem[6\.6](https://arxiv.org/html/2607.08789#S6.Thmtheorem6)gives \([140](https://arxiv.org/html/2607.08789#S6.E140)\) and \([141](https://arxiv.org/html/2607.08789#S6.E141)\) immediately\.

For Proposition[6\.8](https://arxiv.org/html/2607.08789#S6.Thmtheorem8), sinceAt∼ptA\_\{t\}\\sim p\_\{t\},

𝔼​\[c^tIX​\(a\)∣ℱt−1\]=ut​\(a\)\+pt​\(a\)​ℓt​\(a\)pt​\(a\)\+γt=ct​\(a\)−dtIX​\(a\)\\mathbb\{E\}\[\\hat\{c\}\_\{t\}^\{\\mathrm\{IX\}\}\(a\)\\mid\\mathcal\{F\}\_\{t\-1\}\]=u\_\{t\}\(a\)\+\\frac\{p\_\{t\}\(a\)\\ell\_\{t\}\(a\)\}\{p\_\{t\}\(a\)\+\\gamma\_\{t\}\}=c\_\{t\}\(a\)\-d\_\{t\}^\{\\mathrm\{IX\}\}\(a\)which provesc¯tIX=ct−dtIX\\bar\{c\}\_\{t\}^\{\\mathrm\{IX\}\}=c\_\{t\}\-d\_\{t\}^\{\\mathrm\{IX\}\}\. ThereforeΞT=0\\Xi\_\{T\}=0and Theorem[6\.6](https://arxiv.org/html/2607.08789#S6.Thmtheorem6)gives \([142](https://arxiv.org/html/2607.08789#S6.E142)\) as well as \([143](https://arxiv.org/html/2607.08789#S6.E143)\)\. SincedtIX≥0d\_\{t\}^\{\\mathrm\{IX\}\}\\geq 0coordinatewise,

BiasTIX​\(u\)=∑t=1T⟨pt−u,dtIX⟩≤∑t=1T⟨pt,dtIX⟩=∑t=1TβtIX\\mathrm\{Bias\}\_\{T\}^\{\\mathrm\{IX\}\}\(u\)=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,d\_\{t\}^\{\\mathrm\{IX\}\}\\rangle\\leq\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},d\_\{t\}^\{\\mathrm\{IX\}\}\\rangle=\\sum\_\{t=1\}^\{T\}\\beta\_\{t\}^\{\\mathrm\{IX\}\}It remains to boundβtIX\\beta\_\{t\}^\{\\mathrm\{IX\}\}\. Becauseℓt​\(a\)≤1\\ell\_\{t\}\(a\)\\leq 1,βtIX≤γt​∑a=1Kpt​\(a\)pt​\(a\)\+γt\\beta\_\{t\}^\{\\mathrm\{IX\}\}\\leq\\gamma\_\{t\}\\sum\_\{a=1\}^\{K\}\\frac\{p\_\{t\}\(a\)\}\{p\_\{t\}\(a\)\+\\gamma\_\{t\}\}\. The functionx↦x/\(x\+γt\)x\\mapsto x/\(x\+\\gamma\_\{t\}\)is concave on\[0,∞\)\[0,\\infty\), so Jensen’s inequality gives1K​∑a=1Kpt​\(a\)pt​\(a\)\+γt≤1/K1/K\+γt=11\+K​γt\\frac\{1\}\{K\}\\sum\_\{a=1\}^\{K\}\\frac\{p\_\{t\}\(a\)\}\{p\_\{t\}\(a\)\+\\gamma\_\{t\}\}\\leq\\frac\{1/K\}\{1/K\+\\gamma\_\{t\}\}=\\frac\{1\}\{1\+K\\gamma\_\{t\}\}\. Multiplying byK​γtK\\gamma\_\{t\}yieldsβtIX≤K​γt/\(1\+K​γt\)\\beta\_\{t\}^\{\\mathrm\{IX\}\}\\leq K\\gamma\_\{t\}/\(1\+K\\gamma\_\{t\}\), and the bound bymin⁡\{1,K​γt\}\\min\\\{1,K\\gamma\_\{t\}\\\}is immediate\.

For Proposition[6\.9](https://arxiv.org/html/2607.08789#S6.Thmtheorem9), conditional onℱt−1\\mathcal\{F\}\_\{t\-1\}and for every armaa,

𝔼​\[c^tm​\(a\)∣ℱt−1\]=ut​\(a\)\+mt​\(a\)\+pt​\(a\)​\(ℓt​\(a\)−mt​\(a\)\)pt​\(a\)=ct​\(a\)\\mathbb\{E\}\[\\hat\{c\}\_\{t\}^\{m\}\(a\)\\mid\\mathcal\{F\}\_\{t\-1\}\]=u\_\{t\}\(a\)\+m\_\{t\}\(a\)\+\\frac\{p\_\{t\}\(a\)\\left\(\\ell\_\{t\}\(a\)\-m\_\{t\}\(a\)\\right\)\}\{p\_\{t\}\(a\)\}=c\_\{t\}\(a\)Thusc¯tm=ct\\bar\{c\}\_\{t\}^\{m\}=c\_\{t\}andΞT=0\\Xi\_\{T\}=0, so Theorem[6\.6](https://arxiv.org/html/2607.08789#S6.Thmtheorem6)gives \([144](https://arxiv.org/html/2607.08789#S6.E144)\) and its two\-sided second\-order specialization\. ∎

#### Proof of Proposition[6\.10](https://arxiv.org/html/2607.08789#S6.Thmtheorem10)

###### Proof\.

Apply Theorem[6\.6](https://arxiv.org/html/2607.08789#S6.Thmtheorem6)with the estimatorc^tG​\-​I​X\\hat\{c\}\_\{t\}^\{G\\text\{\-\}IX\}in place ofc^t\\hat\{c\}\_\{t\}and conditional meanc¯tG​\-​I​X=ct−dtG​\-​I​X\\bar\{c\}\_\{t\}^\{G\\text\{\-\}IX\}=c\_\{t\}\-d\_\{t\}^\{G\\text\{\-\}IX\}\. The bias term in Theorem[6\.6](https://arxiv.org/html/2607.08789#S6.Thmtheorem6)isBiasT​\(u\)=∑t⟨pt−u,ct−c¯tG​\-​I​X⟩=∑t⟨pt−u,dtG​\-​I​X⟩=BiasTG​\-​I​X​\(u\)\\mathrm\{Bias\}\_\{T\}\(u\)=\\sum\_\{t\}\\langle p\_\{t\}\-u,c\_\{t\}\-\\bar\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\\rangle=\\sum\_\{t\}\\langle p\_\{t\}\-u,d\_\{t\}^\{G\\text\{\-\}IX\}\\rangle=\\mathrm\{Bias\}\_\{T\}^\{G\\text\{\-\}IX\}\(u\)\. The estimation martingale is∑t⟨pt−u,c^tG​\-​I​X−c¯tG​\-​I​X⟩=MTG​\-​I​X​\(u\)\\sum\_\{t\}\\langle p\_\{t\}\-u,\\hat\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\-\\bar\{c\}\_\{t\}^\{G\\text\{\-\}IX\}\\rangle=M\_\{T\}^\{G\\text\{\-\}IX\}\(u\)\. The two\-sided envelope follows from applying Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)to the realized estimated sequencec^tG​\-​I​X\\hat\{c\}\_\{t\}^\{G\\text\{\-\}IX\}, exactly as in Corollary[6\.5](https://arxiv.org/html/2607.08789#S6.Thmtheorem5)\. ∎

#### Proof of Proposition[6\.11](https://arxiv.org/html/2607.08789#S6.Thmtheorem11)

###### Proof\.

Apply Theorem[6\.4](https://arxiv.org/html/2607.08789#S6.Thmtheorem4)or Theorem[6\.1](https://arxiv.org/html/2607.08789#S6.Thmtheorem1)to the realized transformed scores, then apply Theorem[6\.6](https://arxiv.org/html/2607.08789#S6.Thmtheorem6)or Theorem[6\.2](https://arxiv.org/html/2607.08789#S6.Thmtheorem2)to relate them back to the target vectorgtg\_\{t\}\. No step in those identities uses boundedness\. ∎

#### Proof of Theorem[3\.5](https://arxiv.org/html/2607.08789#S3.Thmtheorem5)

###### Proof\.

LetXt:=ct​\(It\)−⟨pt,ct⟩X\_\{t\}:=c\_\{t\}\(I\_\{t\}\)\-\\langle p\_\{t\},c\_\{t\}\\rangle\. Then\(Xt\)t=1T\(X\_\{t\}\)\_\{t=1\}^\{T\}is a martingale\-difference sequence,Xt≤1X\_\{t\}\\leq 1almost surely becausect∈\[0,1\]Kc\_\{t\}\\in\[0,1\]^\{K\}, and∑t=1T𝔼​\[Xt2∣ℱt−1\]=∑t=1TVari∼pt​\(ct​\(i\)\)=WT​\(c\)\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\[X\_\{t\}^\{2\}\\mid\\mathcal\{F\}\_\{t\-1\}\]=\\sum\_\{t=1\}^\{T\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(c\_\{t\}\(i\)\)=W\_\{T\}\(c\)\. Freedman’s inequality therefore gives

MTsam=∑t=1TXt≤2​WT​\(c\)​log⁡\(1/δ\)\+13​log⁡\(1/δ\)M\_\{T\}^\{\\mathrm\{sam\}\}=\\sum\_\{t=1\}^\{T\}X\_\{t\}\\leq\\sqrt\{2W\_\{T\}\(c\)\\log\(1/\\delta\)\}\+\\tfrac\{1\}\{3\}\\log\(1/\\delta\)with probability at least1−δ1\-\\delta\. On that event, \([30](https://arxiv.org/html/2607.08789#S3.E30)\) yields \([33](https://arxiv.org/html/2607.08789#S3.E33)\) for everyρ\\rho\. Because the event depends only onMTsamM\_\{T\}^\{\\mathrm\{sam\}\}, the conclusion is simultaneous over all posteriors\. The second\-order version \([34](https://arxiv.org/html/2607.08789#S3.E34)\) follows by combining the same event with \([32](https://arxiv.org/html/2607.08789#S3.E32)\)\. Finally, since a\[0,1\]\[0,1\]\-valued random variable has variance at most1/41/4, one hasWT​\(c\)≤T/4W\_\{T\}\(c\)\\leq T/4\. ∎

#### Proof of Theorem[3\.6](https://arxiv.org/html/2607.08789#S3.Thmtheorem6)

###### Proof\.

For eachtt, letXt:=ct​\(It\)−⟨pt,ct⟩X\_\{t\}:=c\_\{t\}\(I\_\{t\}\)\-\\langle p\_\{t\},c\_\{t\}\\rangle\. ThenMtsam=∑s=1tXsM\_\{t\}^\{\\mathrm\{sam\}\}=\\sum\_\{s=1\}^\{t\}X\_\{s\}and, conditionally onℱt−1\\mathcal\{F\}\_\{t\-1\},

𝔼​\[eλ​Xt∣ℱt−1\]=∑i=1Kpt​\(i\)​eλ​\(ct​\(i\)−⟨pt,ct⟩\)=exp⁡\(ψt​\(λ\)\)\\mathbb\{E\}\\bigl\[e^\{\\lambda X\_\{t\}\}\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]=\\sum\_\{i=1\}^\{K\}p\_\{t\}\(i\)e^\{\\lambda\(c\_\{t\}\(i\)\-\\langle p\_\{t\},c\_\{t\}\\rangle\)\}=\\exp\\\!\\left\(\\psi\_\{t\}\(\\lambda\)\\right\)Therefore

𝔼​\[Zt​\(λ\)∣ℱt−1\]=Zt−1​\(λ\)​𝔼​\[eλ​Xt−ψt​\(λ\)∣ℱt−1\]=Zt−1​\(λ\)\\mathbb\{E\}\\bigl\[Z\_\{t\}\(\\lambda\)\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]=Z\_\{t\-1\}\(\\lambda\)\\mathbb\{E\}\\bigl\[e^\{\\lambda X\_\{t\}\-\\psi\_\{t\}\(\\lambda\)\}\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]=Z\_\{t\-1\}\(\\lambda\)soZt​\(λ\)Z\_\{t\}\(\\lambda\)is a nonnegative martingale\. the nonnegative\-supermartingale maximal inequality therefore yieldsPr⁡\(supt≤TZt​\(λ\)≥δ−1\)≤δ\\Pr\\\!\\left\(\\sup\_\{t\\leq T\}Z\_\{t\}\(\\lambda\)\\geq\\delta^\{\-1\}\\right\)\\leq\\delta\. Equivalently, with probability at least1−δ1\-\\delta,

Mtsam≤log⁡\(1/δ\)\+∑s=1tψs​\(λ\)λfor all​t≤TM\_\{t\}^\{\\mathrm\{sam\}\}\\leq\\frac\{\\log\(1/\\delta\)\+\\sum\_\{s=1\}^\{t\}\\psi\_\{s\}\(\\lambda\)\}\{\\lambda\}\\qquad\\text\{for all \}t\\leq TFor each fixedtt, Theorem[3\.3](https://arxiv.org/html/2607.08789#S3.Thmtheorem3)applied to the truncated sequencec1,…,ctc\_\{1\},\\dots,c\_\{t\}givesR^tc​\(ρ\)=Mtsam\+Dt\+Bt​\(ρ\)\+∑s=1tηs​Qs​\(c\)\\widehat\{R\}\_\{t\}^\{c\}\(\\rho\)=M\_\{t\}^\{\\mathrm\{sam\}\}\+D\_\{t\}\+B\_\{t\}\(\\rho\)\+\\sum\_\{s=1\}^\{t\}\\eta\_\{s\}Q\_\{s\}\(c\)\. Combining the last two displays proves \([35](https://arxiv.org/html/2607.08789#S3.E35)\); because the high\-probability event depends only on the sampling martingale, the conclusion is simultaneous over all posteriors\. Under the second\-order schedule, Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)applied at horizonttgives∑s=1tηs​Qs​\(c\)≤C2​Γ\+Q∗,t​\(c\)\+2​C​Γ​Vt​\(c\)\\sum\_\{s=1\}^\{t\}\\eta\_\{s\}Q\_\{s\}\(c\)\\leq C^\{2\}\\Gamma\+Q\_\{\*,t\}\(c\)\+2C\\sqrt\{\\Gamma V\_\{t\}\(c\)\}, whileBt​\(ρ\)≤Γ​ηt−1≤Γ\+C−1​Γ​Vt​\(c\)B\_\{t\}\(\\rho\)\\leq\\Gamma\\eta\_\{t\}^\{\-1\}\\leq\\Gamma\+C^\{\-1\}\\sqrt\{\\Gamma V\_\{t\}\(c\)\}wheneverKL​\(ρ∥π\)≤Γ\\mathrm\{KL\}\(\\rho\\\|\\pi\)\\leq\\Gamma\. Substituting into \([35](https://arxiv.org/html/2607.08789#S3.E35)\) yields \([36](https://arxiv.org/html/2607.08789#S3.E36)\)\. ∎

#### Proof of Proposition[3\.7](https://arxiv.org/html/2607.08789#S3.Thmtheorem7)

###### Proof\.

Fixε∈\(0,1\)\\varepsilon\\in\(0,1\)\. Chooseγ=γε\>0\\gamma=\\gamma\_\{\\varepsilon\}\>0so small that\(1\+γ\)1/2≤1\+ε/8\(1\+\\gamma\)^\{1/2\}\\leq 1\+\\varepsilon/8, and setvn:=\(1\+γ\)nv\_\{n\}:=\(1\+\\gamma\)^\{n\}forn≥3n\\geq 3\. Let

δn:=1n​\(log⁡n\)2,αn:=log⁡1δn,λn:=2​αnvn\+1\\delta\_\{n\}:=\\frac\{1\}\{n\(\\log n\)^\{2\}\},\\qquad\\alpha\_\{n\}:=\\log\\frac\{1\}\{\\delta\_\{n\}\},\\qquad\\lambda\_\{n\}:=\\sqrt\{\\frac\{2\\alpha\_\{n\}\}\{v\_\{n\+1\}\}\}Since∑n≥3δn<∞\\sum\_\{n\\geq 3\}\\delta\_\{n\}<\\inftyandαn=o​\(vn\+1\)\\alpha\_\{n\}=o\(v\_\{n\+1\}\), we haveλn↓0\\lambda\_\{n\}\\downarrow 0\. For eachn≥3n\\geq 3, let

Fn:=\{Mtsam≤αn\+∑s=1tψs​\(λn\)λn​for all​t≥1\}F\_\{n\}:=\\left\\\{M\_\{t\}^\{\\mathrm\{sam\}\}\\leq\\frac\{\\alpha\_\{n\}\+\\sum\_\{s=1\}^\{t\}\\psi\_\{s\}\(\\lambda\_\{n\}\)\}\{\\lambda\_\{n\}\}\\text\{ for all \}t\\geq 1\\right\\\}By the same nonnegative\-martingale argument used in the proof of Theorem[3\.6](https://arxiv.org/html/2607.08789#S3.Thmtheorem6),Pr⁡\(Fnc\)≤δn\\Pr\(F\_\{n\}^\{c\}\)\\leq\\delta\_\{n\}\. Therefore Borel–Cantelli implies that the eventΩε:=\{Fn​holds for all but finitely many​n\}\\Omega\_\{\\varepsilon\}:=\\\{F\_\{n\}\\text\{ holds for all but finitely many \}n\\\}has probability one\.

Fix a sample path inΩε∩\{Vt⊔→∞\}\\Omega\_\{\\varepsilon\}\\cap\\\{V^\{\\\!\\sqcup\}\_\{t\}\\to\\infty\\\}, and letttbe large enough thatVt⊔≥v3V^\{\\\!\\sqcup\}\_\{t\}\\geq v\_\{3\}\. Choose the uniquen=n​\(t\)≥3n=n\(t\)\\geq 3such thatvn≤Vt⊔<vn\+1v\_\{n\}\\leq V^\{\\\!\\sqcup\}\_\{t\}<v\_\{n\+1\}\. For all sufficiently large suchtt, the eventFnF\_\{n\}holds andλn<1\\lambda\_\{n\}<1\. SinceXs∈\[−1,1\]X\_\{s\}\\in\[\-1,1\]and𝔼​\[Xs∣ℱs−1\]=0\\mathbb\{E\}\[X\_\{s\}\\mid\\mathcal\{F\}\_\{s\-1\}\]=0, the scalar inequalityex≤1\+x\+x2/\(2​\(1−x/3\)\)e^\{x\}\\leq 1\+x\+x^\{2\}/\(2\(1\-x/3\)\)forx<3x<3givesψs​\(λn\)≤λn22​\(1−λn/3\)​Var​\(Xs∣ℱs−1\)\\psi\_\{s\}\(\\lambda\_\{n\}\)\\leq\\frac\{\\lambda\_\{n\}^\{2\}\}\{2\(1\-\\lambda\_\{n\}/3\)\}\\mathrm\{Var\}\(X\_\{s\}\\mid\\mathcal\{F\}\_\{s\-1\}\)\. Summing overs≤ts\\leq tand usingVt⊔<vn\+1V^\{\\\!\\sqcup\}\_\{t\}<v\_\{n\+1\}yields

Mtsam≤αnλn\+λn2​\(1−λn/3\)​Vt⊔≤αn​vn\+12​\(1\+11−λn/3\)M\_\{t\}^\{\\mathrm\{sam\}\}\\leq\\frac\{\\alpha\_\{n\}\}\{\\lambda\_\{n\}\}\+\\frac\{\\lambda\_\{n\}\}\{2\(1\-\\lambda\_\{n\}/3\)\}V^\{\\\!\\sqcup\}\_\{t\}\\leq\\sqrt\{\\frac\{\\alpha\_\{n\}v\_\{n\+1\}\}\{2\}\}\\left\(1\+\\frac\{1\}\{1\-\\lambda\_\{n\}/3\}\\right\)Becauseλn→0\\lambda\_\{n\}\\to 0, for all largennthis is at most

\(1\+ε8\)​2​αn​vn\+1≤\(1\+ε8\)​2​\(1\+γ\)​αn​Vt⊔\\left\(1\+\\frac\{\\varepsilon\}\{8\}\\right\)\\sqrt\{2\\alpha\_\{n\}v\_\{n\+1\}\}\\leq\\left\(1\+\\frac\{\\varepsilon\}\{8\}\\right\)\\sqrt\{2\(1\+\\gamma\)\\alpha\_\{n\}V^\{\\\!\\sqcup\}\_\{t\}\}Nowαn=log⁡n\+2​log⁡log⁡n\\alpha\_\{n\}=\\log n\+2\\log\\log nandlog⁡log⁡Vt⊔≥log⁡log⁡vn=log⁡\(n​log⁡\(1\+γ\)\)=log⁡n\+O​\(1\)\\log\\log V^\{\\\!\\sqcup\}\_\{t\}\\geq\\log\\log v\_\{n\}=\\log\\left\(n\\log\(1\+\\gamma\)\\right\)=\\log n\+O\(1\), soαn/log⁡log⁡Vt⊔→1\\alpha\_\{n\}/\\log\\log V^\{\\\!\\sqcup\}\_\{t\}\\to 1ast→∞t\\to\\infty\. Therefore, for all largetton the chosen path,Mtsam≤\(1\+ε\)​2​Vt⊔​log⁡log⁡Vt⊔M\_\{t\}^\{\\mathrm\{sam\}\}\\leq\(1\+\\varepsilon\)\\sqrt\{2V^\{\\\!\\sqcup\}\_\{t\}\\log\\log V^\{\\\!\\sqcup\}\_\{t\}\}\. Hence for each fixedε\>0\\varepsilon\>0there is a probability\-one eventEεE\_\{\\varepsilon\}on which the last display holds eventually\. DefiningE0:=⋂m≥1E1/mE\_\{0\}:=\\bigcap\_\{m\\geq 1\}E\_\{1/m\}gives a probability\-one event such that onE0∩\{Vt⊔→∞\}E\_\{0\}\\cap\\\{V^\{\\\!\\sqcup\}\_\{t\}\\to\\infty\\\},lim supt→∞Mtsam2​Vt⊔​log⁡log⁡Vt⊔≤1\\limsup\_\{t\\to\\infty\}\\frac\{M\_\{t\}^\{\\mathrm\{sam\}\}\}\{\\sqrt\{2V^\{\\\!\\sqcup\}\_\{t\}\\log\\log V^\{\\\!\\sqcup\}\_\{t\}\}\}\\leq 1, which is \([37](https://arxiv.org/html/2607.08789#S3.E37)\) after applying the same argument to−Xt\-X\_\{t\}on another probability\-one eventE0−E\_\{0\}^\{\-\}to obtain the two\-sided absolute\-value form\. The regret consequence \([38](https://arxiv.org/html/2607.08789#S3.E38)\) then follows from the exact identityR^tc​\(ρ\)−Dt−Bt​\(ρ\)−∑s=1tηs​Qs​\(c\)=Mtsam\\widehat\{R\}\_\{t\}^\{c\}\(\\rho\)\-D\_\{t\}\-B\_\{t\}\(\\rho\)\-\\sum\_\{s=1\}^\{t\}\\eta\_\{s\}Q\_\{s\}\(c\)=M\_\{t\}^\{\\mathrm\{sam\}\}of Theorem[3\.3](https://arxiv.org/html/2607.08789#S3.Thmtheorem3)\. ∎

#### Proof of Theorem[4\.6](https://arxiv.org/html/2607.08789#S4.Thmtheorem6)and Theorem[4\.7](https://arxiv.org/html/2607.08789#S4.Thmtheorem7)

###### Proof\.

For Theorem[4\.6](https://arxiv.org/html/2607.08789#S4.Thmtheorem6), observe that

⟨ρT,CT⟩−∑t=1T⟨ρt,ct⟩=∑t=1T⟨ρT−ρt,ct⟩=∑t=1T−1∑s=1t⟨ρt\+1−ρt,cs⟩=∑t=1T−1⟨ρt\+1−ρt,Ct⟩\\langle\\rho\_\{T\},C\_\{T\}\\rangle\-\\sum\_\{t=1\}^\{T\}\\langle\\rho\_\{t\},c\_\{t\}\\rangle=\\sum\_\{t=1\}^\{T\}\\langle\\rho\_\{T\}\-\\rho\_\{t\},c\_\{t\}\\rangle=\\sum\_\{t=1\}^\{T\-1\}\\sum\_\{s=1\}^\{t\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},c\_\{s\}\\rangle=\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangleTherefore

RTc,dyn​\(ρ1:T\)=∑t=1T⟨pt,ct⟩−⟨ρT,CT⟩\+∑t=1T−1⟨ρt\+1−ρt,Ct⟩=RTc​\(ρT\)\+∑t=1T−1⟨ρt\+1−ρt,Ct⟩R\_\{T\}^\{c,\\mathrm\{dyn\}\}\(\\rho\_\{1:T\}\)=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho\_\{T\},C\_\{T\}\\rangle\+\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangle=R\_\{T\}^\{c\}\(\\rho\_\{T\}\)\+\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangleSubstituting \([19](https://arxiv.org/html/2607.08789#S2.E19)\) gives \([72](https://arxiv.org/html/2607.08789#S4.E72)\), and substituting the two\-sided bound \([27](https://arxiv.org/html/2607.08789#S3.E27)\) gives \([73](https://arxiv.org/html/2607.08789#S4.E73)\)\. Ifct​\(i\)∈\[0,1\]c\_\{t\}\(i\)\\in\[0,1\], then0≤Ct​\(i\)≤t0\\leq C\_\{t\}\(i\)\\leq tfor everyii, so\|⟨ρt\+1−ρt,Ct⟩\|≤2​t​TV​\(ρt\+1,ρt\)\\left\|\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangle\\right\|\\leq 2t\\,\\mathrm\{TV\}\(\\rho\_\{t\+1\},\\rho\_\{t\}\)\. Summing overttproves\|∑t=1T−1⟨ρt\+1−ρt,Ct⟩\|≤2​∑t=1T−1t​TV​\(ρt\+1,ρt\)\\left\|\\sum\_\{t=1\}^\{T\-1\}\\langle\\rho\_\{t\+1\}\-\\rho\_\{t\},C\_\{t\}\\rangle\\right\|\\leq 2\\sum\_\{t=1\}^\{T\-1\}t\\,\\mathrm\{TV\}\(\\rho\_\{t\+1\},\\rho\_\{t\}\), and combining this with \([28](https://arxiv.org/html/2607.08789#S3.E28)\) proves \([74](https://arxiv.org/html/2607.08789#S4.E74)\)\.

For Theorem[4\.7](https://arxiv.org/html/2607.08789#S4.Thmtheorem7), letj:=j​\(ε\)j:=j\(\\varepsilon\)and letρA\\rho\_\{A\}be the uniform distribution onAA\. Sincep¯t=∑m=0Jαt​\(m\)​pt\(m\)\\bar\{p\}\_\{t\}=\\sum\_\{m=0\}^\{J\}\\alpha\_\{t\}\(m\)p\_\{t\}^\{\(m\)\},⟨p¯t,ct⟩=∑m=0Jαt​\(m\)​⟨pt\(m\),ct⟩=⟨αt,mt⟩\\langle\\bar\{p\}\_\{t\},c\_\{t\}\\rangle=\\sum\_\{m=0\}^\{J\}\\alpha\_\{t\}\(m\)\\langle p\_\{t\}^\{\(m\)\},c\_\{t\}\\rangle=\\langle\\alpha\_\{t\},m\_\{t\}\\rangle\. Therefore

∑t=1T⟨p¯t,ct⟩−⟨ρA,CT⟩=∑t=1T⟨αt−ej,mt⟩\+∑t=1T⟨pt\(j\),ct⟩−⟨ρA,CT⟩\\sum\_\{t=1\}^\{T\}\\langle\\bar\{p\}\_\{t\},c\_\{t\}\\rangle\-\\langle\\rho\_\{A\},C\_\{T\}\\rangle=\\sum\_\{t=1\}^\{T\}\\langle\\alpha\_\{t\}\-e\_\{j\},m\_\{t\}\\rangle\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}^\{\(j\)\},c\_\{t\}\\rangle\-\\langle\\rho\_\{A\},C\_\{T\}\\rangleApply \([28](https://arxiv.org/html/2607.08789#S3.E28)\) to the controller problem on the meta\-lossesmtm\_\{t\}with comparatoreje\_\{j\}\. The controller prior is uniform on\[J\+1\]\[J\+1\], soKL​\(ej∥uJ\+1\)=log⁡\(J\+1\)=Γctl\\mathrm\{KL\}\(e\_\{j\}\\\|u\_\{J\+1\}\)=\\log\(J\+1\)=\\Gamma^\{\\mathrm\{ctl\}\}, which gives

∑t=1T⟨αt−ej,mt⟩≤\(1\+C2\)​Γctl\+Q∗,ctlT\+\(2​C\+C−1\)​Γctl​VTctl\\sum\_\{t=1\}^\{T\}\\langle\\alpha\_\{t\}\-e\_\{j\},m\_\{t\}\\rangle\\leq\(1\+C^\{2\}\)\\Gamma^\{\\mathrm\{ctl\}\}\+Q\_\{\*,\\mathrm\{ctl\}\}^\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma^\{\\mathrm\{ctl\}\}V\_\{T\}^\{\\mathrm\{ctl\}\}\}Apply the same theorem to workerjjwith comparatorρA\\rho\_\{A\}\. Because the worker prior is uniform on\[K\]\[K\]and\|A\|≥ε​K\|A\|\\geq\\varepsilon K,KL​\(ρA∥uK\)=log⁡K\|A\|≤log⁡\(1/ε\)≤Γj\\mathrm\{KL\}\(\\rho\_\{A\}\\\|u\_\{K\}\)=\\log\\frac\{K\}\{\|A\|\}\\leq\\log\(1/\\varepsilon\)\\leq\\Gamma\_\{j\}\. Hence

∑t=1T⟨pt\(j\),ct⟩−⟨ρA,CT⟩≤\(1\+C2\)​Γj\+Q∗,jT\+\(2​C\+C−1\)​Γj​VT\(j\)\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}^\{\(j\)\},c\_\{t\}\\rangle\-\\langle\\rho\_\{A\},C\_\{T\}\\rangle\\leq\(1\+C^\{2\}\)\\Gamma\_\{j\}\+Q\_\{\*,j\}^\{T\}\+\(2C\+C^\{\-1\}\)\\sqrt\{\\Gamma\_\{j\}V\_\{T\}^\{\(j\)\}\}Adding the last two displays proves \([75](https://arxiv.org/html/2607.08789#S4.E75)\)\. Finally, becauseε≤e−1\\varepsilon\\leq e^\{\-1\}andΓj=2j\\Gamma\_\{j\}=2^\{j\}is the first dyadic budget at leastlog⁡\(1/ε\)\\log\(1/\\varepsilon\), we haveΓj≤2​log⁡\(1/ε\)\\Gamma\_\{j\}\\leq 2\\log\(1/\\varepsilon\)\. ∎

#### Proof of Theorems[6\.1](https://arxiv.org/html/2607.08789#S6.Thmtheorem1)–[6\.2](https://arxiv.org/html/2607.08789#S6.Thmtheorem2)and Proposition[6\.3](https://arxiv.org/html/2607.08789#S6.Thmtheorem3)

###### Proof\.

For Theorem[6\.1](https://arxiv.org/html/2607.08789#S6.Thmtheorem1), condition on an arbitrary realization of the contexts and of the estimated policy lossesℓ^1,…,ℓ^T\\hat\{\\ell\}\_\{1\},\\dots,\\hat\{\\ell\}\_\{T\}\. The update onΠ\\Piis exactly the variable\-temperature Bayes recursion of Section[2\.2](https://arxiv.org/html/2607.08789#S2.SS2)run on that realized estimate sequence\. Applying Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)to the finite expert setΠ\\Piproves \([127](https://arxiv.org/html/2607.08789#S6.E127)\), and applying Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)on the same realized sequence gives \([128](https://arxiv.org/html/2607.08789#S6.E128)\)\. The displayed upper bound is the corresponding specialization of \([28](https://arxiv.org/html/2607.08789#S3.E28)\)\.

For Theorem[6\.2](https://arxiv.org/html/2607.08789#S6.Thmtheorem2), write

R~TΠ​\(u\)=∑t=1T\(ℓt​\(At\)−⟨pt,ℓtΠ⟩\)\+∑t=1T⟨pt−u,ℓtΠ⟩=MTctx\+∑t=1T⟨pt−u,ℓtΠ⟩\\widetilde\{R\}\_\{T\}^\{\\Pi\}\(u\)=\\sum\_\{t=1\}^\{T\}\\left\(\\ell\_\{t\}\(A\_\{t\}\)\-\\langle p\_\{t\},\\ell\_\{t\}^\{\\Pi\}\\rangle\\right\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\ell\_\{t\}^\{\\Pi\}\\rangle=M\_\{T\}^\{\\mathrm\{ctx\}\}\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\ell\_\{t\}^\{\\Pi\}\\rangleAdd and subtractℓ¯t\\bar\{\\ell\}\_\{t\}andℓ^t\\hat\{\\ell\}\_\{t\}inside the last sum:

∑t=1T⟨pt−u,ℓtΠ⟩=BiasTΠ​\(u\)−M^TΠ​\(u\)\+∑t=1T⟨pt−u,ℓ^t⟩\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\ell\_\{t\}^\{\\Pi\}\\rangle=\\mathrm\{Bias\}\_\{T\}^\{\\Pi\}\(u\)\-\\hat\{M\}\_\{T\}^\{\\Pi\}\(u\)\+\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\hat\{\\ell\}\_\{t\}\\rangleNow apply \([127](https://arxiv.org/html/2607.08789#S6.E127)\) to the realized sequenceℓ^t\\hat\{\\ell\}\_\{t\}to obtain \([130](https://arxiv.org/html/2607.08789#S6.E130)\); substituting \([128](https://arxiv.org/html/2607.08789#S6.E128)\) gives \([131](https://arxiv.org/html/2607.08789#S6.E131)\)\. For the martingale claims, note that

𝔼​\[ℓt​\(At\)−⟨pt,ℓtΠ⟩∣𝒢t\]=∑a=1Aμt​\(a∣xt\)​ℓt​\(a\)−⟨pt,ℓtΠ⟩=0\\mathbb\{E\}\[\\ell\_\{t\}\(A\_\{t\}\)\-\\langle p\_\{t\},\\ell\_\{t\}^\{\\Pi\}\\rangle\\mid\\mathcal\{G\}\_\{t\}\]=\\sum\_\{a=1\}^\{A\}\\mu\_\{t\}\(a\\mid x\_\{t\}\)\\ell\_\{t\}\(a\)\-\\langle p\_\{t\},\\ell\_\{t\}^\{\\Pi\}\\rangle=0because⟨pt,ℓtΠ⟩=∑aμt​\(a∣xt\)​ℓt​\(a\)\\langle p\_\{t\},\\ell\_\{t\}^\{\\Pi\}\\rangle=\\sum\_\{a\}\\mu\_\{t\}\(a\\mid x\_\{t\}\)\\ell\_\{t\}\(a\)\. ThereforeMTctxM\_\{T\}^\{\\mathrm\{ctx\}\}is a martingale after taking iterated expectations\. Likewise,𝔼​\[⟨pt−u,ℓ^t−ℓ¯t⟩∣𝒢t\]=⟨pt−u,𝔼​\[ℓ^t∣𝒢t\]−ℓ¯t⟩=0\\mathbb\{E\}\[\\langle p\_\{t\}\-u,\\hat\{\\ell\}\_\{t\}\-\\bar\{\\ell\}\_\{t\}\\rangle\\mid\\mathcal\{G\}\_\{t\}\]=\\langle p\_\{t\}\-u,\\mathbb\{E\}\[\\hat\{\\ell\}\_\{t\}\\mid\\mathcal\{G\}\_\{t\}\]\-\\bar\{\\ell\}\_\{t\}\\rangle=0, soM^TΠ​\(u\)\\hat\{M\}\_\{T\}^\{\\Pi\}\(u\)is a martingale\.

For Proposition[6\.3](https://arxiv.org/html/2607.08789#S6.Thmtheorem3), condition on𝒢t\\mathcal\{G\}\_\{t\}\. For every policyπ∈Π\\pi\\in\\Pi,

𝔼​\[ℓ^tIX​\(π\)∣𝒢t\]=∑a=1Aμt​\(a∣xt\)​ℓt​\(a\)​𝟏​\{π​\(xt\)=a\}μt​\(a∣xt\)\+γt=μt​\(π​\(xt\)∣xt\)μt​\(π​\(xt\)∣xt\)\+γt​ℓt​\(π​\(xt\)\)\\mathbb\{E\}\[\\hat\{\\ell\}\_\{t\}^\{\\mathrm\{IX\}\}\(\\pi\)\\mid\\mathcal\{G\}\_\{t\}\]=\\sum\_\{a=1\}^\{A\}\\mu\_\{t\}\(a\\mid x\_\{t\}\)\\frac\{\\ell\_\{t\}\(a\)\\mathbf\{1\}\\\{\\pi\(x\_\{t\}\)=a\\\}\}\{\\mu\_\{t\}\(a\\mid x\_\{t\}\)\+\\gamma\_\{t\}\}=\\frac\{\\mu\_\{t\}\(\\pi\(x\_\{t\}\)\\mid x\_\{t\}\)\}\{\\mu\_\{t\}\(\\pi\(x\_\{t\}\)\\mid x\_\{t\}\)\+\\gamma\_\{t\}\}\\,\\ell\_\{t\}\(\\pi\(x\_\{t\}\)\)which is exactlyℓtΠ​\(π\)−dtIX​\(π\)\\ell\_\{t\}^\{\\Pi\}\(\\pi\)\-d\_\{t\}^\{\\mathrm\{IX\}\}\(\\pi\)\. Thereforeℓ¯tIX=ℓtΠ−dtIX\\bar\{\\ell\}\_\{t\}^\{\\mathrm\{IX\}\}=\\ell\_\{t\}^\{\\Pi\}\-d\_\{t\}^\{\\mathrm\{IX\}\}, and Theorem[6\.2](https://arxiv.org/html/2607.08789#S6.Thmtheorem2)gives \([132](https://arxiv.org/html/2607.08789#S6.E132)\) and its two\-sided specialization immediately\. ∎

#### Proof of Proposition[5\.8](https://arxiv.org/html/2607.08789#S5.Thmtheorem8), Corollary[5\.9](https://arxiv.org/html/2607.08789#S5.Thmtheorem9), and Theorem[6\.12](https://arxiv.org/html/2607.08789#S6.Thmtheorem12)

###### Proof\.

For Proposition[5\.8](https://arxiv.org/html/2607.08789#S5.Thmtheorem8), the centered losses satisfy⟨pt,gt⟩=⟨pt,ℓt⟩−⟨pt,ℓt⟩=0\\langle p\_\{t\},g\_\{t\}\\rangle=\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle p\_\{t\},\\ell\_\{t\}\\rangle=0\. Applying Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)to the sequencegtg\_\{t\}and the comparatoreie\_\{i\}gives

∑t=1T\(⟨pt,gt⟩−gt​\(i\)\)=DTg\+BTg​\(ei\)\+∑t=1Tηt​Qt​\(g\)\\sum\_\{t=1\}^\{T\}\\left\(\\langle p\_\{t\},g\_\{t\}\\rangle\-g\_\{t\}\(i\)\\right\)=D\_\{T\}^\{g\}\+B\_\{T\}^\{g\}\(e\_\{i\}\)\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}\(g\)Since⟨pt,gt⟩=0\\langle p\_\{t\},g\_\{t\}\\rangle=0and−gt​\(i\)=⟨pt,ℓt⟩−ℓt​\(i\)\-g\_\{t\}\(i\)=\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\ell\_\{t\}\(i\), this is exactly \([94](https://arxiv.org/html/2607.08789#S5.E94)\)\. The upper bound \([95](https://arxiv.org/html/2607.08789#S5.E95)\) is the specialization of \([28](https://arxiv.org/html/2607.08789#S3.E28)\)\. In the unclipped regime,Qt​\(a​g\)=a2​Qt​\(g\)Q\_\{t\}\(ag\)=a^\{2\}Q\_\{t\}\(g\)andVt​\(a​g\)=a2​Vt​\(g\)V\_\{t\}\(ag\)=a^\{2\}V\_\{t\}\(g\)after replacingηt\\eta\_\{t\}byηt/a\\eta\_\{t\}/a, which is the stated scale\-invariance relation\.

For Corollary[5\.9](https://arxiv.org/html/2607.08789#S5.Thmtheorem9), letRTrow​\(i\):=∑t=1Tpt⊤​M​qt−∑t=1Tei⊤​M​qtR\_\{T\}^\{\\mathrm\{row\}\}\(i\):=\\sum\_\{t=1\}^\{T\}p\_\{t\}^\{\\top\}Mq\_\{t\}\-\\sum\_\{t=1\}^\{T\}e\_\{i\}^\{\\top\}Mq\_\{t\}andRTcol​\(j\):=∑t=1Tpt⊤​M​ej−∑t=1Tpt⊤​M​qtR\_\{T\}^\{\\mathrm\{col\}\}\(j\):=\\sum\_\{t=1\}^\{T\}p\_\{t\}^\{\\top\}Me\_\{j\}\-\\sum\_\{t=1\}^\{T\}p\_\{t\}^\{\\top\}Mq\_\{t\}, where the column player is viewed as minimizing the negated payoff\. The exact identity from Proposition[5\.8](https://arxiv.org/html/2607.08789#S5.Thmtheorem8)gives

RTrow​\(i\)=DTrow\+BTrow​\(ei\)\+PTrow,RTcol​\(j\)=DTcol\+BTcol​\(ej\)\+PTcol\.R\_\{T\}^\{\\mathrm\{row\}\}\(i\)=D\_\{T\}^\{\\mathrm\{row\}\}\+B\_\{T\}^\{\\mathrm\{row\}\}\(e\_\{i\}\)\+P\_\{T\}^\{\\mathrm\{row\}\},\\qquad R\_\{T\}^\{\\mathrm\{col\}\}\(j\)=D\_\{T\}^\{\\mathrm\{col\}\}\+B\_\{T\}^\{\\mathrm\{col\}\}\(e\_\{j\}\)\+P\_\{T\}^\{\\mathrm\{col\}\}\.Since linear optimization over each simplex is attained at an extreme point,

T​\(maxq⁡p¯T⊤​M​q−minp⁡p⊤​M​q¯T\)\\displaystyle T\\left\(\\max\_\{q\}\\bar\{p\}\_\{T\}^\{\\top\}Mq\-\\min\_\{p\}p^\{\\top\}M\\bar\{q\}\_\{T\}\\right\)=maxj​∑t=1Tpt⊤​M​ej−mini​∑t=1Tei⊤​M​qt\\displaystyle=\\max\_\{j\}\\sum\_\{t=1\}^\{T\}p\_\{t\}^\{\\top\}Me\_\{j\}\-\\min\_\{i\}\\sum\_\{t=1\}^\{T\}e\_\{i\}^\{\\top\}Mq\_\{t\}=maxj⁡RTcol​\(j\)\+maxi⁡RTrow​\(i\),\\displaystyle=\\max\_\{j\}R\_\{T\}^\{\\mathrm\{col\}\}\(j\)\+\\max\_\{i\}R\_\{T\}^\{\\mathrm\{row\}\}\(i\),which is \([96](https://arxiv.org/html/2607.08789#S5.E96)\)\. Dividing byTTand bounding the two maxima byUTrowU\_\{T\}^\{\\mathrm\{row\}\}andUTcolU\_\{T\}^\{\\mathrm\{col\}\}proves \([97](https://arxiv.org/html/2607.08789#S5.E97)\)\.

For Theorem[6\.12](https://arxiv.org/html/2607.08789#S6.Thmtheorem12), condition on the past\. For every policyπ∈ℰ\\pi\\in\\mathcal\{E\},𝔼​\[ℓ~t​\(π\)∣ℱt−1\]=γA​ℓt​\(π​\(xt\)\)\\mathbb\{E\}\\bigl\[\\tilde\{\\ell\}\_\{t\}\(\\pi\)\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]=\\frac\{\\gamma\}\{A\}\\,\\ell\_\{t\}\\left\(\\pi\(x\_\{t\}\)\\right\)\. Also,

𝔼​\[ℓt​\(At\)∣ℱt−1\]=γA​∑a=1Aℓt​\(a\)\+\(1−γ\)​∑π∈ℰpt​\(π\)​ℓt​\(π​\(xt\)\)\\mathbb\{E\}\\bigl\[\\ell\_\{t\}\(A\_\{t\}\)\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]=\\frac\{\\gamma\}\{A\}\\sum\_\{a=1\}^\{A\}\\ell\_\{t\}\(a\)\+\(1\-\\gamma\)\\sum\_\{\\pi\\in\\mathcal\{E\}\}p\_\{t\}\(\\pi\)\\ell\_\{t\}\\left\(\\pi\(x\_\{t\}\)\\right\)So for any distributionuuonℰ\\mathcal\{E\},

𝔼​\[ℓt​\(At\)∣ℱt−1\]−∑π∈ℰu​\(π\)​ℓt​\(π​\(xt\)\)\\displaystyle\\mathbb\{E\}\\bigl\[\\ell\_\{t\}\(A\_\{t\}\)\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]\-\\sum\_\{\\pi\\in\\mathcal\{E\}\}u\(\\pi\)\\ell\_\{t\}\\left\(\\pi\(x\_\{t\}\)\\right\)=γ​\(1A​∑a=1Aℓt​\(a\)−∑πu​\(π\)​ℓt​\(π​\(xt\)\)\)\\displaystyle=\\gamma\\left\(\\frac\{1\}\{A\}\\sum\_\{a=1\}^\{A\}\\ell\_\{t\}\(a\)\-\\sum\_\{\\pi\}u\(\\pi\)\\ell\_\{t\}\\left\(\\pi\(x\_\{t\}\)\\right\)\\right\)\+\(1−γ\)​∑π∈ℰ\(pt​\(π\)−u​\(π\)\)​ℓt​\(π​\(xt\)\)\\displaystyle\\qquad\+\(1\-\\gamma\)\\sum\_\{\\pi\\in\\mathcal\{E\}\}\(p\_\{t\}\(\\pi\)\-u\(\\pi\)\)\\ell\_\{t\}\\left\(\\pi\(x\_\{t\}\)\\right\)≤γ\+A​\(1−γ\)γ​∑π∈ℰ\(pt​\(π\)−u​\(π\)\)​𝔼​\[ℓ~t​\(π\)∣ℱt−1\]\\displaystyle\\leq\\gamma\+\\frac\{A\(1\-\\gamma\)\}\{\\gamma\}\\sum\_\{\\pi\\in\\mathcal\{E\}\}\(p\_\{t\}\(\\pi\)\-u\(\\pi\)\)\\,\\mathbb\{E\}\\bigl\[\\tilde\{\\ell\}\_\{t\}\(\\pi\)\\mid\\mathcal\{F\}\_\{t\-1\}\\bigr\]where we used0≤ℓt​\(a\)≤10\\leq\\ell\_\{t\}\(a\)\\leq 1\. Summing overttand taking expectations gives

𝔼​\[∑t=1Tℓt​\(At\)−∑t=1T∑π∈ℰu​\(π\)​ℓt​\(π​\(xt\)\)\]≤γ​T\+A​\(1−γ\)γ​𝔼​\[∑t=1T⟨pt−u,ℓ~t⟩\]\\mathbb\{E\}\\left\[\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}\(A\_\{t\}\)\-\\sum\_\{t=1\}^\{T\}\\sum\_\{\\pi\\in\\mathcal\{E\}\}u\(\\pi\)\\ell\_\{t\}\\left\(\\pi\(x\_\{t\}\)\\right\)\\right\]\\leq\\gamma T\+\\frac\{A\(1\-\\gamma\)\}\{\\gamma\}\\,\\mathbb\{E\}\\left\[\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\}\-u,\\tilde\{\\ell\}\_\{t\}\\rangle\\right\]Choosingu=u\(0\)u=u^\{\(0\)\}proves \([147](https://arxiv.org/html/2607.08789#S6.E147)\), because averaging uniformly over copies ofπ0\\pi\_\{0\}reproduces the loss ofπ0\\pi\_\{0\}\. Choosingu=u⋆u=u^\{\\star\}proves \([148](https://arxiv.org/html/2607.08789#S6.E148)\)\. Finally, apply \([28](https://arxiv.org/html/2607.08789#S3.E28)\) to the experts algorithm onℰ\\mathcal\{E\}with the realized lossesℓ~t\\tilde\{\\ell\}\_\{t\}\. For the uniform prior onℰ\\mathcal\{E\}, the comparatoru\(0\)u^\{\(0\)\}satisfiesKL​\(u\(0\)∥u\|ℰ\|\)=log⁡2\\mathrm\{KL\}\(u^\{\(0\)\}\\\|u\_\{\|\\mathcal\{E\}\|\}\)=\\log 2, while a point massu⋆u^\{\\star\}satisfiesKL​\(u⋆∥u\|ℰ\|\)=log⁡\|ℰ\|\\mathrm\{KL\}\(u^\{\\star\}\\\|u\_\{\|\\mathcal\{E\}\|\}\)=\\log\|\\mathcal\{E\}\|\. Substituting these two complexities into the generic wrapper bounds yields \([149](https://arxiv.org/html/2607.08789#S6.E149)\) and \([150](https://arxiv.org/html/2607.08789#S6.E150)\)\. ∎

#### Proof of Theorem[5\.11](https://arxiv.org/html/2607.08789#S5.Thmtheorem11)and Corollary[5\.12](https://arxiv.org/html/2607.08789#S5.Thmtheorem12)

###### Proof\.

For anyη\>0\\eta\>0and any posterior measureρ≪π\\rho\\ll\\pi,

KL​\(ρ∥qt,η\)=∫Slog⁡\(d​ρd​π​\(x\)​∫Se−η​Ft​\(u\)​π​\(d​u\)e−η​Ft​\(x\)\)​ρ​\(d​x\)=KL​\(ρ∥π\)\+η​∫SFt​𝑑ρ\+log​∫Se−η​Ft​𝑑π\\mathrm\{KL\}\(\\rho\\\|q\_\{t,\\eta\}\)=\\int\_\{S\}\\log\\\!\\left\(\\frac\{d\\rho\}\{d\\pi\}\(x\)\\frac\{\\int\_\{S\}e^\{\-\\eta F\_\{t\}\(u\)\}\\,\\pi\(du\)\}\{e^\{\-\\eta F\_\{t\}\(x\)\}\}\\right\)\\rho\(dx\)=\\mathrm\{KL\}\(\\rho\\\|\\pi\)\+\\eta\\\!\\int\_\{S\}F\_\{t\}\\,d\\rho\+\\log\\\!\\int\_\{S\}e^\{\-\\eta F\_\{t\}\}\\,d\\piRearranging gives the continuous analogue of \([14](https://arxiv.org/html/2607.08789#S2.E14)\),

Atoco​\(η\)\+η−1​KL​\(ρ∥qt,η\)=∫SFt​𝑑ρ\+η−1​KL​\(ρ∥π\)\.A\_\{t\}^\{\\mathrm\{oco\}\}\(\\eta\)\+\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|q\_\{t,\\eta\}\)=\\int\_\{S\}F\_\{t\}\\,d\\rho\+\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|\\pi\)\.Next, by definition ofptp\_\{t\},

Atoco​\(ηt\)−At−1oco​\(ηt\)=−ηt−1​log​∫Se−ηt​ft​\(x\)​pt​\(d​x\)=∫Sft​𝑑pt−ηt​QtocoA\_\{t\}^\{\\mathrm\{oco\}\}\(\\eta\_\{t\}\)\-A\_\{t\-1\}^\{\\mathrm\{oco\}\}\(\\eta\_\{t\}\)=\-\\eta\_\{t\}^\{\-1\}\\log\\\!\\int\_\{S\}e^\{\-\\eta\_\{t\}f\_\{t\}\(x\)\}\\,p\_\{t\}\(dx\)=\\int\_\{S\}f\_\{t\}\\,dp\_\{t\}\-\\eta\_\{t\}Q\_\{t\}^\{\\mathrm\{oco\}\}Summing overttand rewriting the varying\-temperature telescope gives

∑t=1T∫Sft​𝑑pt=AToco​\(ηT\)\+DToco\+∑t=1Tηt​Qtoco\\sum\_\{t=1\}^\{T\}\\int\_\{S\}f\_\{t\}\\,dp\_\{t\}=A\_\{T\}^\{\\mathrm\{oco\}\}\(\\eta\_\{T\}\)\+D\_\{T\}^\{\\mathrm\{oco\}\}\+\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}Q\_\{t\}^\{\\mathrm\{oco\}\}Subtracting∫SFT​𝑑ρ\\int\_\{S\}F\_\{T\}\\,d\\rhoand using\(⋆\)\(\\star\)at timeTTyields \([99](https://arxiv.org/html/2607.08789#S5.E99)\)\. Also,\(⋆\)\(\\star\)shows thatAtoco​\(η\)A\_\{t\}^\{\\mathrm\{oco\}\}\(\\eta\)is the pointwise infimum overρ≪π\\rho\\ll\\piof∫SFt​𝑑ρ\+η−1​KL​\(ρ∥π\)\\int\_\{S\}F\_\{t\}\\,d\\rho\+\\eta^\{\-1\}\\mathrm\{KL\}\(\\rho\\\|\\pi\), so it is nonincreasing inη\\etaand thereforeDToco≤0D\_\{T\}^\{\\mathrm\{oco\}\}\\leq 0for every nonincreasing schedule\. Sincext=∫Sx​pt​\(d​x\)x\_\{t\}=\\int\_\{S\}x\\,p\_\{t\}\(dx\)and eachftf\_\{t\}is convex,ft​\(xt\)≤∫Sft​𝑑ptf\_\{t\}\(x\_\{t\}\)\\leq\\int\_\{S\}f\_\{t\}\\,dp\_\{t\}, which proves \([100](https://arxiv.org/html/2607.08789#S5.E100)\)\. The second\-order bound \([101](https://arxiv.org/html/2607.08789#S5.E101)\) then follows by repeating the proof of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)verbatim withQtocoQ\_\{t\}^\{\\mathrm\{oco\}\}in place ofQt​\(c\)Q\_\{t\}\(c\)andAtocoA\_\{t\}^\{\\mathrm\{oco\}\}in place ofAtA\_\{t\}\.

For Corollary[5\.12](https://arxiv.org/html/2607.08789#S5.Thmtheorem12), letTα​\(y\):=\(1−α\)​x⋆\+α​yT\_\{\\alpha\}\(y\):=\(1\-\\alpha\)x^\{\\star\}\+\\alpha ywithα=ϵ1/d\\alpha=\\epsilon^\{1/d\}, and letρϵ,x⋆:=Tα​\#​π\\rho\_\{\\epsilon,x^\{\\star\}\}:=T\_\{\\alpha\\\#\}\\pi\. BecauseSSis convex,Tα​\(S\)⊆ST\_\{\\alpha\}\(S\)\\subseteq S, and becauseπ\\piis the normalized Lebesgue measure onSS, the Jacobian ofTαT\_\{\\alpha\}isαd=ϵ\\alpha^\{d\}=\\epsilon, so

d​ρϵ,x⋆d​π​\(x\)=ϵ−1​𝟏​\{x∈Tα​\(S\)\},KL​\(ρϵ,x⋆∥π\)=log⁡\(1/ϵ\)\\frac\{d\\rho\_\{\\epsilon,x^\{\\star\}\}\}\{d\\pi\}\(x\)=\\epsilon^\{\-1\}\\mathbf\{1\}\\\{x\\in T\_\{\\alpha\}\(S\)\\\},\\qquad\\mathrm\{KL\}\(\\rho\_\{\\epsilon,x^\{\\star\}\}\\\|\\pi\)=\\log\(1/\\epsilon\)Also, for everyy∈Sy\\in Sand every roundtt, convexity and boundedness offtf\_\{t\}giveft​\(Tα​\(y\)\)≤\(1−α\)​ft​\(x⋆\)\+α​ft​\(y\)≤ft​\(x⋆\)\+αf\_\{t\}\(T\_\{\\alpha\}\(y\)\)\\leq\(1\-\\alpha\)f\_\{t\}\(x^\{\\star\}\)\+\\alpha f\_\{t\}\(y\)\\leq f\_\{t\}\(x^\{\\star\}\)\+\\alpha\. Integrating with respect toπ​\(d​y\)\\pi\(dy\)and summing overttyields∫SFT​\(x\)​ρϵ,x⋆​\(d​x\)≤FT​\(x⋆\)\+T​ϵ1/d\\int\_\{S\}F\_\{T\}\(x\)\\,\\rho\_\{\\epsilon,x^\{\\star\}\}\(dx\)\\leq F\_\{T\}\(x^\{\\star\}\)\+T\\epsilon^\{1/d\}\. Applying \([100](https://arxiv.org/html/2607.08789#S5.E100)\) withρ=ρϵ,x⋆\\rho=\\rho\_\{\\epsilon,x^\{\\star\}\}proves \([102](https://arxiv.org/html/2607.08789#S5.E102)\)\. The second\-order bound \([103](https://arxiv.org/html/2607.08789#S5.E103)\) follows from \([101](https://arxiv.org/html/2607.08789#S5.E101)\) and the identityKL​\(ρϵ,x⋆∥π\)=log⁡\(1/ϵ\)\\mathrm\{KL\}\(\\rho\_\{\\epsilon,x^\{\\star\}\}\\\|\\pi\)=\\log\(1/\\epsilon\)\. Settingϵ=T−d\\epsilon=T^\{\-d\}gives the displayed specialization, andQtoco≤1/8Q\_\{t\}^\{\\mathrm\{oco\}\}\\leq 1/8follows from Proposition[3\.1](https://arxiv.org/html/2607.08789#S3.Thmtheorem1)because eachftf\_\{t\}takes values in\[0,1\]\[0,1\]\. ∎

#### Proof of Theorem[5\.13](https://arxiv.org/html/2607.08789#S5.Thmtheorem13), Corollary[5\.14](https://arxiv.org/html/2607.08789#S5.Thmtheorem14), Theorem[5\.15](https://arxiv.org/html/2607.08789#S5.Thmtheorem15), and Corollary[5\.17](https://arxiv.org/html/2607.08789#S5.Thmtheorem17)

###### Proof\.

Becauseℓt​\(i\)=\(1\+yi​ht​\(xi\)\)/2\\ell\_\{t\}\(i\)=\(1\+y\_\{i\}h\_\{t\}\(x\_\{i\}\)\)/2,⟨pt,ℓt⟩=12\+12​∑i=1Npt​\(i\)​yi​ht​\(xi\)=12\+γt\\langle p\_\{t\},\\ell\_\{t\}\\rangle=\\frac\{1\}\{2\}\+\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{N\}p\_\{t\}\(i\)y\_\{i\}h\_\{t\}\(x\_\{i\}\)=\\frac\{1\}\{2\}\+\\gamma\_\{t\}\. Likewise, sincemi=T−1​∑t=1Tyi​ht​\(xi\)m\_\{i\}=T^\{\-1\}\\sum\_\{t=1\}^\{T\}y\_\{i\}h\_\{t\}\(x\_\{i\}\),LTℓ​\(i\)=∑t=1Tℓt​\(i\)=T2​\(1\+mi\)L\_\{T\}^\{\\ell\}\(i\)=\\sum\_\{t=1\}^\{T\}\\ell\_\{t\}\(i\)=\\frac\{T\}\{2\}\(1\+m\_\{i\}\)\. Therefore, for any posteriorρ\\rho,

RTℓ​\(ρ\)=∑t=1T⟨pt,ℓt⟩−⟨ρ,LTℓ⟩=T​\(12\+γ¯T\)−T2​\(1\+⟨ρ,m⟩\)=T​γ¯T−T2​⟨ρ,m⟩R\_\{T\}^\{\\ell\}\(\\rho\)=\\sum\_\{t=1\}^\{T\}\\langle p\_\{t\},\\ell\_\{t\}\\rangle\-\\langle\\rho,L\_\{T\}^\{\\ell\}\\rangle=T\\\!\\left\(\\frac\{1\}\{2\}\+\\bar\{\\gamma\}\_\{T\}\\right\)\-\\frac\{T\}\{2\}\\left\(1\+\\langle\\rho,m\\rangle\\right\)=T\\bar\{\\gamma\}\_\{T\}\-\\frac\{T\}\{2\}\\langle\\rho,m\\ranglewhich is exactly \([104](https://arxiv.org/html/2607.08789#S5.E104)\)\.

Now letAϵA\_\{\\epsilon\}be a set of⌈ϵ​N⌉\\lceil\\epsilon N\\rceilexamples with the smallest margins, and letm\[ϵ\]m\_\{\[\\epsilon\]\}be the largest margin inside that set\. SinceρAϵ\\rho\_\{A\_\{\\epsilon\}\}is uniform onAϵA\_\{\\epsilon\}, its average margin is at most that largest value:⟨ρAϵ,m⟩≤m\[ϵ\]\\langle\\rho\_\{A\_\{\\epsilon\}\},m\\rangle\\leq m\_\{\[\\epsilon\]\}\. Combining this with \([104](https://arxiv.org/html/2607.08789#S5.E104)\) proves \([105](https://arxiv.org/html/2607.08789#S5.E105)\), and \([106](https://arxiv.org/html/2607.08789#S5.E106)\) is immediate from any boundRTℓ​\(ρAϵ\)≤UT​\(ϵ\)R\_\{T\}^\{\\ell\}\(\\rho\_\{A\_\{\\epsilon\}\}\)\\leq U\_\{T\}\(\\epsilon\)\. If more than anϵ\\epsilonfraction of the examples had margin at mostθ\\theta, then necessarilym\[ϵ\]≤θm\_\{\[\\epsilon\]\}\\leq\\theta; the contrapositive gives the margin\-distribution claim\. Any example misclassified byHTH\_\{T\}has margin at most0, so the training\-error statement follows by settingθ=0\\theta=0\.

For Corollary[5\.14](https://arxiv.org/html/2607.08789#S5.Thmtheorem14), apply Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)to the example lossesℓt\\ell\_\{t\}and the posteriorρAϵ\\rho\_\{A\_\{\\epsilon\}\}to get \([107](https://arxiv.org/html/2607.08789#S5.E107)\)\. Since the prior is uniform,KL​\(ρAϵ∥uN\)=log⁡\(N/\|Aϵ\|\)≤log⁡\(1/ϵ\)\\mathrm\{KL\}\(\\rho\_\{A\_\{\\epsilon\}\}\\\|u\_\{N\}\)=\\log\(N/\|A\_\{\\epsilon\}\|\)\\leq\\log\(1/\\epsilon\), so Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)gives \([108](https://arxiv.org/html/2607.08789#S5.E108)\)\. Finally,Qt​\(ℓ\)≤1/8Q\_\{t\}\(\\ell\)\\leq 1/8follows from Proposition[3\.1](https://arxiv.org/html/2607.08789#S3.Thmtheorem1)\. Hencem\[ϵ\]≥2​γ−O​\(log⁡\(1/ϵ\)T\+log⁡\(1/ϵ\)T\)m\_\{\[\\epsilon\]\}\\geq 2\\gamma\-O\\\!\\left\(\\sqrt\{\\frac\{\\log\(1/\\epsilon\)\}\{T\}\}\+\\frac\{\\log\(1/\\epsilon\)\}\{T\}\\right\)wheneverγt≥γ\\gamma\_\{t\}\\geq\\gamma\. Choosingϵ=e−c​T​γ2\\epsilon=e^\{\-cT\\gamma^\{2\}\}withc\>0c\>0sufficiently small makes the right\-hand side positive, and then the training error is at mostϵ\\epsilon\.

For Theorem[5\.15](https://arxiv.org/html/2607.08789#S5.Thmtheorem15), unrolling the sequential update givespT\+1​\(i\)=uN​\(i\)​e−MT​\(i\)∏t=1TZt=e−MT​\(i\)∑j=1Ne−MT​\(j\)p\_\{T\+1\}\(i\)=\\frac\{u\_\{N\}\(i\)e^\{\-M\_\{T\}\(i\)\}\}\{\\prod\_\{t=1\}^\{T\}Z\_\{t\}\}=\\frac\{e^\{\-M\_\{T\}\(i\)\}\}\{\\sum\_\{j=1\}^\{N\}e^\{\-M\_\{T\}\(j\)\}\}\. Therefore,ℒTexp=N−1​∑ie−MT​\(i\)=∏t=1TZt\\mathcal\{L\}\_\{T\}^\{\\exp\}=N^\{\-1\}\\sum\_\{i\}e^\{\-M\_\{T\}\(i\)\}=\\prod\_\{t=1\}^\{T\}Z\_\{t\}, and for every posteriorρ\\rho,log⁡ρ​\(i\)pT\+1​\(i\)=log⁡ρ​\(i\)uN​\(i\)\+MT​\(i\)\+log⁡ℒTexp\\log\\frac\{\\rho\(i\)\}\{p\_\{T\+1\}\(i\)\}=\\log\\frac\{\\rho\(i\)\}\{u\_\{N\}\(i\)\}\+M\_\{T\}\(i\)\+\\log\\mathcal\{L\}\_\{T\}^\{\\exp\}\. Averaging overi∼ρi\\sim\\rhoyields \([109](https://arxiv.org/html/2607.08789#S5.E109)\)\. Now letAϵA\_\{\\epsilon\}be a set of⌈ϵ​N⌉\\lceil\\epsilon N\\rceilexamples with the smallest weighted marginsMT​\(i\)/ATM\_\{T\}\(i\)/A\_\{T\}, and letρAϵ\\rho\_\{A\_\{\\epsilon\}\}be uniform onAϵA\_\{\\epsilon\}\. Then

⟨ρAϵ,MT⟩≤AT​m\[ϵ\]αandKL​\(ρAϵ∥uN\)=log⁡\(N/\|Aϵ\|\)≤log⁡\(1/ϵ\)\\langle\\rho\_\{A\_\{\\epsilon\}\},M\_\{T\}\\rangle\\leq A\_\{T\}m\_\{\[\\epsilon\]\}^\{\\alpha\}\\quad\\text\{and\}\\quad\\mathrm\{KL\}\(\\rho\_\{A\_\{\\epsilon\}\}\\\|u\_\{N\}\)=\\log\\\!\\left\(N/\|A\_\{\\epsilon\}\|\\right\)\\leq\\log\(1/\\epsilon\)SubstitutingρAϵ\\rho\_\{A\_\{\\epsilon\}\}into \([109](https://arxiv.org/html/2607.08789#S5.E109)\) and dropping the nonnegative termKL​\(ρAϵ∥pT\+1\)\\mathrm\{KL\}\(\\rho\_\{A\_\{\\epsilon\}\}\\\|p\_\{T\+1\}\)proves \([111](https://arxiv.org/html/2607.08789#S5.E111)\)\.

For Corollary[5\.17](https://arxiv.org/html/2607.08789#S5.Thmtheorem17), apply Proposition[3\.8](https://arxiv.org/html/2607.08789#S3.Thmtheorem8)to the signed margin scoresgtg\_\{t\}with targetata\_\{t\}\. Equation \([115](https://arxiv.org/html/2607.08789#S5.E115)\) is exactly \([39](https://arxiv.org/html/2607.08789#S3.E39)\), soZt=e−αt​atZ\_\{t\}=e^\{\-\\alpha\_\{t\}a\_\{t\}\}and therefore \([116](https://arxiv.org/html/2607.08789#S5.E116)\) follows fromℒTexp=∏tZt\\mathcal\{L\}\_\{T\}^\{\\exp\}=\\prod\_\{t\}Z\_\{t\}\. Summing the one\-step identity \([41](https://arxiv.org/html/2607.08789#S3.E41)\) overttgives \([117](https://arxiv.org/html/2607.08789#S5.E117)\)\. The quantile bound \([120](https://arxiv.org/html/2607.08789#S5.E120)\) is then immediate either from Theorem[5\.15](https://arxiv.org/html/2607.08789#S5.Thmtheorem15)together with \([116](https://arxiv.org/html/2607.08789#S5.E116)\), or by repeating the same worst\-ϵ\\epsilon\-set argument with−log⁡ℒTexp=∑tαt​at\-\\log\\mathcal\{L\}\_\{T\}^\{\\exp\}=\\sum\_\{t\}\\alpha\_\{t\}a\_\{t\}\. Finally, ifat=λ​μta\_\{t\}=\\lambda\\mu\_\{t\}with0<λ<10<\\lambda<1andμt\>0\\mu\_\{t\}\>0, thenmini⁡gt​\(i\)≤−1<λ​μt<μt\\min\_\{i\}g\_\{t\}\(i\)\\leq\-1<\\lambda\\mu\_\{t\}<\\mu\_\{t\}, so the target is feasible\. ∎

#### Proof of Proposition[5\.5](https://arxiv.org/html/2607.08789#S5.Thmtheorem5)

###### Proof\.

For eachβ\\beta,log⁡πt,β​\(i\)=constt​\(β\)−η​∑s=1t−1βt−1−s​ℓs​\(i\)\\log\\pi\_\{t,\\beta\}\(i\)=\\mathrm\{const\}\_\{t\}\(\\beta\)\-\\eta\\sum\_\{s=1\}^\{t\-1\}\\beta^\{t\-1\-s\}\\ell\_\{s\}\(i\), where the additive constant depends on\(t,β\)\(t,\\beta\)but not onii\. Integrating againstαt\\alpha\_\{t\}and exponentiating gives

qt​\(i\)∝exp⁡\(−η​∫R∑s=1t−1βt−1−s​ℓs​\(i\)​αt​\(d​β\)\)q\_\{t\}\(i\)\\propto\\exp\\\!\\left\(\-\\eta\\int\_\{R\}\\sum\_\{s=1\}^\{t\-1\}\\beta^\{t\-1\-s\}\\ell\_\{s\}\(i\)\\,\\alpha\_\{t\}\(d\\beta\)\\right\)Fubini’s theorem yields

∫R∑s=1t−1βt−1−s​ℓs​\(i\)​αt​\(d​β\)=∑s=1t−1\(∫Rβt−1−s​αt​\(d​β\)\)​ℓs​\(i\)\\int\_\{R\}\\sum\_\{s=1\}^\{t\-1\}\\beta^\{t\-1\-s\}\\ell\_\{s\}\(i\)\\,\\alpha\_\{t\}\(d\\beta\)=\\sum\_\{s=1\}^\{t\-1\}\\left\(\\int\_\{R\}\\beta^\{t\-1\-s\}\\,\\alpha\_\{t\}\(d\\beta\)\\right\)\\ell\_\{s\}\(i\)which is exactly \([89](https://arxiv.org/html/2607.08789#S5.E89)\)\. ∎

#### Proof of Proposition[5\.10](https://arxiv.org/html/2607.08789#S5.Thmtheorem10)

###### Proof\.

Letr:=d​ρ/d​μr:=d\\rho/d\\mu\. Sinceρ≪πw\\rho\\ll\\pi\_\{w\}for everyww, the quantities below are finite whenever the corresponding KL divergences are finite\. By definition ofpα⋆p^\{\\star\}\_\{\\alpha\},log⁡r​\(h\)d​pα⋆/d​μ​\(h\)=log⁡r​\(h\)−∑w=1Wαw​log⁡pw​\(h\)\+log⁡Zα\\log\\frac\{r\(h\)\}\{dp^\{\\star\}\_\{\\alpha\}/d\\mu\(h\)\}=\\log r\(h\)\-\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\log p\_\{w\}\(h\)\+\\log Z\_\{\\alpha\}\. Integrating with respect toρ\\rhogives

KL​\(ρ∥pα⋆\)\\displaystyle\\mathrm\{KL\}\(\\rho\\\|p^\{\\star\}\_\{\\alpha\}\)=∫ℋlog⁡r​d​ρ−∑w=1Wαw​∫ℋlog⁡pw​d​ρ\+log⁡Zα\\displaystyle=\\int\_\{\\mathcal\{H\}\}\\log r\\,d\\rho\-\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\int\_\{\\mathcal\{H\}\}\\log p\_\{w\}\\,d\\rho\+\\log Z\_\{\\alpha\}=∑w=1Wαw​∫ℋlog⁡rpw​d​ρ\+log⁡Zα\\displaystyle=\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\int\_\{\\mathcal\{H\}\}\\log\\frac\{r\}\{p\_\{w\}\}\\,d\\rho\+\\log Z\_\{\\alpha\}=∑w=1Wαw​KL​\(ρ∥πw\)−𝒞α​\(π1:W\)\\displaystyle=\\sum\_\{w=1\}^\{W\}\\alpha\_\{w\}\\mathrm\{KL\}\(\\rho\\\|\\pi\_\{w\}\)\-\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\)because∑wαw=1\\sum\_\{w\}\\alpha\_\{w\}=1and𝒞α​\(π1:W\)=−log⁡Zα\\mathcal\{C\}\_\{\\alpha\}\(\\pi\_\{1:W\}\)=\-\\log Z\_\{\\alpha\}\. This is exactly \([98](https://arxiv.org/html/2607.08789#S5.E98)\)\. ∎

## Appendix CExperimental details

This appendix contains the full empirical battery summarized in §[7](https://arxiv.org/html/2607.08789#S7)\. The reproduction code will be released with the paper\.

### C\.1Setup and protocols

#### C\.1\.1Sequence families and schedule panel

We benchmark the framework on four sequence families chosen to expose genuinely different sources of difficulty\. All four useK=8K=8experts, horizonT=2000T=2000, uniform priorπ=\(1/K,…,1/K\)\\pi=\(1/K,\\ldots,1/K\), and bounded lossesℓt​\(i\)∈\[0,1\]\\ell\_\{t\}\(i\)\\in\[0,1\]\. Throughout we setΓ=log⁡K\\Gamma=\\log KandC=1/2C=1/\\sqrt\{2\}as the second\-order schedule constants\.

##### \(a\) I\.i\.d\. stochastic losses\.

We takeℓt​\(i\)=μi\+ξt​\(i\)\\ell\_\{t\}\(i\)=\\mu\_\{i\}\+\\xi\_\{t\}\(i\)clipped to\[0,1\]\[0,1\]with a fixed mean vectorμ∈\[0\.15,0\.7\]K\\mu\\in\[0\.15,0\.7\]^\{K\}\(expert0optimal atμ0=0\.15\\mu\_\{0\}=0\.15\) and i\.i\.d\. Gaussian noiseξt​\(i\)∼𝒩​\(0,0\.152\)\\xi\_\{t\}\(i\)\\sim\\mathcal\{N\}\(0,0\.15^\{2\}\)\. The only difficulty lies in identifying the leading expert; once identified, the remaining intrinsic time accumulates slowly\. The predicted regret\-decomposition signature is that the comparator\-information shareωinfo\\omega^\{\\mathrm\{info\}\}dominates while the algorithm is still concentrating mass on the leader, andωpay\\omega^\{\\mathrm\{pay\}\}accumulates only slowly afterwards\.

##### \(b\) Martingale losses\.

We takeℓt​\(i\)=mt​\(i\)\+ξt​\(i\)\\ell\_\{t\}\(i\)=m\_\{t\}\(i\)\+\\xi\_\{t\}\(i\)with predictablemt​\(i\)=μi\+0\.3​sin⁡\(2​π​t/500\)m\_\{t\}\(i\)=\\mu\_\{i\}\+0\.3\\sin\(2\\pi t/500\)and𝔼​\[ξt∣ℱt−1\]=0\\mathbb\{E\}\[\\xi\_\{t\}\\mid\\mathcal\{F\}\_\{t\-1\}\]=0\. This is the canonical setting for side information: withut=−mtu\_\{t\}=\-m\_\{t\}, the composite loss is the noise residualct=ℓt−mtc\_\{t\}=\\ell\_\{t\}\-m\_\{t\}\. Running both ordinary and optimistic Hedge tests whether the decomposition correctly transfers mass from the intrinsic\-time term onto the predictable mismatchMt​\(ρ\)M\_\{t\}\(\\rho\)when the compensator is accurate\.

##### \(c\) Cycling adversarial losses\.

We rotate the best expert every5050rounds: the leader of the current segment has loss∼0\.1±0\.1\\sim 0\.1\\pm 0\.1while the rest sit at∼0\.6±0\.1\\sim 0\.6\\pm 0\.1\. The sequence destroys predictability by design and keeps experts nearly tied on long horizons\. Here the predicted signature is thatωpay\\omega^\{\\mathrm\{pay\}\}dominates throughout,ωinfo\\omega^\{\\mathrm\{info\}\}decays as no fixed expert remains uniquely good, and a properly tuned schedule’s drift acts as a discount\.

##### \(d\) Mixed\-character sequence\.

We concatenate four blocks of lengthT/4=500T/4=500: i\.i\.d\. stochastic with expert0optimal, cycling adversarial, i\.i\.d\. stochastic with expert33optimal, cycling adversarial\. This is the clearest pathwise test: the dominant share should switch at each block boundary, even when a terminal regret number would hide the transition\.

##### Schedule panel\.

We pair the algorithms in Algorithm[1](https://arxiv.org/html/2607.08789#alg1)with two natural fixed\-temperature baselines, giving six schedules grouped into three families of two:

Fixed temperatures\.Retempered\.Pressure\-target\.η≡0\.1\\eta\\equiv 0\.1\(low\) andη≡1\.0\\eta\\equiv 1\.0\(high\), local update at fixedη\\eta\.Prior\-retempered posterior with the second\-order ruleηt=min⁡\{1,C​Γ/Vt−1\}\\eta\_\{t\}=\\min\\\{1,C\\sqrt\{\\Gamma/V\_\{t\-1\}\}\\\}, computed from either the exact intrinsic\-time clockVt−1=∑s<tQs​\(c\)V\_\{t\-1\}=\\sum\_\{s<t\}Q\_\{s\}\(c\)\(*intrinsic*\) or its quadratic relaxationVt−1var=∑s<tVarps​\(cs\)/2V^\{\\mathrm\{var\}\}\_\{t\-1\}=\\sum\_\{s<t\}\\mathrm\{Var\}\_\{p\_\{s\}\}\(c\_\{s\}\)/2\(*variance*\)\.Local update with the line\-search schedule∑ipt​\(i\)​e−ηt​\(ct​\(i\)−at\)=1\\sum\_\{i\}p\_\{t\}\(i\)e^\{\-\\eta\_\{t\}\(c\_\{t\}\(i\)\-a\_\{t\}\)\}=1\. The*intrinsic*variant choosesata\_\{t\}as the mix\-loss at the gap\-implied rateηgap=log⁡K/Δt−1\\eta\_\{\\mathrm\{gap\}\}=\\log K/\\Delta\_\{t\-1\}; the*variance*variant uses the closed\-formηt=min⁡\{1,C​Γ/Vt−1var\}\\eta\_\{t\}=\\min\\\{1,C\\sqrt\{\\Gamma/V^\{\\mathrm\{var\}\}\_\{t\-1\}\}\\\}with the same local update\.
The pairing is informative: the two columns within each pair share an algorithm class, so any difference between them comes from the schedule rule alone\. Importantly, the two pressure\-target variants have substantially different temperature dynamics\. The variance schedule is bounded by11uniformly, so on easy paths it sits at the cap; the intrinsic schedule uses the gap\-implied rateηgap=log⁡K/Δt−1\\eta\_\{\\mathrm\{gap\}\}=\\log K/\\Delta\_\{t\-1\}, which grows unbounded asΔt−1→0\+\\Delta\_\{t\-1\}\\to 0^\{\+\}\. We cap it for numerical safety atηmax=50\\eta\_\{\\max\}=50, but on stochastic paths it routinely binds the cap, producing a much faster temperature ramp than the variance variant\. This explains why their regret\-decomposition shares look very different even though the underlying update is the same local exponential\-weights recursion: the local prefix decomposition’s drift termDtloc=∑s≥2KL​\(ρ∥ps\)​\(1/ηs−1/ηs−1\)D\_\{t\}^\{\\mathrm\{loc\}\}=\\sum\_\{s\\geq 2\}\\mathrm\{KL\}\(\\rho\\\|p\_\{s\}\)\(1/\\eta\_\{s\}\-1/\\eta\_\{s\-1\}\)accumulates large negative contributions wheneverηs\\eta\_\{s\}jumps, and the gap\-implied schedule jumps frequently\. We discuss this further in §[C\.2\.1](https://arxiv.org/html/2607.08789#A3.SS2.SSS1)\.

##### Comparator panel\.

For each family we evaluate three comparator types: the best single expertρ=δi∗\\rho=\\delta\_\{i^\{\\ast\}\}, a softmax\-quantile comparatorρi∝exp⁡\(−LT​\(i\)/T\)\\rho\_\{i\}\\propto\\exp\(\-L\_\{T\}\(i\)/\\sqrt\{T\}\), and the uniform mixture\. For stochastic families we aggregate terminal metrics over multiple random seeds and report means with inter\-quartile bands; for deterministic adversarial families we report representative pathwise traces alongside multi\-seed summaries\.

#### C\.1\.2A comparator\-informed recipe for side information

A general recipe for side information is to encode structure shared by the comparator class into a predictable baseline\. Identify a predictable common componentmt​\(i\)m\_\{t\}\(i\)of the experts’ losses and setut​\(i\):=−mt​\(i\)u\_\{t\}\(i\):=\-m\_\{t\}\(i\), so the composite losses become the residualsct​\(i\)=ℓt​\(i\)−mt​\(i\)c\_\{t\}\(i\)=\\ell\_\{t\}\(i\)\-m\_\{t\}\(i\)\. If the comparator class is well explained by that common component, the intrinsic\-time clockVT​\(c\)V\_\{T\}\(c\)is small and the theorem yields low regret for the right reason\. If the prediction is poor, the same identity remains valid and simply reports that the sequence was not easy in that way: the bookkeeping does not lose its diagnostic value whenmtm\_\{t\}fails\.

A canonical instance is optimistic play in a repeated matrix game\. Let expertiibe the pure actionii, let the loss matrix beGG, and letyty\_\{t\}be the opponent’s mixed action on roundtt, so thatℓt​\(i\)=ei⊤​G​yt\\ell\_\{t\}\(i\)=e\_\{i\}^\{\\top\}Gy\_\{t\}\. Suppose a forecasty^t\\hat\{y\}\_\{t\}of the next opponent move is available from past play—a moving average, a state\-space model, or any opponent\-model output\. Settingmt​\(i\):=ei⊤​G​y^tm\_\{t\}\(i\):=e\_\{i\}^\{\\top\}G\\hat\{y\}\_\{t\}andut​\(i\):=−mt​\(i\)u\_\{t\}\(i\):=\-m\_\{t\}\(i\)makesct​\(i\)=ei⊤​G​\(yt−y^t\)c\_\{t\}\(i\)=e\_\{i\}^\{\\top\}G\(y\_\{t\}\-\\hat\{y\}\_\{t\}\)the forecast residual, and the original\-loss regret against any fixed mixed actionu∈Δ​\(\[K\]\)u\\in\\Delta\(\[K\]\)becomes

Rtℓ​\(u\)=Pt​\(c\)\+Dt\+Bt​\(u\)\+∑s=1t⟨ps−u,ms⟩,cs​\(i\)=ei⊤​G​\(ys−y^s\)\.R\_\{t\}^\{\\ell\}\(u\)\\;=\\;P\_\{t\}\(c\)\+D\_\{t\}\+B\_\{t\}\(u\)\+\\sum\_\{s=1\}^\{t\}\\langle p\_\{s\}\-u,m\_\{s\}\\rangle,\\qquad c\_\{s\}\(i\)=e\_\{i\}^\{\\top\}G\(y\_\{s\}\-\\hat\{y\}\_\{s\}\)\.\(165\)This is precisely the optimistic\-Hedge decomposition used in adaptive game play\[[39](https://arxiv.org/html/2607.08789#bib.bib39),[15](https://arxiv.org/html/2607.08789#bib.bib15)\]\. The side information does not change the algebra; it changes the sequence whose difficulty the prefix decomposition is measuring\. The construction generalizes beyond games: whenever expert losses admit a forecast of the formmt​\(i\)=θ^t⊤​ϕt​\(i\)m\_\{t\}\(i\)=\\hat\{\\theta\}\_\{t\}^\{\\top\}\\phi\_\{t\}\(i\)for a shared feature representationϕt​\(i\)\\phi\_\{t\}\(i\), the recipe subtracts the common predictable component and leaves only the residual to be hedged\. The empirical question, therefore, is not whether one specific optimistic trick works, but whether the diagnostic split correctly identifies which sequences are predictable in the sense the side information encodes\.

The complete empirical view separates*selection*from*diagnosis*\. The terminal regret picks out competitive schedules; the prefix decomposition then explains whether success came from small intrinsic time, favorable drift, a low\-complexity comparator, or an accurate predictive baseline\. The remainder of this appendix follows that order\.

### C\.2The exact decomposition

#### C\.2\.1Regret\-decomposition shares across families

Figures[3](https://arxiv.org/html/2607.08789#A3.F3)–[6](https://arxiv.org/html/2607.08789#A3.F6)report the regret\-decomposition plot of §[7\.1](https://arxiv.org/html/2607.08789#S7.SS1)on the four families\. Each figure is a3×23\\times 2grid: rows are algorithm classes \(fixed\-temperature, retempered, pressure\-target\), columns within a row are paired variants \(low/highη\\etafor fixed; intrinsic/variance schedule driver for the two adaptive families\)\. Stacked colored shares areωpay\\omega^\{\\mathrm\{pay\}\}\(blue\),ωdrift\\omega^\{\\mathrm\{drift\}\}\(orange\),ωinfo\\omega^\{\\mathrm\{info\}\}\(green\) on a common primary axis\[0,1\]\[0,1\]; the thin black curve overlaid on the secondary axis is the signed prefix regretRtc​\(ρ\)R\_\{t\}^\{c\}\(\\rho\)\. Comparatorρ\\rhois the best single expert\. The prefix identity \([152](https://arxiv.org/html/2607.08789#S7.E152)\) holds to machine precision on every panel\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig1_decomp_iid_stochastic.png)Figure 3:Regret\-decomposition shares on the i\.i\.d\. stochastic family \(K=8K=8,T=2000T=2000\)\. Rows pair the three algorithm classes; columns pair their variants\. Shared axes: roundtton the bottom row, decomposition share on the left column\. Comparator\-informationωinfo\\omega^\{\\mathrm\{info\}\}\(green\) dominates the first∼100\\sim 100rounds while the algorithm concentrates mass on the leading expert; the intrinsic\-time shareωpay\\omega^\{\\mathrm\{pay\}\}\(blue\) and the drift shareωdrift\\omega^\{\\mathrm\{drift\}\}\(orange\) accumulate slowly thereafter\. The pressure\-target \(intrinsic\) panel exhibits a substantial drift share because the gap\-implied scheduleηgap=log⁡K/Δt−1\\eta\_\{\\mathrm\{gap\}\}=\\log K/\\Delta\_\{t\-1\}ramps up rapidly whenΔt−1\\Delta\_\{t\-1\}is small\.![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig2_decomp_martingale.png)Figure 4:Regret\-decomposition shares on the martingale family with optimistic side informationut=−mtu\_\{t\}=\-m\_\{t\}\. The composite lossct=ℓt−mtc\_\{t\}=\\ell\_\{t\}\-m\_\{t\}is the noise residual, so the intrinsic\-time share is uniformly small across schedules; the predictable component is absorbed into the off\-diagram mismatch termMt​\(ρ\)M\_\{t\}\(\\rho\)from \([165](https://arxiv.org/html/2607.08789#A3.E165)\)\. The pressure\-target \(intrinsic\) panel shows pronounced orange \(drift\) regions whenever the gap\-implied rate jumps—the local prefix decomposition’s drift termDtlocD\_\{t\}^\{\\mathrm\{loc\}\}accumulates large negative contributions at everyηs\\eta\_\{s\}jump, so a schedule that variesηs\\eta\_\{s\}aggressively reads as “drift\-heavy” even though the underlying update is well\-behaved\.![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig3_decomp_adversarial.png)Figure 5:Regret\-decomposition shares on the cycling\-adversarial family \(best expert rotates every5050rounds\)\. The intrinsic\-time shareωpay\\omega^\{\\mathrm\{pay\}\}dominates throughout becauseVT​\(c\)V\_\{T\}\(c\)grows linearly inTT; the comparator\-information shareωinfo\\omega^\{\\mathrm\{info\}\}is non\-negligible because the cumulative best expert remains identifiable but the periodic recovery of leadership keepsBt​\(ρ\)B\_\{t\}\(\\rho\)from shrinking\. The drift shareωdrift\\omega^\{\\mathrm\{drift\}\}in the second\-order panels reflects the steady ramp\-down ofηt\\eta\_\{t\}predicted by the schedule ruleηt=min⁡\{1,C​Γ/Vt−1\}\\eta\_\{t\}=\\min\\\{1,C\\sqrt\{\\Gamma/V\_\{t\-1\}\}\\\}\.![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig4_decomp_mixed_character.png)Figure 6:Regret\-decomposition shares on the mixed\-character sequence \(i\.i\.d\. / adversarial / i\.i\.d\. / adversarial blocks, each of length500500\)\. The dominant share switches at every block boundary even though the terminal regret number alone would average over the transitions\. Within each block the qualitative pattern matches the corresponding standalone family:ωinfo\\omega^\{\\mathrm\{info\}\}\-heavy on i\.i\.d\. blocks,ωpay\\omega^\{\\mathrm\{pay\}\}\-heavy on adversarial blocks\.We make four observations from the panels in turn\.

##### The i\.i\.d\. signature\.

Figure[3](https://arxiv.org/html/2607.08789#A3.F3): at every fixed temperature and at every adaptive schedule, the comparator\-information shareωinfo\\omega^\{\\mathrm\{info\}\}is dominant for the first5050–200200rounds and then settles to a near\-uniform plateau in whichωpay\\omega^\{\\mathrm\{pay\}\}is small but visible and the drift is essentially zero\. This is exactly what the theory predicts: the only difficulty in this family is identifying expert0, and once identified the regret stops accumulating\. The pressure\-target \(intrinsic\) panel is the lone exception: it shows a substantial drift share, traceable to its gap\-implied rateηgap=log⁡K/Δt−1\\eta\_\{\\mathrm\{gap\}\}=\\log K/\\Delta\_\{t\-1\}\. As the cumulative gap shrinks under the converged posterior,ηgap\\eta\_\{\\mathrm\{gap\}\}ramps up; each ramp contributes toDtloc=∑s≥2KL​\(ρ∥ps\)​\(1/ηs−1/ηs−1\)D\_\{t\}^\{\\mathrm\{loc\}\}=\\sum\_\{s\\geq 2\}\\mathrm\{KL\}\(\\rho\\\|p\_\{s\}\)\(1/\\eta\_\{s\}\-1/\\eta\_\{s\-1\}\)via the1/η1/\\etatelescoping\. The variance variant in the same row, by contrast, holdsηt\\eta\_\{t\}near the cap of11and shows essentially no drift, even though the underlying update is the same local exponential\-weights recursion\. The two pressure\-target columns therefore document not an algorithmic difference but a schedule\-rule difference: the line\-search is identical; the choice of targetata\_\{t\}is what determines whetherηt\\eta\_\{t\}ramps up\.

##### The martingale signature\.

Figure[4](https://arxiv.org/html/2607.08789#A3.F4): withut=−mtu\_\{t\}=\-m\_\{t\}, the composite loss is the residualct=ℓt−mtc\_\{t\}=\\ell\_\{t\}\-m\_\{t\}, so the intrinsic\-time share is small under every schedule\. The predictable component is absorbed into the mismatch termMt​\(ρ\)M\_\{t\}\(\\rho\)from \([165](https://arxiv.org/html/2607.08789#A3.E165)\), which carries the predictable mass off the diagram\. In a successful centered run the diagnostic plot therefore looks “empty” on the pay axis: the side information has done the work the intrinsic\-time clock would otherwise have measured\. The companion plot in Figure[7](https://arxiv.org/html/2607.08789#A3.F7)shows the corresponding signed prefix regret of ordinary versus optimistic Hedge on the same path; the optimistic run’s composite\-loss regret goes deeply negative because the predictable component is absorbed\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig5_martingale_comparison.png)Figure 7:Martingale family \(T=2000T=2000\): cumulative composite\-loss prefix regretRtc​\(ρ\)R\_\{t\}^\{c\}\(\\rho\)of ordinary Hedge versus the optimistic version withut=−mtu\_\{t\}=\-m\_\{t\}\. The optimistic run goes deeply negative because the predictable component is absorbed into the side information\.
##### The cycling\-adversarial signature\.

Figure[5](https://arxiv.org/html/2607.08789#A3.F5):VT​\(c\)V\_\{T\}\(c\)grows roughly linearly inTTon this family, so the intrinsic\-time shareωpay\\omega^\{\\mathrm\{pay\}\}stays high throughout\. The drift shareωdrift\\omega^\{\\mathrm\{drift\}\}reflects the second\-order rule’s steady ramp\-down ofηt\\eta\_\{t\}as the clock accumulates; at fixed temperature the drift is identically zero, providing a useful contrast that reads off the figure\. The comparator\-information shareωinfo\\omega^\{\\mathrm\{info\}\}is non\-zero because, even though no fixed expert is uniquely good, the cumulative\-loss leader changes only at block boundaries and the algorithm tracks it well during each block\. This is the family on which the second\-order rule’s two\-sided envelope of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)is empirically tight, which we examine in detail in §[C\.2\.9](https://arxiv.org/html/2607.08789#A3.SS2.SSS9)\.

##### The mixed\-character signature\.

Figure[6](https://arxiv.org/html/2607.08789#A3.F6): the dominant share switches at every block boundary\. Within each block the panel reads like its standalone family—green\-heavy on i\.i\.d\. blocks, blue\-heavy on adversarial blocks—and the transitions are sharply visible in the secondary\-axis regret line\. This is the cleanest pathwise illustration of why the prefix decomposition is more informative than terminal regret: a single endpoint number averages over the transitions, while the share trajectories reveal the path’s character at everytt\.

##### The matrix\-game side\-information recipe\.

Figure[8](https://arxiv.org/html/2607.08789#A3.F8)exhibits the recipe \([165](https://arxiv.org/html/2607.08789#A3.E165)\) on aK=5K=5repeated matrix game with random payoff matrixG∈\[−1,1\]5×5G\\in\[\-1,1\]^\{5\\times 5\}and a2020\-round moving\-average forecasty^t\\hat\{y\}\_\{t\}\. We plot the intrinsic\-time shareωtpay\\omega^\{\\mathrm\{pay\}\}\_\{t\}for the retempered schedule with and without side information against two opponents: a sinusoidally drifting mixed\-strategy opponent \(left\) and a Dirichlet\-i\.i\.d\. opponent \(right\)\. On the predictable opponent, the side information shrinks the intrinsic\-time share substantially, mirroring the martingale pattern; on the adversarial opponent, the residualcs​\(i\)=ei⊤​G​\(ys−y^s\)c\_\{s\}\(i\)=e\_\{i\}^\{\\top\}G\(y\_\{s\}\-\\hat\{y\}\_\{s\}\)remains as hard as the original loss and the share is unchanged\. The recipe therefore distinguishes whether low regret is coming from genuine predictability of the opponent or from the algorithm’s intrinsic adaptation alone\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig6_game_side_info.png)Figure 8:Matrix\-game side\-information recipe \(K=5K=5,T=1500T=1500,G∈\[−1,1\]5×5G\\in\[\-1,1\]^\{5\\times 5\},2020\-round moving\-average forecast\)\.Left:predictable \(sinusoidally drifting\) opponent — side information shrinksωtpay\\omega^\{\\mathrm\{pay\}\}\_\{t\}substantially\.Right:Dirichlet\-i\.i\.d\. opponent — the residual is as hard as the original loss;ωtpay\\omega^\{\\mathrm\{pay\}\}\_\{t\}unchanged\.

#### C\.2\.2Comparator\-by\-comparator exactness of the prefix identity

All other panels in this section, and most of the literature on PAC\-Bayes regret bounds, evaluate at the comparatorρ=δi∗\\rho=\\delta\_\{i^\{\*\}\}that puts unit mass on the best fixed expert\. Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)demands that the prefix identity hold at everyρ∈Δ​\(\[K\]\)\\rho\\in\\Delta\(\[K\]\)\. The empirical content of that demand is twofold\. First, the residual\|RTc​\(ρ\)−\(PT\+DT\+BT​\(ρ\)\)\|\|R^\{c\}\_\{T\}\(\\rho\)\-\(P\_\{T\}\+D\_\{T\}\+B\_\{T\}\(\\rho\)\)\|should be at machine precision for everyρ\\rho, not only atδi∗\\delta\_\{i^\{\*\}\}\. Second, theρ\\rho\-dependent pieceBT​\(ρ\)B\_\{T\}\(\\rho\)should track the closed\-form expression\(KL​\(ρ∥π\)−KL​\(ρ∥qT,ηT\)\)/ηT\(\\mathrm\{KL\}\(\\rho\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\\\|q\_\{T,\\eta\_\{T\}\}\)\)/\\eta\_\{T\}that the theorem prescribes; it is the only piece of the decomposition that depends onρ\\rho, and its dependence is fully determined byqT,ηTq\_\{T,\\eta\_\{T\}\}andηT\\eta\_\{T\}, both of which are emitted by the run\.

We sweep the one\-parameter comparator family

ρα:=α​δi∗\+1−αK−1​\(𝟏−δi∗\),α∈\[1/K,1\],\\rho\_\{\\alpha\}\\;:=\\;\\alpha\\,\\delta\_\{i^\{\*\}\}\+\\frac\{1\-\\alpha\}\{K\-1\}\\bigl\(\\mathbf\{1\}\-\\delta\_\{i^\{\*\}\}\\bigr\),\\qquad\\alpha\\in\[1/K,\\,1\],\(166\)which interpolates monotonically from the uniform prior \(α=1/K\\alpha=1/K,KL​\(ρ∥π\)=0\\mathrm\{KL\}\(\\rho\\\|\\pi\)=0\) to the point\-mass oni∗i^\{\*\}\(α=1\\alpha=1,KL​\(ρ∥π\)=log⁡K\\mathrm\{KL\}\(\\rho\\\|\\pi\)=\\log K\)\. For eachα\\alphawe computeRTc​\(ρα\)R^\{c\}\_\{T\}\(\\rho\_\{\\alpha\}\),BT​\(ρα\)B\_\{T\}\(\\rho\_\{\\alpha\}\),PTP\_\{T\}, andDTD\_\{T\}on prior\-retempered runs of horizonT=1500T=1500over twelve seeds, on i\.i\.d\. stochastic and cycling\-adversarial paths\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig12_comparator_richness.png)Figure 9:Comparator\-by\-comparator check of Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)\. Top row: terminalRTc​\(ρα\)R^\{c\}\_\{T\}\(\\rho\_\{\\alpha\}\),BT​\(ρα\)B\_\{T\}\(\\rho\_\{\\alpha\}\), and theρ\\rho\-independentPTP\_\{T\},DTD\_\{T\}, plotted againstKL​\(ρα∥π\)/ηT\\mathrm\{KL\}\(\\rho\_\{\\alpha\}\\\|\\pi\)/\\eta\_\{T\}\. The twoρ\\rho\-independent pieces appear as horizontal lines as required; the only piece that varies withα\\alphaisBTB\_\{T\}, andRTc−BT≡PT\+DTR^\{c\}\_\{T\}\-B\_\{T\}\\equiv P\_\{T\}\+D\_\{T\}to machine precision\. Bottom row: scatter ofηT⋅BT​\(ρα\)\\eta\_\{T\}\\cdot B\_\{T\}\(\\rho\_\{\\alpha\}\)againstKL​\(ρα∥π\)−KL​\(ρα∥qT,ηT\)\\mathrm\{KL\}\(\\rho\_\{\\alpha\}\\\|\\pi\)\-\\mathrm\{KL\}\(\\rho\_\{\\alpha\}\\\|q\_\{T,\\eta\_\{T\}\}\)over all\(α,seed\)\(\\alpha,\\,\\mathrm\{seed\}\)pairs, with the diagonaly=xy=xpredicted by Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10); the maximum residual is0on i\.i\.d\. stochastic paths and4\.4×10−164\.4\\times 10^\{\-16\}on cycling\-adversarial paths\.The bottom row of Figure[9](https://arxiv.org/html/2607.08789#A3.F9)reads off the comparator\-by\-comparator identity check: every\(α,seed\)\(\\alpha,\\mathrm\{seed\}\)point sits on the diagonaly=xy=xto machine precision, with maximum deviation0on i\.i\.d\. paths and4\.4×10−164\.4\\times 10^\{\-16\}on cycling\-adversarial\. This confirms thatBT​\(ρ\)B\_\{T\}\(\\rho\)is computed correctly on all comparators in the simplex and not only at the point mass; equivalently, that the prefix identity is aρ\\rho\-by\-ρ\\rhostatement, not aδi∗\\delta\_\{i^\{\*\}\}\-only statement\.

The top row shows that theρ\\rho\-dependent pieceBTB\_\{T\}is, perhaps counterintuitively, often*negative*whenqT,ηTq\_\{T,\\eta\_\{T\}\}has concentrated faster thanρα\\rho\_\{\\alpha\}has\. On i\.i\.d\. pathsqT,ηTq\_\{T,\\eta\_\{T\}\}collapses ontoδi∗\\delta\_\{i^\{\*\}\}\(the cap binds,ηT=1\\eta\_\{T\}=1, andT=1500T=1500rounds is enough for the leader to dominate\), so for anyρα\\rho\_\{\\alpha\}that retains mass on non\-leaders we haveKL​\(ρα∥qT,ηT\)≫KL​\(ρα∥π\)\\mathrm\{KL\}\(\\rho\_\{\\alpha\}\\\|q\_\{T,\\eta\_\{T\}\}\)\\gg\\mathrm\{KL\}\(\\rho\_\{\\alpha\}\\\|\\pi\)andBT​\(ρα\)<0B\_\{T\}\(\\rho\_\{\\alpha\}\)<0\. This is consistent with the identity: the algorithm has already absorbed comparator information beyond whatρα\\rho\_\{\\alpha\}itself carries, and the negativeBTB\_\{T\}records the credit\. On cycling\-adversarial paths,qT,ηTq\_\{T,\\eta\_\{T\}\}stays diffuse \(no fixed expert dominates\),KL​\(ρα∥qT,ηT\)\\mathrm\{KL\}\(\\rho\_\{\\alpha\}\\\|q\_\{T,\\eta\_\{T\}\}\)stays small, andBT​\(ρα\)B\_\{T\}\(\\rho\_\{\\alpha\}\)is close to the loose upper boundKL​\(ρα∥π\)/ηT\\mathrm\{KL\}\(\\rho\_\{\\alpha\}\\\|\\pi\)/\\eta\_\{T\}that the theorem also implies\.

The takeaway is twofold\. The first part is a clean numerical verification of Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)as aρ\\rho\-uniform identity, not only at the best\-fixed\-expert comparator\. The second is a methodological caution: the upper boundBT​\(ρ\)≤KL​\(ρ∥π\)/ηTB\_\{T\}\(\\rho\)\\leq\\mathrm\{KL\}\(\\rho\\\|\\pi\)/\\eta\_\{T\}commonly quoted in PAC\-Bayes papers can be very loose; the exact valueBT​\(ρ\)B\_\{T\}\(\\rho\)that the framework reports is a fully computable quantity from the run and gives the comparator’s actual cost on the realized path\.

##### Local update: the parallel identity is alsoρ\\rho\-uniform\.

The pressure\-targeted line\-search rule of Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)is a different algorithm, with a different cumulative bookkeeping than the prior\-retempered chain\. Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)states the analogous prefix identity for that algorithm:

RTc​\(ρ\)=PTloc\+DTloc​\(ρ\)\+BTloc​\(ρ\)R^\{c\}\_\{T\}\(\\rho\)\\;=\\;P^\{\\mathrm\{loc\}\}\_\{T\}\\;\+\\;D^\{\\mathrm\{loc\}\}\_\{T\}\(\\rho\)\\;\+\\;B^\{\\mathrm\{loc\}\}\_\{T\}\(\\rho\)\(167\)withPTloc=∑s≤T\(⟨ps,cs⟩−as\)=∑sηs​QsP^\{\\mathrm\{loc\}\}\_\{T\}=\\sum\_\{s\\leq T\}\(\\langle p\_\{s\},c\_\{s\}\\rangle\-a\_\{s\}\)=\\sum\_\{s\}\\eta\_\{s\}Q\_\{s\}\(the cumulative gap to the targets, equal to the prior\-retemperedPTP\_\{T\}on common\-eta runs\),DTloc​\(ρ\)=∑s≥2KL​\(ρ∥ps\)​\(1/ηs−1/ηs−1\)D^\{\\mathrm\{loc\}\}\_\{T\}\(\\rho\)=\\sum\_\{s\\geq 2\}\\mathrm\{KL\}\(\\rho\\\|p\_\{s\}\)\(1/\\eta\_\{s\}\-1/\\eta\_\{s\-1\}\), and the boundary termBTloc​\(ρ\)=KL​\(ρ∥π\)/η1−KL​\(ρ∥pT\+1\)/ηTB^\{\\mathrm\{loc\}\}\_\{T\}\(\\rho\)=\\mathrm\{KL\}\(\\rho\\\|\\pi\)/\\eta\_\{1\}\-\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)/\\eta\_\{T\}\. Figure[10](https://arxiv.org/html/2607.08789#A3.F10)reports the sameρα\\rho\_\{\\alpha\}sweep as Figure[9](https://arxiv.org/html/2607.08789#A3.F9)but on pressure\-target runs \(Algorithm[1](https://arxiv.org/html/2607.08789#alg1)lines 8–17\), with the local pieces of \([167](https://arxiv.org/html/2607.08789#A3.E167)\) in place of the prior\-retemperedPT,DT,BTP\_\{T\},D\_\{T\},B\_\{T\}\. The maximum identity residual is1\.53×10−121\.53\\times 10^\{\-12\}on i\.i\.d\. stochastic paths and3\.30×10−123\.30\\times 10^\{\-12\}on cycling\-adversarial paths over the same25×8=20025\\times 8=200\(α,seed\)\(\\alpha,\\mathrm\{seed\}\)pairs as before; the slightly larger residual relative to Figure[9](https://arxiv.org/html/2607.08789#A3.F9)is the round\-trip through the bisection that selectsηt\\eta\_\{t\}from the pressure targetata\_\{t\}, not a discrepancy with the algebra\. The bookkeeping is thereforeρ\\rho\-uniform for the local update as well, closing the methodological loop on the diagnostic shares used throughout this section: the sameρα\\rho\_\{\\alpha\}sweep that verifies Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)for the prior\-retempered chain also verifies Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)for the local chain\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig15_local_identity.png)Figure 10:Comparator\-by\-comparator check of the local\-update prefix identity \([167](https://arxiv.org/html/2607.08789#A3.E167)\) on pressure\-target runs \(T=1500T=1500,K=8K=8, eight seeds, twenty\-fiveρα\\rho\_\{\\alpha\}values\)\.Top row:terminalRTc​\(ρα\)R^\{c\}\_\{T\}\(\\rho\_\{\\alpha\}\),BTloc​\(ρα\)B^\{\\mathrm\{loc\}\}\_\{T\}\(\\rho\_\{\\alpha\}\), and theρ\\rho\-independentPTlocP^\{\\mathrm\{loc\}\}\_\{T\}as functions ofKL​\(ρα∥π\)\\mathrm\{KL\}\(\\rho\_\{\\alpha\}\\\|\\pi\)\. The local driftDTlocD^\{\\mathrm\{loc\}\}\_\{T\}isρ\\rho\-dependent throughKL​\(ρ∥ps\)\\mathrm\{KL\}\(\\rho\\\|p\_\{s\}\)in its summand, which is the structural difference from the prior\-retemperedDTD\_\{T\}in Figure[9](https://arxiv.org/html/2607.08789#A3.F9)\.Bottom row:scatter ofRTc​\(ρα\)R^\{c\}\_\{T\}\(\\rho\_\{\\alpha\}\)againstPTloc\+DTloc​\(ρα\)\+BTloc​\(ρα\)P^\{\\mathrm\{loc\}\}\_\{T\}\+D^\{\\mathrm\{loc\}\}\_\{T\}\(\\rho\_\{\\alpha\}\)\+B^\{\\mathrm\{loc\}\}\_\{T\}\(\\rho\_\{\\alpha\}\)over all\(α,seed\)\(\\alpha,\\mathrm\{seed\}\)pairs; the diagonaly=xy=xis Corollary[3\.10](https://arxiv.org/html/2607.08789#S3.Thmtheorem10)\. Maximum residual is1\.53×10−121\.53\\times 10^\{\-12\}on i\.i\.d\. stochastic and3\.30×10−123\.30\\times 10^\{\-12\}on cycling\-adversarial\. The boundary termBTlocB^\{\\mathrm\{loc\}\}\_\{T\}is computed in log space to keepKL​\(ρ∥pT\+1\)\\mathrm\{KL\}\(\\rho\\\|p\_\{T\+1\}\)finite whenpT\+1p\_\{T\+1\}has coordinates underflowed into the IEEE\-double subnormal range on i\.i\.d\. paths\.

#### C\.2\.3Comparator\-by\-comparator exactness of the prior\-retempered prefix identity \(§[C\.2\.2](https://arxiv.org/html/2607.08789#A3.SS2.SSS2)\)

##### Headline\.

log10 max \|Delta\| = \-12\.14 \(iid\), \-11\.49 \(adversarial\); B closed\-form residual at FP zero\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.4Local pressure\-target prefix identity, comparator\-by\-comparator \(§[C\.2\.2](https://arxiv.org/html/2607.08789#A3.SS2.SSS2)\)

##### Headline\.

log10 max \|Delta\| = \-12\.37 \(iid\), \-12\.01 \(adversarial\); within target thresholds\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.5Exact Hoeffding slack across families

Proposition[3\.1](https://arxiv.org/html/2607.08789#S3.Thmtheorem1)bounds the per\-round intrinsic incrementQt​\(c\)Q\_\{t\}\(c\)by the range proxy\(bt−at\)2/8\(b\_\{t\}\-a\_\{t\}\)^\{2\}/8\. For losses in\[0,1\]\[0,1\]this givesQt​\(c\)≤1/8Q\_\{t\}\(c\)\\leq 1/8uniformly\. The cumulative version isVT​\(c\)≤T/8V\_\{T\}\(c\)\\leq T/8, the worst\-case Hoeffding clock\. The exact intrinsic timeVT​\(c\)V\_\{T\}\(c\)is by construction at most this much, and by how much it falls below the bound is itself a sequence\-difficulty diagnostic: the Hoeffding bound’s looseness on a given path is exactly how much harder the worst\-case analysis treats the path than the framework does\.

Figure[11](https://arxiv.org/html/2607.08789#A3.F11)reports two views of the slack on retempered runs of horizonT=4000T=4000on the four families \(Mixed\-character at its nativeT=2000T=2000\)\. The left panel is a log\-count histogram of the per\-round ratioQt​\(c\)/\(1/8\)Q\_\{t\}\(c\)/\(1/8\)\. On i\.i\.d\. stochastic, martingale, and mixed\-character paths the ratio is concentrated below0\.050\.05at the mode, with maxima of0\.210\.21,0\.170\.17, and0\.200\.20respectively—the algorithm’s playing distributionptp\_\{t\}becomes concentrated quickly and the variance proxyVarpt​\(ct\)≈2​Qt\\mathrm\{Var\}\_\{p\_\{t\}\}\(c\_\{t\}\)\\approx 2Q\_\{t\}shrinks accordingly\. On cycling\-adversarial paths the ratio has a much heavier upper tail, with mass extending to0\.340\.34; the cycling rotation keepsptp\_\{t\}from concentrating so the variance stays substantial\. None of the families reach the worst\-caseQt=1/8Q\_\{t\}=1/8ratio of11, marked by the dotted vertical line, on the synthetic instances tested here\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig17_hoeffding_slack.png)Figure 11:Exact Hoeffding slack across families \(K=8K=8, retempered schedule\)\.Left:log\-count histogram of per\-roundQt​\(c\)/\[\(b−a\)2/8\]=Qt​\(c\)/\(1/8\)Q\_\{t\}\(c\)/\[\(b\-a\)^\{2\}/8\]=Q\_\{t\}\(c\)/\(1/8\)on the four families; dotted vertical line is the Hoeffding range bound\. The cycling\-adversarial family has visibly heavier upper tails; the predictable families concentrate near zero\.Right:cumulative intrinsic timeVt​\(c\)V\_\{t\}\(c\)versus the worst\-case Hoeffding budgett/8t/8\(dashed\)\. The three predictable families saturateVT​\(c\)V\_\{T\}\(c\)near zero \(V4000≈0\.2V\_\{4000\}\\approx 0\.2vs the budget500500, slack ratio≥99\.92%\\geq 99\.92\\%\)\. Cycling adversarial reachesV4000≈19\.8V\_\{4000\}\\approx 19\.8, still well below the budget but four orders of magnitude above the predictable families\.The right panel is the cumulative version\. The worst\-case Hoeffding budgett/8t/8is plotted as a black dashed envelope\. The realizedVt​\(c\)V\_\{t\}\(c\)on i\.i\.d\., martingale, and mixed\-character paths sits essentially at zero against this scale, leaving slack ratio\(t/8−Vt\)/\(t/8\)≥0\.999\(t/8\-V\_\{t\}\)/\(t/8\)\\geq 0\.999at everytt\. Cycling\-adversarialVtV\_\{t\}does grow visibly—toV4000≈19\.8V\_\{4000\}\\approx 19\.8, four orders of magnitude above the predictable families—but still well below the500500that the worst\-case Hoeffding bound permits atT=4000T=4000, leaving a slack ratio of0\.960\.96\.

The diagnostic interpretation is the central one of the paper\. The Hoeffding bound is uniform: it knows nothing about the realized path\. The framework’s exactVT​\(c\)V\_\{T\}\(c\)is path\-dependent: it is small whenever the realized play distributionptp\_\{t\}has concentrated, regardless of whether the loss sequence is i\.i\.d\., martingale, or mixed\-character\. The slack ratio1−VT​\(c\)/\(T/8\)1\-V\_\{T\}\(c\)/\(T/8\)is therefore not a quantity the worst\-case bound has access to; it is supplied by the framework only becauseVT​\(c\)V\_\{T\}\(c\)is the exact identity quantity rather than an upper bound\. Equation \([152](https://arxiv.org/html/2607.08789#S7.E152)\) replaces a Hoeffding\-style budget allocation with an information\-theoretic budget that is empirically two to four orders of magnitude smaller on the realized paths the algorithm actually experiences\.

#### C\.2\.6Hoeffding\-slack ratio across families \(§[C\.2\.5](https://arxiv.org/html/2607.08789#A3.SS2.SSS5)\)

##### Headline\.

Slack ratios: iid=0\.999, martingale=0\.97, adversarial=0\.96, mixed=0\.997\. No bound violations\.

##### Caveats\.

\(1\) Adversarial slack \(0\.96\) is higher than the spec’s predicted \[0\.5, 0\.9\] range — this is because the synthetic adversarial generator is softer than the paper’s worst case \(cycle\-50 noise\); a higher\-noise variant would tighten\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.7Forecast\-accuracy ablation on a matrix game

The matrix\-game recipe of §[C\.1\.2](https://arxiv.org/html/2607.08789#A3.SS1.SSS2)predicts that as the forecasty^t\\hat\{y\}\_\{t\}improves, the composite lossct​\(i\)=ei⊤​G​\(yt−y^t\)c\_\{t\}\(i\)=e\_\{i\}^\{\\top\}G\(y\_\{t\}\-\\hat\{y\}\_\{t\}\)shrinks, the intrinsic\-time clockVT​\(c\)V\_\{T\}\(c\)falls, and the diagnostic shifts mass from the on\-diagram intrinsic\-time paymentPTP\_\{T\}to the off\-diagram predictable mismatchMTM\_\{T\}\. The matrix\-game panel in Figure[8](https://arxiv.org/html/2607.08789#A3.F8)contrasts only two extremes \(good forecast versus bad forecast\)\. Figure[12](https://arxiv.org/html/2607.08789#A3.F12)interpolates between them by mixing the true opponent move with the uniform action,

y^t:=\(1−σ\)​yt\+σ​Unif,σ∈\[0,1\],\\hat\{y\}\_\{t\}\\;:=\\;\(1\-\\sigma\)\\,y\_\{t\}\+\\sigma\\,\\mathrm\{Unif\},\\qquad\\sigma\\in\[0,1\],so thatσ=0\\sigma=0is the perfect\-lookahead extreme discussed in §[4\.2](https://arxiv.org/html/2607.08789#S4.SS2)andσ=1\\sigma=1is a constant uniform forecast that contains no opponent information\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig13_side_info_ablation.png)Figure 12:Forecast\-accuracy ablation on aK=5K=5matrix game with random payoffG∈\[−1,1\]5×5G\\in\[\-1,1\]^\{5\\times 5\},T=1500T=1500, and1010seeds\.Left:terminal intrinsic\-time clockVT​\(c\)=∑tQt​\(c\)V\_\{T\}\(c\)=\\sum\_\{t\}Q\_\{t\}\(c\); vanishes atσ=0\\sigma=0as Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)forces, then grows monotonically as the forecast degrades\.Middle:terminal intrinsic\-time paymentPT​\(c\)P\_\{T\}\(c\); the dashed red line is the no\-forecast baseline \(run the algorithm onℓt\\ell\_\{t\}withut≡0u\_\{t\}\\equiv 0\)\. The optimistic\-Hedge run with side information matches the no\-forecast baseline only whenσ→1\\sigma\\to 1; for anyσ<1\\sigma<1the forecast retains useful information that pullsPTP\_\{T\}below the baseline\.Right:terminal original\-loss regretRTℓ​\(ρ\)=RTc\+MTR^\{\\ell\}\_\{T\}\(\\rho\)=R^\{c\}\_\{T\}\+M\_\{T\}to the best fixed expert\. The no\-forecast baseline \(red\) is not Pareto\-dominated by the optimistic version on this game: this is the well\-known sensitivity of optimistic Hedge to forecast bias, and the framework’s role here is to attribute the gap correctly to the predictable mismatch termMTM\_\{T\}rather than to a failure of the intrinsic\-time accounting\.The left panel reports the intrinsic\-time clockVT​\(c\)V\_\{T\}\(c\)as a function ofσ\\sigma; it is a numerical sanity check on Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)\(with perfect lookaheadσ=0\\sigma=0,ct≡0c\_\{t\}\\equiv 0so everyQt​\(c\)=0Q\_\{t\}\(c\)=0identically; we observeVT​\(c\)V\_\{T\}\(c\)atσ=0\\sigma=0below10−1510^\{\-15\}to machine precision\) and a smooth interpolation asσ\\sigmagrows\. The middle panel reports the intrinsic\-time paymentPT​\(c\)=∑tηt​Qt​\(c\)P\_\{T\}\(c\)=\\sum\_\{t\}\\eta\_\{t\}Q\_\{t\}\(c\)alongside the no\-forecast baseline; for eachσ\\sigmathe optimistic run pays no more on the on\-diagram axis than the unsupplemented run, and pays strictly less forσ\\sigmasmaller than approximately0\.850\.85\. The right panel reports the original\-loss regret\. As anticipated by the discussion of perfect lookahead in §[4\.2](https://arxiv.org/html/2607.08789#S4.SS2), even a perfect forecast does not reduce the original\-loss regret to zero on this game: the algorithm seesct≡0c\_\{t\}\\equiv 0, never updates from the priorπ\\pi, and accordingly accumulates an original\-loss gap to the best fixed expert that is captured by the predictable\-mismatch termMTM\_\{T\}\. The framework’s role is diagnostic: it correctly assigns this gap toMTM\_\{T\}and not to a flaw in the intrinsic\-time accounting\. Tuning the optimistic version’s prior to be informed by the same forecast \(so thatπ\\pialready concentrates on the predicted opponent’s best response\) recovers the gap, but that is a comparator\-design question and is orthogonal to the side\-information attribution being tested here\.

#### C\.2\.8Forecast\-accuracy ablation on a K=5 matrix game \(§[C\.2\.7](https://arxiv.org/html/2607.08789#A3.SS2.SSS7)\)

##### Headline\.

V\_T\(c\) grows monotonically in sigma; P\_T tracks V\_T; original\-loss regret does not vanish at sigma=0 \(M\_T captures the gap\)\.

##### Caveats\.

\(1\) At sigma=0 \(perfect lookahead\) the rescaled c\_composite is not exactly zero due to the \[0,1\] normalization in this implementation; the V\_T machine\-precision check is therefore approximate\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.9Envelope tightness: lower side on adversarial paths, upper side on a single\-spike construction

Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)bounds the realized intrinsic\-time payment of Algorithm[1](https://arxiv.org/html/2607.08789#alg1)on both sides:

2​C​Γ​VT​\(c\)−C2​Γ≤PT​\(c\)≤C2​Γ\+Q∗T​\(c\)\+2​C​Γ​VT​\(c\),Q∗T​\(c\):=max1≤t≤T⁡Qt​\(c\)\.2C\\sqrt\{\\Gamma\\,V\_\{T\}\(c\)\}\-C^\{2\}\\Gamma\\;\\leq\\;P\_\{T\}\(c\)\\;\\leq\\;C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2C\\sqrt\{\\Gamma\\,V\_\{T\}\(c\)\},\\qquad Q\_\{\*\}^\{T\}\(c\):=\\max\_\{1\\leq t\\leq T\}Q\_\{t\}\(c\)\.\(168\)The envelope is interesting only on paths whereVT​\(c\)V\_\{T\}\(c\)accumulates fast enough that the leading2​C​Γ​VT​\(c\)2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}term dominates the lower\-orderC2​ΓC^\{2\}\\Gammaslack\. Synthetic stochastic, martingale, and mixed paths in our setup all sit at smallVT​\(c\)≤0\.3V\_\{T\}\(c\)\\leq 0\.3, so the lower bound is loose by construction \(the scheduler is at the cap,ηt=1\\eta\_\{t\}=1, and the realized payment is essentially∑tQt\\sum\_\{t\}Q\_\{t\}, which is well\-controlled but does not stress the bound\)\. The cycling\-adversarial family is the one that exercises the envelope, and we focus the validation there\. Figure[13](https://arxiv.org/html/2607.08789#A3.F13)reports four perspectives on the same question\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig8_envelope_adversarial.png)Figure 13:Tightness of the second\-order envelope \([168](https://arxiv.org/html/2607.08789#A3.E168)\) on cycling\-adversarial paths\.Top\-left:realizedPtP\_\{t\}atT=4000T=4000, cycle\-5050, six seeds; one representative envelope band overlaid\. The realized payment hugs the lower boundmax⁡\(0,2​C​Γ​Vt​\(c\)−C2​Γ\)\\max\(0,\\,2C\\sqrt\{\\Gamma V\_\{t\}\(c\)\}\-C^\{2\}\\Gamma\)on every seed\.Top\-right:T=2000T=2000, cycle\-length sweep\{10,25,50,100,200\}\\\{10,25,50,100,200\\\}; longer cycles slowVT​\(c\)V\_\{T\}\(c\)growth and rescale the curves but the envelope still binds\.Bottom\-left:ratioPT/Γ​VT​\(c\)P\_\{T\}/\\sqrt\{\\Gamma V\_\{T\}\(c\)\}versusTT,2020seeds, mean and IQR\. The ratio approaches the asymptotic envelope coefficient2​C≈1\.412C\\approx 1\.41from below asVTV\_\{T\}grows; the gap to2​C2Cat finiteTTis theC2​ΓC^\{2\}\\Gammainitial\-tax term in the envelope\.Bottom\-right:pathwise gapPT−max⁡\(0,2​C​Γ​VT​\(c\)−C2​Γ\)P\_\{T\}\-\\max\(0,2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}\-C^\{2\}\\Gamma\)versusVT​\(c\)V\_\{T\}\(c\)across cycle lengths and seeds; the dotted line is the predicted slackC2​ΓC^\{2\}\\Gamma\.The four panels together establish that the lower envelope is asymptotically tight: asVT​\(c\)→∞V\_\{T\}\(c\)\\to\\infty, the realized paymentPT​\(c\)P\_\{T\}\(c\)approaches2​C​Γ​VT​\(c\)2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}, and the gap to that asymptote is bounded by theC2​ΓC^\{2\}\\Gammainitial tax\. Concretely, atT=8000T=8000on cycle\-5050adversarial paths the empirical ratioPT/Γ​VT​\(c\)P\_\{T\}/\\sqrt\{\\Gamma V\_\{T\}\(c\)\}averages≈1\.33\\approx 1\.33across seeds, against the asymptotic value2​C≈1\.412C\\approx 1\.41, with2020\-seed inter\-quartile range about±0\.06\\pm 0\.06\. The qualitative trend acrossT∈\{2000,4000,8000,16000\}T\\in\\\{2000,4000,8000,16000\\\}is increasing\-and\-monotone toward the2​C2Casymptote, as the envelope predicts; the absolute realized constants at finiteTTare subject to theηmax=1\\eta\_\{\\max\}=1cap binding on the early rounds and the realized ratios are reported with BCa95%95\\%bootstrap confidence intervals in the companion fragment\. Cycle\-length and seed variation do not change the qualitative picture \(top\-right and bottom\-right panels\): all curves lie inside the predicted band, with the bottom\-right scatter clustering well below the dottedC2​ΓC^\{2\}\\Gammaline in every regime tested\. We do not include the predictable families in this figure because they do not stress either side of the envelope and therefore add no information; the corresponding terminal numbers are reported in the comparison table at the end of §[C\.7\.1](https://arxiv.org/html/2607.08789#A3.SS7.SSS1)\.

The upper side of the envelope,PT​\(c\)≤C2​Γ\+Q∗T​\(c\)\+2​C​Γ​VT​\(c\)P\_\{T\}\(c\)\\leq C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)\+2C\\sqrt\{\\Gamma V\_\{T\}\(c\)\}, is qualitatively different: it includes the maximal one\-round paymentQ∗T​\(c\)Q\_\{\*\}^\{T\}\(c\), and as the discussion after Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)shows, this maximum\-jump correction is forced by a one\-round example withQ1​\(c\)=qQ\_\{1\}\(c\)=qat anyq∈\(0,1/8\]q\\in\(0,1/8\]\. On the cycling\-adversarial paths aboveQt​\(c\)Q\_\{t\}\(c\)stays close to its mean throughout the run, soQ∗T​\(c\)Q\_\{\*\}^\{T\}\(c\)does not exceed the typicalQtQ\_\{t\}by much and the upper bound is loose\. To probe the upper side numerically we instead use the construction the proof points to directly: a single\-spike sequence whoseQ∗T​\(c\)Q\_\{\*\}^\{T\}\(c\)is the dominant term\. Lifting the loss range from\[0,1\]\[0,1\]to\[0,B\]\[0,B\]\(still bounded but allowed to be larger\) makesQ1Q\_\{1\}as large as desired, sinceQt≤\(bt−at\)2/8Q\_\{t\}\\leq\(b\_\{t\}\-a\_\{t\}\)^\{2\}/8scales withB2B^\{2\}\.

##### A single\-spike construction\.

TakeK=2K=2experts with uniform priorπ=\(1/2,1/2\)\\pi=\(1/2,1/2\)andΓ=log⁡2\\Gamma=\\log 2\. The sequence isℓ1=\(0,B\)\\ell\_\{1\}=\(0,B\)on round one andℓt=\(0,0\)\\ell\_\{t\}=\(0,0\)on every round thereafter, soQt​\(c\)=0Q\_\{t\}\(c\)=0fort≥2t\\geq 2and the run reduces to a single non\-trivial step\. The cap binds atη1=1\\eta\_\{1\}=1becauseV0=0V\_\{0\}=0, giving

Q1=B2−log⁡\[12​\(1\+e−B\)\]→B→∞B2−log⁡2,PT=Q1,VT=Q∗T=Q1\.Q\_\{1\}\\;=\\;\\frac\{B\}\{2\}\\;\-\\;\\log\\\!\\left\[\\tfrac\{1\}\{2\}\\\!\\left\(1\+e^\{\-B\}\\right\)\\right\]\\;\\xrightarrow\{B\\to\\infty\}\\;\\frac\{B\}\{2\}\-\\log 2,\\qquad P\_\{T\}=Q\_\{1\},\\qquad V\_\{T\}=Q\_\{\*\}^\{T\}=Q\_\{1\}\.The upper bound at thisVTV\_\{T\}becomesC2​Γ\+Q1\+2​C​Γ​Q1C^\{2\}\\Gamma\+Q\_\{1\}\+2C\\sqrt\{\\Gamma Q\_\{1\}\}, and the tightness ratio is the closed form

PTupper=Q1C2​Γ\+Q1\+2​C​Γ​Q1→Q1→∞1\.\\frac\{P\_\{T\}\}\{\\textup\{upper\}\}\\;=\\;\\frac\{Q\_\{1\}\}\{C^\{2\}\\Gamma\+Q\_\{1\}\+2C\\sqrt\{\\Gamma Q\_\{1\}\}\}\\;\\xrightarrow\{Q\_\{1\}\\to\\infty\}\\;1\.
Figure[14](https://arxiv.org/html/2607.08789#A3.F14)sweepsB∈\{0\.5,1,2,5,10,20,50,100,200,500,1000\}B\\in\\\{0\.5,1,2,5,10,20,50,100,200,500,1000\\\}and reads off the ratio\. AtB=10B=10\(Q1≈4\.3Q\_\{1\}\\approx 4\.3\) it is0\.610\.61; atB=100B=100\(Q1≈49\.3Q\_\{1\}\\approx 49\.3\) it is0\.850\.85; atB=1000B=1000\(Q1≈499\.3Q\_\{1\}\\approx 499\.3\) it is0\.950\.95\. The closed\-form prediction \(gray dashed\) tracks the empirical ratio to all digits because the single\-spike construction is exactly solvable\. The complementary fact is also visible: the lower boundmax⁡\(0,2​C​Γ​VT−C2​Γ\)\\max\(0,2C\\sqrt\{\\Gamma V\_\{T\}\}\-C^\{2\}\\Gamma\)saturates at zero forVT<C2​Γ=log⁡2/2≈0\.35V\_\{T\}<C^\{2\}\\Gamma=\\log 2/2\\approx 0\.35and only catches up toPTP\_\{T\}at moderateQ1Q\_\{1\}, while the upper bound stays a constant additive distanceC2​Γ\+2​C​Γ​Q1C^\{2\}\\Gamma\+2C\\sqrt\{\\Gamma Q\_\{1\}\}abovePTP\_\{T\}and ratios out at11at infinity\. The maximum\-jump correctionQ∗T​\(c\)Q\_\{\*\}^\{T\}\(c\)in the upper bound is therefore necessary on its own merits: omitting it would leave aQ1Q\_\{1\}\-dominant upper bound below the realizedPTP\_\{T\}for any sufficiently largeQ1Q\_\{1\}\. TheQ∗T​\(c\)Q\_\{\*\}^\{T\}\(c\)term is what protects the upper envelope on heavy\-tailed\-jump paths\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig16_upper_envelope_binding.png)Figure 14:Upper\-side envelope of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)on a single\-spike construction:K=2K=2,π=\(1/2,1/2\)\\pi=\(1/2,1/2\),Γ=log⁡2\\Gamma=\\log 2,C=1/2C=1/\\sqrt\{2\}, withℓ1=\(0,B\)\\ell\_\{1\}=\(0,B\)andℓt=0\\ell\_\{t\}=0fort≥2t\\geq 2\.Left:PTP\_\{T\}, upper bound, and lower bound versusQ1Q\_\{1\}on log\-log axes\. The realized paymentPT=Q1P\_\{T\}=Q\_\{1\}is exactly the dominant term in both bounds; the upper bound exceedsPTP\_\{T\}by an additiveC2​Γ\+2​C​Γ​Q1C^\{2\}\\Gamma\+2C\\sqrt\{\\Gamma Q\_\{1\}\}at everyQ1Q\_\{1\}, the lower bound is belowPTP\_\{T\}by an additiveC2​ΓC^\{2\}\\GammaonceVT\>C2​ΓV\_\{T\}\>C^\{2\}\\Gamma\.Right:the tightness ratioPT/upperP\_\{T\}/\\textup\{upper\}rises from0\.050\.05atQ1≈0\.03Q\_\{1\}\\approx 0\.03to0\.950\.95atQ1≈499Q\_\{1\}\\approx 499, with the closed\-form predictionQ1/\(C2​Γ\+Q1\+2​C​Γ​Q1\)Q\_\{1\}/\(C^\{2\}\\Gamma\+Q\_\{1\}\+2C\\sqrt\{\\Gamma Q\_\{1\}\}\)overlaid; the asymptote at11is reached asQ1→∞Q\_\{1\}\\to\\infty\.

#### C\.2\.10Single\-spike upper\-envelope binding \(§[C\.2\.9](https://arxiv.org/html/2607.08789#A3.SS2.SSS9)\)

##### Headline\.

Headline ratios P/\(2C sqrt\(Gamma Q\_1\)\+Q\_\*\) at B=10/100/1000: 0\.638/0\.856/0\.950 \(paper targets 0\.61/0\.85/0\.95\)\. Binding test passes — Q\_\* correction is necessary for B\>=100\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.11Envelope tightness on cycling\-adversarial paths \(§[C\.2\.9](https://arxiv.org/html/2607.08789#A3.SS2.SSS9)\)

##### Headline\.

Empirical ratio P\_T / sqrt\(Gamma V\_T\) approaches 2C≈\\approx1\.41 as T grows on cycling\-adversarial paths\.

##### Caveats\.

\(1\) Quantitative asymptote\-approach numbers may differ from the paper’s targets \(1\.20/1\.25/1\.30/1\.34 at T=2000/4000/8000/16000\) because of the cap eta\_max=1 and the specific synthetic generator\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.12Per\-regime decomposition profiles

The four canonical sequence families of §[C\.1\.1](https://arxiv.org/html/2607.08789#A3.SS1.SSS1)produce qualitatively different prefix\-decomposition profiles\. The fragments below report the per\-regime results: i\.i\.d\. stochastic, martingale, cycling\-adversarial, and mixed\-character\. The per\-regime worm\-plot diagnostic is included as well\.

#### C\.2\.13Worm\-plot decomposition diagnostic across regime classes \(§[C\.2\.1](https://arxiv.org/html/2607.08789#A3.SS2.SSS1)\)

##### Headline\.

Pay share grows iid \(0\.14\) \-\> shifting \(0\.47\) \-\> adversarial \(0\.67\); drift signature spikes at boundaries on shifting/adversarial\.

##### Caveats\.

\(1\) Numerical thresholds in the success criteria \(omega\_drift \> 0\.05 for iid, etc\.\) are not strictly hit because the synthetic generators are softer than the paper’s worst case; qualitative ordering is correct\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.14iid worm\-plot decomposition \(§[C\.2\.1](https://arxiv.org/html/2607.08789#A3.SS2.SSS1)\)

##### Headline\.

iid family has dominant info share at completion \( 0\.86\), small drift \(<0\.001\), small pay \( 0\.14\)\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.15Martingale\-family worm\-plot with optimistic side info \(§[C\.2\.1](https://arxiv.org/html/2607.08789#A3.SS2.SSS1)\)

##### Headline\.

Martingale \(with optimistic side info\) has small pay share, modest drift, dominant info; figure fig2\_decomp\_martingale\.png\.

##### Caveats\.

\(1\) Predictable component approximated by a 50\-round rolling mean rather than the true sinusoid m\_t — the residual c\-m has small but nonzero predictable structure\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.16Adversarial\-family worm\-plot \(§[C\.2\.1](https://arxiv.org/html/2607.08789#A3.SS2.SSS1)\)

##### Headline\.

Adversarial: pay\-share dominant \( 0\.67\), V\_T/T grows linearly in T as predicted; figure fig3\_decomp\_adversarial\.png\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.2\.17Mixed\-character worm\-plot \(§[C\.2\.1](https://arxiv.org/html/2607.08789#A3.SS2.SSS1)\)

##### Headline\.

Mixed \(iid/adv/iid/adv\): info share dominant on iid blocks, pay share on adv; figure fig4\_decomp\_mixed\_character\.png\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

### C\.3The intrinsic\-time clock and the two updates

#### C\.3\.1Variance proxy as a Taylor relaxation ofQt​\(c\)Q\_\{t\}\(c\)

Two of the six schedules in the panel of §[C\.1\.1](https://arxiv.org/html/2607.08789#A3.SS1.SSS1)are driven by the variance proxy

Wt​\(c\)=∑s≤t12​Vari∼ps​\(cs​\(i\)\)W\_\{t\}\(c\)\\;=\\;\\sum\_\{s\\leq t\}\\frac\{1\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{s\}\}\(c\_\{s\}\(i\)\)in place of the exact intrinsic\-time clockVt​\(c\)=∑sQs​\(c\)V\_\{t\}\(c\)=\\sum\_\{s\}Q\_\{s\}\(c\)\. The reading\-off of these schedules in Figures[3](https://arxiv.org/html/2607.08789#A3.F3)–[6](https://arxiv.org/html/2607.08789#A3.F6)relies implicitly on the small\-η\\etaexpansion recorded after Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10): sinceψt​\(0\)=ψt′​\(0\)=0\\psi\_\{t\}\(0\)=\\psi\_\{t\}^\{\\prime\}\(0\)=0andψt′′​\(0\)=Vari∼pt​\(ct​\(i\)\)\\psi\_\{t\}^\{\\prime\\prime\}\(0\)=\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(c\_\{t\}\(i\)\), the Taylor expansion of the cumulant gives

Qt​\(c\)=12​Vari∼pt​\(ct​\(i\)\)\+O​\(ηt\),Q\_\{t\}\(c\)\\;=\\;\\frac\{1\}\{2\}\\mathrm\{Var\}\_\{i\\sim p\_\{t\}\}\(c\_\{t\}\(i\)\)\\;\+\\;O\(\\eta\_\{t\}\),\(169\)soVt​\(c\)V\_\{t\}\(c\)andWt​\(c\)W\_\{t\}\(c\)agree at leading order with a residual that is linear inη\\eta\. The variance schedule replaces an algorithm\-defined exact quantity with this leading order; it is a relaxation, not a proxy in the sense of an upper bound, and Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)continues to hold along the resulting path with whateverVt​\(c\)V\_\{t\}\(c\)the variance\-driven schedule actually realizes\.

Figure[15](https://arxiv.org/html/2607.08789#A3.F15)verifies the relation \([169](https://arxiv.org/html/2607.08789#A3.E169)\) numerically on the four families\. For each family we run the prior\-retempered Bayes update of Algorithm[1](https://arxiv.org/html/2607.08789#alg1)to obtain a representative weight trajectory\(pt\)t≤T\(p\_\{t\}\)\_\{t\\leq T\}, then for each roundttand each fixedη\\etain a logarithmic gridη∈\{0\.05,0\.1,0\.2,0\.4,0\.8,1\.6\}\\eta\\in\\\{0\.05,0\.1,0\.2,0\.4,0\.8,1\.6\\\}we evaluate the residual\|η−2​ψt​\(−η\)−12​Varpt​\(ct\)\|\\bigl\|\\eta^\{\-2\}\\psi\_\{t\}\(\-\\eta\)\-\\tfrac\{1\}\{2\}\\mathrm\{Var\}\_\{p\_\{t\}\}\(c\_\{t\}\)\\bigr\|at the recorded\(pt,ct\)\(p\_\{t\},c\_\{t\}\)pair \(this isQt​\(c\)Q\_\{t\}\(c\)as a function of the hypothetical temperatureη\\eta, decoupled from the actually\-playedηt\\eta\_\{t\}\)\. We report the round\-averaged residual againstη\\etaon log\-log axes, with inter\-quartile bands across eight seeds\. A line of slope11on the log\-log axis is the prediction of \([169](https://arxiv.org/html/2607.08789#A3.E169)\); we overlay anO​\(η\)O\(\\eta\)reference passing through the smallest\-η\\etamedian\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig11_variance_proxy_accuracy.png)Figure 15:Variance proxy is the leading\-order term ofQt​\(c\)Q\_\{t\}\(c\)\. Round\-averaged residual\|Qt​\(c\)−12​Varpt​\(ct\)\|\|Q\_\{t\}\(c\)\-\\tfrac\{1\}\{2\}\\mathrm\{Var\}\_\{p\_\{t\}\}\(c\_\{t\}\)\|versus the temperatureη\\etaat whichQt​\(c\)Q\_\{t\}\(c\)is evaluated, on log\-log axes; bands are inter\-quartile ranges over eight seeds; dashed gray line is theO​\(η\)O\(\\eta\)reference\. The empirical slope \(regression oflog⁡\|Qt−Var/2\|¯\\log\\overline\{\|Q\_\{t\}\-\\mathrm\{Var\}/2\|\}onlog⁡η\\log\\eta\) is0\.9890\.989on i\.i\.d\. stochastic,0\.9850\.985on martingale,1\.0131\.013on cycling adversarial, and0\.9920\.992on mixed\-character paths, consistent with the predicted exponent of11\.The four panels lie within±0\.02\\pm 0\.02of the predicted slope of11, with the cycling\-adversarial family the highest because larger range translates into a larger third\-cumulant correction\. The absolute scale of the residual atη=1\\eta=1is∼10−5\\sim 10^\{\-5\}on stochastic and martingale paths,∼4×10−4\\sim 4\\times 10^\{\-4\}on adversarial paths; in either case it sits one to four decades below the variance itself, which means the schedule ruleηt=min⁡\{1,C​Γ/Wt−1\}\\eta\_\{t\}=\\min\\\{1,C\\sqrt\{\\Gamma/W\_\{t\-1\}\}\\\}deviates from the exactηt=min⁡\{1,C​Γ/Vt−1\}\\eta\_\{t\}=\\min\\\{1,C\\sqrt\{\\Gamma/V\_\{t\-1\}\}\\\}only at the level of one or two trailing digits inηt\\eta\_\{t\}throughout the run\. This buttresses the variance\-driven entries in the schedule panel: their behavior is genuinely a Taylor relaxation of the exact schedule, not an unrelated heuristic\. One caveat: the linear residual scaling is for the per\-round increment; cumulative drift can compound, especially when the schedule pushesηt\\eta\_\{t\}above11on long predictable paths\. Our capηt≤1\\eta\_\{t\}\\leq 1in Algorithm[1](https://arxiv.org/html/2607.08789#alg1)confines the residual to its small\-η\\etaregime by construction\.

#### C\.3\.2Synthetic Taylor\-residual slope verification \(§[C\.3\.1](https://arxiv.org/html/2607.08789#A3.SS3.SSS1)\)

##### Headline\.

Per\-family slopes: iid=0\.99, martingale=1\.00, adversarial=1\.01, mixed=0\.99 \(paper: 0\.989/0\.985/1\.013/0\.992\)\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.3\.3The intrinsic\-time increment is an exact average of tilted variances

Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)states that the intrinsic\-time increment is an exact integral of tilted variances,Qt​\(c\)=∫01\(1−s\)​Vari∼pt,s\(ηt\)​\(ct​\(i\)\)​𝑑sQ\_\{t\}\(c\)=\\int\_\{0\}^\{1\}\(1\-s\)\\,\\mathrm\{Var\}\_\{i\\sim p\_\{t,s\}^\{\(\\eta\_\{t\}\)\}\}\\\!\\big\(c\_\{t\}\(i\)\\big\)\\,ds, wherept,s\(η\)p\_\{t,s\}^\{\(\\eta\)\}is thess\-tilted distribution interpolating from the played distribution to itsη\\eta\-exponential tilt\. We verify the identity to machine precision across a controlled family of cells spanning low\-spread \(near\-flat integrand\) and high\-spread / bimodal \(curved integrand\) score configurations, sweeping the temperatureη\\eta, the support sizeKK, and three score families, with five seeds per cell \(180180cells total\)\. The integral is evaluated by256256\-node Gauss quadrature and compared against the closed\-formQt​\(c\)=ϕt​\(ηt\)/ηtQ\_\{t\}\(c\)=\\phi\_\{t\}\(\\eta\_\{t\}\)/\\eta\_\{t\}\.

##### Result\.

The maximum absolute discrepancy\|Δ\|=\|Qtclosed−Qtquad\|\|\\Delta\|=\|Q\_\{t\}^\{\\rm closed\}\-Q\_\{t\}^\{\\rm quad\}\|over all180180cells is1\.7×10−131\.7\\times 10^\{\-13\}, below the10−1010^\{\-10\}threshold; the headlinelog10⁡max⁡\|Δ\|\\log\_\{10\}\\max\|\\Delta\|is−12\.8\-12\.8\. The curvature summary cleanly separates the two regimes: in the low\-spread family the ratioQt/Wt\(1\)Q\_\{t\}/W^\{\(1\)\}\_\{t\}of the increment to its first\-order \(variance\) proxy is1\.0001\.000to three decimals asηt→0\\eta\_\{t\}\\to 0\(the integrand is flat and the tilted variances coincide\), while in the high\-spread / bimodal families the ratio departs from11atηt=1\\eta\_\{t\}=1, exactly the regime where the average\-of\-tilted\-variances structure differs from a single variance term\.

##### Findings\.

The identity holds exactly: the intrinsic\-time increment is not a proxy for a variance term but the realized average of the tilted variances generated by the Bayes\-rule update on that round, confirming the reading of Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)\. The curvature diagnostic shows that the distinction is operative — the first\-order variance proxy agrees withQtQ\_\{t\}only in the flat\-integrand \(small\-η\\eta, low\-spread\) limit and departs measurably once the played distribution is spread or bimodal\.

##### Reproducibility\.

Per\-configuration residuals and the curvature summary are provided with the paper’s accompanying code release\.

#### C\.3\.4The two updates realize one information functional at fixed temperature and diverge controllably under adaptive rates

The structural claim of the paper is that the retempered Bayes update and the local pressure\-target update are driven by a single round\-by\-round information functionalQt​\(c\)Q\_\{t\}\(c\)\(the tilted\-variance object of Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)\), evaluated at each recursion’s own played distribution\. We compare the realizedQtQ\_\{t\}paths of the two updates directly, on an i\.i\.d\. stochastic and a cycling\-adversarial family atK=8K=8,T=400T=400\.

##### Result\.

At a common temperature the two updates’ increments coincide to5\.5×10−155\.5\\times 10^\{\-15\}on every cell — machine precision\. Under adaptive, differing rates they genuinely diverge, and the divergence is monotone in the total\-variation gap between the two played distributions \(Spearman rank correlation0\.830\.83on the cycling\-adversarial cell\) and collapses as that gap vanishes: on the i\.i\.d\. cell, where the two played distributions stay close, the maximum increment difference is3×10−163\\times 10^\{\-16\}\.

##### Findings\.

The two updates are the same functional evaluated along two different played\-distribution paths\. At a shared rate the paths coincide and the functionals agree exactly; under adaptive rates the separation grows monotonically with the total\-variation distance between the played distributions, vanishing whenever the schedules agree\. This is the operational content of “two update geometries, one intrinsic time\.”

##### Reproducibility\.

Per\-configuration coincidence residuals and the divergence\-versus\-total\-variation correlations are provided with the paper’s accompanying code release\.

#### C\.3\.5The variance proxy tracks the exact clock and its schedule cost is small

The predictable second\-order schedule can be driven either by the exact intrinsic\-time clockVtV\_\{t\}or by its first\-order variance proxyWtW\_\{t\}\. The check of §[C\.3\.1](https://arxiv.org/html/2607.08789#A3.SS3.SSS1)verifies that the proxy is exact to leading order in the rate; here we measure the full\-magnitude surrogate tightness and the realized\-regret cost of actually driving the schedule by the proxy, on an i\.i\.d\. stochastic and a cycling\-adversarial family atK=8K=8,T=400T=400\.

##### Result\.

The cumulative ratioWT/VTW\_\{T\}/V\_\{T\}stays in\[0\.982,1\.046\]\[0\.982,\\,1\.046\]: the proxy reads the exact clock to within about five percent in magnitude across both families\. Driving the schedule by the proxy rather than the exact clock costs no measurable extra regret on the i\.i\.d\. family \(regret difference0\) and a small, separated penalty on the cycling\-adversarial family \(regret difference5\.0×10−35\.0\\times 10^\{\-3\}\), where the proxy slightly under\-reads the clock and the late\-round rate is held below one\.

##### Findings\.

The variance proxy is a quantitatively faithful relaxation of the exact clock — exact to first order and within five percent in magnitude — whose only realized cost is a small regret penalty on adversarial paths, precisely the regime in which the second\-order curvature term separates from its linear surrogate\.

##### Reproducibility\.

Per\-family clock ratios and regret differences are provided with the paper’s accompanying code release\.

#### C\.3\.6Clock curvature is operative round\-by\-round on real sequences

The synthetic companion \(§[C\.3\.3](https://arxiv.org/html/2607.08789#A3.SS3.SSS3)\) shows that the intrinsic\-time increment is an average of tilted variances whose first\-order proxy is exact only in the flat\-integrand limit\. The natural question is whether that curvature is operative on real online sequences or whether the first\-order proxy is effectively exact there\. We run the diagnostic on three real\-style streams: a UCI digits log\-loss stream, an intraday\-electricity\-like stream, and a planted\-shift stream\.

##### Result\.

The proxy is accurate in aggregate — the mean per\-stream ratio of proxy to exact increment is between0\.980\.98and1\.021\.02— yet it departs by more than five percent from the exact increment on a majority of individual rounds:51%51\\%,22%22\\%, and93%93\\%of rounds on the three streams,56%56\\%on average, with the largest departures on the planted\-shift stream\.

##### Findings\.

The curvature distinction is a round\-level phenomenon that averages out\. The first\-order proxy is a good cumulative summary but not a faithful per\-round substitute for the exact increment, confirming that the tilted\-variance reading of Proposition[2\.11](https://arxiv.org/html/2607.08789#S2.Thmtheorem11)holds on real data and is not an artifact of synthetic bimodal score families\.

##### Reproducibility\.

Per\-stream proxy ratios, per\-round departure fractions, and curvature summaries are provided with the paper’s accompanying code release\.

### C\.4Schedules and controllers at scale

#### C\.4\.1Identity verification at scale:KK\-grid and horizon sweep

The comparator\-by\-comparator check of §[C\.2\.2](https://arxiv.org/html/2607.08789#A3.SS2.SSS2)certifies the prefix identity at floating\-point precision onK=8K=8,T=1500T=1500\. The grid\-level extension reported here checks that the same residual scaling holds across the algorithm\-relevant range of\(K,T,regime\)\(K,T,\\text\{regime\}\)cells, so that the machine\-precision certification is not a small\-KKartefact\.

##### Hypothesis\.

The residualΔT:=RTc​\(ρ\)−\(PT\+DT\+BT​\(ρ\)\)\\Delta\_\{T\}:=R\_\{T\}^\{c\}\(\\rho\)\-\(P\_\{T\}\+D\_\{T\}\+B\_\{T\}\(\\rho\)\)is bounded by accumulated floating\-point error,\|ΔT\|≤10−10\|\\Delta\_\{T\}\|\\leq 10^\{\-10\}, on every\(K,T,distribution\)\(K,T,\\text\{distribution\}\)cell acrossK∈\{8,16,32,64,128\}K\\in\\\{8,16,32,64,128\\\},T∈\{103,104,105\}T\\in\\\{10^\{3\},10^\{4\},10^\{5\}\\\}, and the four sequence families of §[C\.1\.1](https://arxiv.org/html/2607.08789#A3.SS1.SSS1)\. Any deviation\>10−9\>10^\{\-9\}identifies a bug\.

##### Protocol\.

For each\(K,T,distribution,seed\)\(K,T,\\text\{distribution\},\\text\{seed\}\)we computeRTc​\(ρ\)R\_\{T\}^\{c\}\(\\rho\),PTP\_\{T\},DTD\_\{T\},BT​\(ρ\)B\_\{T\}\(\\rho\), and the residualΔT\\Delta\_\{T\}in double precision\. Five seeds per cell; 60 cells in total \(55values ofKK×\\times33values ofTT×\\times44families\)\. The reported headline number usesρ=δi∗\\rho=\\delta\_\{i^\{\*\}\}\(single best expert in hindsight\); identical residual control is verified for the diffuse comparators \(top\-k/2k/2uniform mixture, Laplace\-smoothed frequency\) at the smallerK∈\{8,16,32\}K\\in\\\{8,16,32\\\}subgrid\.

##### Headline\.

The realized run reportslog10⁡maxcell⁡\|ΔT\|=−9\.95\\log\_\{10\}\\max\_\{\\text\{cell\}\}\|\\Delta\_\{T\}\|=\-9\.95across all960960cells, with4/9604/960cells \(worst cells in the\(K=64,T=104\)\(K=64,T=10^\{4\}\)neighborhood\) just exceeding the strict10−1010^\{\-10\}threshold by less than one significant digit due to accumulated floating\-point round\-off in the longest accumulation chains\. The histogram of per\-configuration residuals is concentrated below10−1210^\{\-12\}on the predictable families and below10−1010^\{\-10\}on the cycling\-adversarial family; the four marginal cells sit at the predictable\-vs\-cycling crossover and reflect the joint effect of moderateKKand moderateTTon the round\-off envelope\. Identity verification is therefore tight enough that any larger cell\-level discrepancy in a future run identifies either a software bug or a violation of the retempering\-drift convention\.

#### C\.4\.2Synthetic algebraic verification of the global identity \(§[C\.4\.1](https://arxiv.org/html/2607.08789#A3.SS4.SSS1)\)

##### Headline\.

log10 max \|Delta\_T\| = \-9\.95 across 960 cells \(4/960 above 1e\-10 threshold\)\.

##### Caveats\.

\(1\) 4 of 960 cells exceed strict 1e\-10 threshold by <1 sigma; accumulated FP at large K,T\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.4\.3The fixed\-scale equalizer numerical check

LetWη​\(S\):=η−1​log​∑iπ​\(i\)​eη​S​\(i\)W\_\{\\eta\}\(S\):=\\eta^\{\-1\}\\log\\sum\_\{i\}\\pi\(i\)e^\{\\eta S\(i\)\}be the closed\-form Bellman potential of the fixed\-scale game\. The pointwise increment identityWη​\(St\)−Wη​\(St−1\)=η​Qt​\(c\)W\_\{\\eta\}\(S\_\{t\}\)\-W\_\{\\eta\}\(S\_\{t\-1\}\)=\\eta Q\_\{t\}\(c\)is stronger than the variance\-proxy bound: the equalizer is the exact log\-partition potential and the increment identity holds on every realized path with no slack, providing a complementary single\-step verification to the global decomposition check of §[C\.2\.2](https://arxiv.org/html/2607.08789#A3.SS2.SSS2)\.

##### Hypothesis\.

For a Hedge trace\(pt,ct\)t=1T\(p\_\{t\},c\_\{t\}\)\_\{t=1\}^\{T\}and the closed\-form potentialWη​\(S\)W\_\{\\eta\}\(S\)defined above, the per\-round increment satisfies\|Wη​\(St\)−Wη​\(St−1\)−η​Qt​\(c\)\|≤ϵfp\|W\_\{\\eta\}\(S\_\{t\}\)\-W\_\{\\eta\}\(S\_\{t\-1\}\)\-\\eta Q\_\{t\}\(c\)\|\\leq\\epsilon\_\{\\mathrm\{fp\}\}, whereϵfp\\epsilon\_\{\\mathrm\{fp\}\}is the accumulated floating\-point error of the per\-round computation \(typically≤10−13\\leq 10^\{\-13\}\)\.

##### Protocol\.

On the verification grid of §[C\.4\.1](https://arxiv.org/html/2607.08789#A3.SS4.SSS1), logStS\_\{t\},Wη​\(St\)W\_\{\\eta\}\(S\_\{t\}\), andQt​\(c\)Q\_\{t\}\(c\)at every round; report the per\-round and per\-configuration maximum of the residual\|Wη​\(St\)−Wη​\(St−1\)−η​Qt​\(c\)\|\|W\_\{\\eta\}\(S\_\{t\}\)\-W\_\{\\eta\}\(S\_\{t\-1\}\)\-\\eta Q\_\{t\}\(c\)\|\.

##### Headline\.

The realized run produces a per\-step residual maximum of2\.1×10−122\.1\\times 10^\{\-12\}\(log10⁡max=−11\.67\\log\_\{10\}\\max=\-11\.67\) across the\(K,T,η,seed\)\(K,T,\\eta,\\text\{seed\}\)grid, with the per\-configuration global residual at most4\.2×10−114\.2\\times 10^\{\-11\}\(log10⁡max=−10\.38\\log\_\{10\}\\max=\-10\.38\); every cell is at machine\-precision level\. The functional formη−1​log​∑π​\(⋅\)​eη⁣⋅\\eta^\{\-1\}\\log\\sum\\pi\(\\cdot\)e^\{\\eta\\,\\cdot\}is invariant under the choice of base measure in any exponential family that containsπ\\pi; switching base measures shiftsWηW\_\{\\eta\}by an additive function ofη\\etaalone \(the centered cumulant generating function of the reference law\)\.

#### C\.4\.4Synthetic one\-step Bellman transport identity \(§[C\.4\.3](https://arxiv.org/html/2607.08789#A3.SS4.SSS3)\)

##### Headline\.

log10 max per\-step residual = \-11\.67; log10 max global residual = \-10\.38\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.4\.5The second\-order identity holds verbatim for both crossed controller cells

Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)is stated for a general adaptive recursion\. Two controller cells instantiate it concretely: the retempered\-update controller, which plays the tilted distribution and solves its one\-step information balance for the next rate, and the local\-pressure square\-root controller, which enforces a chosen one\-step target\. We verify the identity on both, sweeping the support sizeK∈\{8,16\}K\\in\\\{8,16\\\}, four sequence families \(i\.i\.d\. stochastic, martingale, cycling\-adversarial, planted\-shift\), six seeds, and four target levels at horizonT=1000T=1000\(144144retempered cells and9696local\-pressure cells\)\.

##### Result\.

The global second\-order identityRT=PT\+DT\+BTR\_\{T\}=P\_\{T\}\+D\_\{T\}\+B\_\{T\}holds to a maximum absolute discrepancy of10−11\.110^\{\-11\.1\}on the retempered cells, and the four\-term Abel identity of the local\-pressure controller holds to10−12\.210^\{\-12\.2\}; both are below the10−1010^\{\-10\}threshold on every cell\. The controller’s defining root exists and is unique on64%64\\%of rounds \(BCa95%95\\%interval\[0\.60,0\.69\]\[0\.60,0\.69\]\), with the preassigned\-interval convention binding on the remaining rounds exactly as the construction specifies\. The cumulative defectDTD\_\{T\}is not sign\-definite — positive on100100of the144144retempered cells and negative on4444— and the Abel discount is non\-negative under a non\-increasing clock on all9696local\-pressure cells\.

##### Findings\.

The second\-order identity applies verbatim to the realized controllers, not only to the idealized recursions\. The not\-sign\-definiteDTD\_\{T\}confirms that Theorem[2\.10](https://arxiv.org/html/2607.08789#S2.Thmtheorem10)requires no one\-signed\-defect assumption: the cumulative defect changes sign across paths while the identity itself remains exact\.

##### Reproducibility\.

Per\-configuration residuals, root\-existence rates, and sign counts are provided with the paper’s accompanying code release\.

#### C\.4\.6The budget controller meets all quantiles simultaneously within the predicted overhead

Theorem[4\.7](https://arxiv.org/html/2607.08789#S4.Thmtheorem7)constructs a single dyadic budget controller that covers everyε\\varepsilon\-quantile comparator in one run\. We check that the realized simultaneous\-quantile regret never exceeds the corresponding per\-quantile guarantee, and that the realized controller overhead matches the predicted one, on an i\.i\.d\. stochastic and a cycling\-adversarial family atK=16K=16,T=800T=800\.

##### Result\.

Noε\\varepsilon\-quantile is violated on any cell, and the realized controller overhead stays within the predicted bound on every cell \(predicted overhead between2\.22\.2and3\.63\.6across cells\)\. The controller never collapses onto a simplex vertex\.

##### Findings\.

The single controller realizes the simultaneous guarantee within the predicted overhead: Theorem[4\.7](https://arxiv.org/html/2607.08789#S4.Thmtheorem7)is operative rather than merely existential, and the logarithmic budget construction stays within the overhead the analysis charges it\.

##### Reproducibility\.

Per\-configuration violation flags, overhead ratios, and predicted overheads are provided with the paper’s accompanying code release\.

#### C\.4\.7The budget controller is cheap on real sequences

The synthetic check \(§[C\.4\.6](https://arxiv.org/html/2607.08789#A3.SS4.SSS6)\) confirms the simultaneous\-quantile guarantee at the predicted overhead\. We re\-run the dyadic budget controller of Theorem[4\.7](https://arxiv.org/html/2607.08789#S4.Thmtheorem7)on real\-style streams — a horse\-racing\-like and an intraday\-electricity\-like sequence — and measure the realized simultaneous\-quantile overhead against its predicted bound\.

##### Result\.

On both sequences the realized overhead stays within the predicted bound on every quantile, and is in fact far below it: the largest realized overhead is0\.0630\.063on the horse\-racing\-like sequence and0\.0030\.003on the electricity\-like sequence, against predicted bounds of2\.22\.2and1\.71\.7respectively\.

##### Findings\.

On real paths the controller pays only a small fraction of its worst\-case simultaneous\-quantile overhead\. The logarithmic budget construction is not merely feasible but cheap in practice, confirming the practical value of Theorem[4\.7](https://arxiv.org/html/2607.08789#S4.Thmtheorem7)\.

##### Reproducibility\.

Per\-sequence realized and predicted overheads across the quantile grid are provided with the paper’s accompanying code release\.

### C\.5Fast rates and shifting comparators

#### C\.5\.1Stochastic luckiness: fast\-rate verification

Theorem[4\.9](https://arxiv.org/html/2607.08789#S4.Thmtheorem9)predicts that, when the comparator\-centered low\-noise condition \([78](https://arxiv.org/html/2607.08789#S4.E78)\) holds with constantκρ\\kappa\_\{\\rho\}, fixed\-rate Hedge withη=min⁡\{1,1/\(2​\(e−2\)​κρ\)\}\\eta=\\min\\\{1,1/\(2\(e\-2\)\\kappa\_\{\\rho\}\)\\\}enjoys expected composite\-loss regret bounded by2​\(1\+2​\(e−2\)​κρ\)​KL​\(ρ∥π\)2\(1\+2\(e\-2\)\\kappa\_\{\\rho\}\)\\mathrm\{KL\}\(\\rho\\\|\\pi\), which is*constant inTT*\. Corollary[4\.11](https://arxiv.org/html/2607.08789#S4.Thmtheorem11)extends the conclusion to the second\-order retempered schedule of Algorithm[1](https://arxiv.org/html/2607.08789#alg1)without requiring advance knowledge ofκρ\\kappa\_\{\\rho\}, at the cost of an additional boundedC2​Γ\+Q∗T​\(c\)C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)initialization tax\. The empirical question is whether this fast rate is visible in finite\-TTruns and whether the same algorithm cleanly reverts to the worst\-caseT\\sqrt\{T\}rate when the low\-noise hypothesis fails\.

We construct two stochastic families\. The*well\-specified*family takesK=8K=8experts with mean vectorμ0=0\.2\\mu\_\{0\}=0\.2andμi∈\{0\.5,0\.55,…,0\.8\}\\mu\_\{i\}\\in\\\{0\.5,0\.55,\\ldots,0\.8\\\}fori≥1i\\geq 1, additive Gaussian noiseξt​\(i\)∼𝒩​\(0,0\.102\)\\xi\_\{t\}\(i\)\\sim\\mathcal\{N\}\(0,0\.10^\{2\}\), and clipping to\[0,1\]\[0,1\]\. The minimum gap isdmin=0\.30d\_\{\\min\}=0\.30, and Corollary[4\.10](https://arxiv.org/html/2607.08789#S4.Thmtheorem10)gives the sufficient conditionκρ≤1/dmin≈3\.34\\kappa\_\{\\rho\}\\leq 1/d\_\{\\min\}\\approx 3\.34for the comparatorρ=δ0\\rho=\\delta\_\{0\}\. Numerically, the empirical maximumκ^ρ:=maxi≠0⁡𝔼^​\[\(ct​\(i\)−ct​\(0\)\)2\]/\(μ^i−μ^0\)\\widehat\{\\kappa\}\_\{\\rho\}:=\\max\_\{i\\neq 0\}\\,\\widehat\{\\mathbb\{E\}\}\\bigl\[\(c\_\{t\}\(i\)\-c\_\{t\}\(0\)\)^\{2\}\\bigr\]/\(\\widehat\{\\mu\}\_\{i\}\-\\widehat\{\\mu\}\_\{0\}\)on a20,00020\{,\}000\-round realization isκ^ρ≈0\.63\\widehat\{\\kappa\}\_\{\\rho\}\\approx 0\.63, well below the1/dmin1/d\_\{\\min\}ceiling and giving a theorem bound of2​\(1\+2​\(e−2\)​κ^ρ\)​log⁡K≈7\.922\(1\+2\(e\-2\)\\widehat\{\\kappa\}\_\{\\rho\}\)\\log K\\approx 7\.92for the constant\-regret prediction\. The*misspecified*family hasμi≡0\.5\\mu\_\{i\}\\equiv 0\.5for everyii; with no separation the per\-expert ratio diverges \(the denominatorμi−μk∗\\mu\_\{i\}\-\\mu\_\{k^\{\\ast\}\}vanishes at population level\), so the low\-noise hypothesis fails\. We run both families at horizonsT∈\{500,1000,2000,4000,8000,16000\}T\\in\\\{500,1000,2000,4000,8000,16000\\\}over eight seeds, with two schedules each: fixed\-rate Hedge at the theorem\-prescribedη\\eta, and the second\-order retempered schedule withΓ=log⁡K\\Gamma=\\log K\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig14_bernstein_luckiness.png)Figure 16:Stochastic\-luckiness verification \(Theorem[4\.9](https://arxiv.org/html/2607.08789#S4.Thmtheorem9)and Corollary[4\.11](https://arxiv.org/html/2607.08789#S4.Thmtheorem11)\)\.Left:per\-expert Bernstein ratios𝔼^​\[\(ct​\(i\)−ct​\(k∗\)\)2\]/\(μ^i−μ^k∗\)\\widehat\{\\mathbb\{E\}\}\[\(c\_\{t\}\(i\)\-c\_\{t\}\(k^\{\\ast\}\)\)^\{2\}\]/\(\\widehat\{\\mu\}\_\{i\}\-\\widehat\{\\mu\}\_\{k^\{\\ast\}\}\)on a20,00020\{,\}000\-round realization, logyy\-axis\. Well\-specified family \(blue\) sits uniformly belowκ^ρ≈0\.63\\widehat\{\\kappa\}\_\{\\rho\}\\approx 0\.63\(dashed blue\) and below the gap\-derived ceiling1/dmin≈3\.341/d\_\{\\min\}\\approx 3\.34\(dotted gray\); misspecified family \(hatched red,μi≡μk∗\\mu\_\{i\}\\equiv\\mu\_\{k^\{\\ast\}\}\) shows ratios one to four orders of magnitude higher because the gap is essentially zero\.Middle:regret vsTTon log–log axes; the well\-specified curves are flat \(empirical slopes\+0\.000\+0\.000for both schedules\) while the misspecified curves grow with empirical slopes\+0\.43\+0\.43\(fixed\) and\+0\.49\+0\.49\(retempered\), versus the predictedT\\sqrt\{T\}slope0\.50\.5\. The dashed horizontal line is the theorem\-prescribed bound on the well\-specified family\.Right:same data on a linear scale with inter\-quartile bands across seeds\. The two schedules are visually indistinguishable on the well\-specified family \(both at the capη=1\\eta=1throughout\); on the misspecified family they coincide because the retempered schedule also sits at the cap untilVtV\_\{t\}accumulates\.Figure[16](https://arxiv.org/html/2607.08789#A3.F16)reports the result\. The well\-specified family produces regret2\.29±0\.022\.29\\pm 0\.02at everyTTtested, an empirical slope of0\.0000\.000on log–log axes, and a value comfortably inside the theorem bound of7\.927\.92\. The misspecified family produces a regret that grows from4\.84\.8atT=500T=500to30\.430\.4atT=16,000T=16\{,\}000, with empirical slope0\.430\.43–0\.490\.49, matching the predictedT\\sqrt\{T\}rate\. The Bernstein\-condition panel on the left makes the qualitative dichotomy visible: every well\-specified expert satisfies the condition with constantκ^ρ≤0\.63\\widehat\{\\kappa\}\_\{\\rho\}\\leq 0\.63, while every misspecified expert violates it by orders of magnitude\.

Three observations are worth flagging\. First, the well\-specified empirical regret of2\.292\.29is approximately0\.29​KL​\(δ0∥π\)/κ^ρ0\.29\\mathrm\{KL\}\(\\delta\_\{0\}\\\|\\pi\)/\\widehat\{\\kappa\}\_\{\\rho\}at the realizedκ^ρ\\widehat\{\\kappa\}\_\{\\rho\}, well inside the theorem’s pre\-factor2​\(1\+2​\(e−2\)​κ^ρ\)≈3\.812\(1\+2\(e\-2\)\\widehat\{\\kappa\}\_\{\\rho\}\)\\approx 3\.81, confirming that the fixed\-rate bound is the right shape but not asymptotically tight\. Second, the second\-order retempered schedule attains the same fast rate as the fixed\-rate Hedge without ever using the value ofκ^ρ\\widehat\{\\kappa\}\_\{\\rho\}, exactly as Corollary[4\.11](https://arxiv.org/html/2607.08789#S4.Thmtheorem11)predicts: on this constructionκ^ρ\\widehat\{\\kappa\}\_\{\\rho\}is small enough that the retempered rule sits at the capηt=1\\eta\_\{t\}=1throughout, and the cap\-binding regime is precisely where the retempered and fixed\-rate updates coincide\. Third, the same retempered schedule cleanly reverts to theT\\sqrt\{T\}rate on the misspecified family\. This is the empirical content of the framework’s worst\-case–fast\-rate duality: one algorithm tracks both regimes without configuration changes, with the prefix decomposition reading off in real time whether the run sits in the lucky or worst\-case regime\.

The retempered schedule’s fast\-rate behavior on this family does not exhibit the additionalC2​Γ\+Q∗T​\(c\)C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)tax that Corollary[4\.11](https://arxiv.org/html/2607.08789#S4.Thmtheorem11)adds to the fixed\-rate bound, becauseVT​\(c\)V\_\{T\}\(c\)never reaches the threshold at which the schedule rule starts cooling\. A construction with largerκ^ρ\\widehat\{\\kappa\}\_\{\\rho\}\(e\.g\., reducing the gap or increasing the noise\) would push the retempered schedule below the cap on a fraction of rounds and expose the additional constant; we ran a sigma sweep that confirms this prediction qualitatively \(the empirical regret tracks2​\(1\+2​\(e−2\)​κ^ρ\)​log⁡K2\(1\+2\(e\-2\)\\widehat\{\\kappa\}\_\{\\rho\}\)\\log Kasκ^ρ\\widehat\{\\kappa\}\_\{\\rho\}grows, with the retempered variant eventually beating the fixed\-rate variant by a small constant onceη<1\\eta<1binds\)\.

#### C\.5\.2Stochastic\-luckiness fast\-rate verification \(§[C\.5\.1](https://arxiv.org/html/2607.08789#A3.SS5.SSS1)\)

##### Headline\.

Empirical slopes: low\-noise family fast rate \(slope < 0\.55 target if margin scaling is sufficient\); worst\-case 0\.5\.

##### Caveats\.

\(1\) The low\-noise margin construction uses a power\-of\-T scaling; if T is too small the margin doesn’t dominate the noise and the fast rate isn’t visible\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.5\.3Sleeping experts and shifting comparators

Section[4](https://arxiv.org/html/2607.08789#S4)extends the prefix identity to comparators that shift acrosskksegments by replacingKL​\(ρ∥π\)\\mathrm\{KL\}\(\\rho\\\|\\pi\)with the sum of segmentwise KL terms\. The empirical check we report is on a planted\-change\-point sequence withk=4k=4segments of lengthT/k=500T/k=500, where each segment has a different leading expert \(loss mean0\.150\.15\) and the others’ loss means lie in\[0\.45,0\.75\]\[0\.45,0\.75\]\. We compare the schedules to two comparators: the best fixed expert and the per\-segment leader \(the optimal shifting comparator\)\. For the paper’s algorithms we additionally evaluate a fixed\-share wrapper\[[40](https://arxiv.org/html/2607.08789#bib.bib40)\]at mixing rateα=0\.02\\alpha=0\.02\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig7_shifting_comparator.png)Figure 17:Sleeping experts on a planted\-change\-point sequence withk=4k=4segments of length500500\(T=2000T=2000,K=8K=8\)\. Left: cumulative regret to the best fixed expert; vertical dotted lines mark segment boundaries\. Right: terminal regret to the best fixed expert\. The pressure\-target schedule attains negative regret because the cycling structure inside each segment lets a leader\-tracking algorithm beat any fixed expert\. The fixed\-share wrapper recovers nearly identical behavior\. Squint’sη≤1/2\\eta\\leq 1/2cap drives its regret to≈86\\approx 86\.The pressure\-target schedule attains regret−3\.8\-3\.8against the best fixed expert: like the cycling\-adversarial family, this is a genuine geometric advantage rather than luck, because the cycling structure inside each segment means the per\-segment leader changes faster than the segment\-aggregate leader, and a leader\-tracking algorithm can exploit the within\-segment dynamics\. AdaHedge and AdaNormalHedge stay slightly positive \(\+1\.5\+1\.5\)\. The fixed\-share wrapper atα=0\.02\\alpha=0\.02recovers the pressure\-target’s regret to within0\.50\.5, demonstrating that the underlying pressure\-target update already adapts well to non\-stationarity without any explicit tracking machinery\. Regret to the per\-segment leader is large and roughly equal across all stationary algorithms \(≈660\\approx 660\), as predicted by the shifting\-comparator lower bound: no algorithm with stationary state can match the segment\-by\-segment optimum on a path with this many planted switches\. The left panel makes the contrast pathwise legible: pressure\-target’s cumulative\-regret curve to the best fixed expert dips below zero in the middle of each segment and only spikes at change\-points\.

##### Shifting\-budget scaling\.

The shifting\-comparator boundary termBTshiftB\_\{T\}^\{\\text\{shift\}\}implied by the multi\-segment KL extension grows linearly in the number of switchesSSacross our planted sweep, matching the leading\-orderS​log⁡KS\\log Krate predicted by the shifting analysis\. The realized absolute scale of the impliedBTshiftB\_\{T\}^\{\\text\{shift\}\}is much larger thanS​log⁡KS\\log Kalone, however, because in the prior\-retempered chainBTshiftB\_\{T\}^\{\\text\{shift\}\}also absorbs the cumulative drift contributed by the leader change at each segment boundary; the rate claim is a leading\-order statement, not a tight constant claim about the absolute scale\.

#### C\.5\.4Shifting\-comparator exact identity verification on planted change\-point sequences \(§[C\.5\.3](https://arxiv.org/html/2607.08789#A3.SS5.SSS3)\)

##### Headline\.

ImpliedBTshiftB\_\{T\}^\{\\text\{shift\}\}grows roughly linearly inSS; absolute value dominated by the segment\-leader’s cumulative loss, not justS​log⁡KS\\log K\.

##### Caveats\.

\(1\) TheBTshift/\(S​log⁡K\)B\_\{T\}^\{\\text\{shift\}\}/\(S\\log K\)ratio is large \(∼70\\sim 70–200200\) becauseBTshiftB\_\{T\}^\{\\text\{shift\}\}here also absorbs the prior\-retempered drift from changing leaders; the spec’s expected\-growth claim is leading\-order, not a tight constant\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

### C\.6Applications

#### C\.6\.1Benchmark\-trace protocol \(synthetic substitutes\): horse\-racing, intraday electricity, NYSE portfolio

The synthetic core \(§[C\.2\.1](https://arxiv.org/html/2607.08789#A3.SS2.SSS1)–§[C\.5\.3](https://arxiv.org/html/2607.08789#A3.SS5.SSS3)\) controls the difficulty regime to expose specific predictions\. The subsection reported here checks that the second\-order schedule and the prefix decomposition behave as predicted on the*protocols*of three benchmark sequences from the adaptive online learning literature: the horse\-racing trace of\[[12](https://arxiv.org/html/2607.08789#bib.bib12),[45](https://arxiv.org/html/2607.08789#bib.bib45)\]\(K=7K=7horses,T≈104T\\approx 10^\{4\}\); the intraday electricity prices of\[[21](https://arxiv.org/html/2607.08789#bib.bib21)\]\(K=24K=24expert forecasts,T≈104T\\approx 10^\{4\}\); and online portfolio selection on New York Stock Exchange \(NYSE\) returns \(K=36K=36stocks,T≈5500T\\approx 5500,\[[16](https://arxiv.org/html/2607.08789#bib.bib16),[1](https://arxiv.org/html/2607.08789#bib.bib1)\]\)\.*These are not real\-data results\.*Every cell reported here uses a*synthetic substitute*whose empirical\-distribution profile matches the reference trace, not the original benchmark data; each cell is marked*\(synthetic substitute\)*in the companion fragment header\. Running the identical protocol on the original benchmark traces is beyond the present scope; we do not claim head\-to\-head results on the real traces here\.

##### Hypothesis\.

The predictable second\-order scheduleηt=min⁡\{1,C​Γ/Vt−1​\(c\)\}\\eta\_\{t\}=\\min\\\{1,C\\sqrt\{\\Gamma/V\_\{t\-1\}\(c\)\}\\\}matches or outperforms \(i\) hand\-tuned fixedη\\eta, \(ii\) Kt\-doubling Hedge, \(iii\) AdaHedge\[[20](https://arxiv.org/html/2607.08789#bib.bib20)\], \(iv\) NormalHedge\[[13](https://arxiv.org/html/2607.08789#bib.bib13)\], \(v\) Squint\[[45](https://arxiv.org/html/2607.08789#bib.bib45)\], \(vi\) Coin\-Betting\[[56](https://arxiv.org/html/2607.08789#bib.bib56)\], \(vii\) MetaGrad\[[72](https://arxiv.org/html/2607.08789#bib.bib72)\], and \(viii\) iProd / FTRL\-style parameter\-free methods\[[18](https://arxiv.org/html/2607.08789#bib.bib18)\]on the three datasets, with a single scale factorCCas its only tuning knob\. Qualitative prediction: the decomposition shareωinfo\\omega^\{\\mathrm\{info\}\}is nearly constant across algorithms on horse\-racing \(an i\.i\.d\. trace at the timescale of interest\), while the split betweenωpay\\omega^\{\\mathrm\{pay\}\}andωdrift\\omega^\{\\mathrm\{drift\}\}is what distinguishes them\.

##### Protocol\.

For each dataset, run all nine schedules \(the second\-order schedule of Algorithm[1](https://arxiv.org/html/2607.08789#alg1)plus the eight baselines listed above\) for5050seeds \(the seed enters via the noise model only; the data sequence is fixed\)\. Report per\-seed terminal regret against the best fixed expert, the pathRt/t​log⁡KR\_\{t\}/\\sqrt\{t\\log K\}, the empirical intrinsic timeVT/TV\_\{T\}/T, and the decomposition shares\(ωpay,ωdrift,ωinfo\)\(\\omega^\{\\mathrm\{pay\}\},\\omega^\{\\mathrm\{drift\}\},\\omega^\{\\mathrm\{info\}\}\)with mean and BCa 95% bootstrap confidence intervals\. Where a real dataset is unavailable in our environment it is simulated using a matching empirical distribution; such cells are marked*\(synthetic substitute\)*in the final table and in the per\-fragment header\.

#### C\.6\.2Variance\-proxy tightness on horse\-racing and intraday\-electricity substitutes \(§[C\.6\.1](https://arxiv.org/html/2607.08789#A3.SS6.SSS1)\)

##### Headline\.

Q/W ratios approach 1 from above as eta \-\> 0; horse\-racing substitute Q/W=1\.018 at eta=0\.1; electricity substitute Q/W=0\.998 at eta=0\.1\.

##### Caveats\.

\(1\) synthetic\_substitute=True for both datasets; the original CBMS07 horse\-racing trace and DGGS13 electricity trace are not locally available\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.3Second\-order schedule on horse\-racing substitute \(§[C\.6\.1](https://arxiv.org/html/2607.08789#A3.SS6.SSS1)\)

##### Headline\.

Second\-order/oracle ratio = 1\.28; second\-order/NormalHedge = 1\.02 \(within 5% of NH\)\.

##### Caveats\.

\(1\) synthetic\_substitute=True \(CBMS07 trace not local\)\. \(2\) NormalHedge / Squint / Coin\-Betting / MetaGrad / iProd are approximated as scaled AdaHedge variants — the canonical reference implementations are not used\. This limits comparability with the paper’s headline numbers\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.4Second\-order schedule on intraday\-electricity substitute \(§[C\.6\.1](https://arxiv.org/html/2607.08789#A3.SS6.SSS1)\)

##### Headline\.

Second\-order/oracle ratio = 1\.32 \(within 1\.4x target\)\.

##### Caveats\.

\(1\) synthetic\_substitute=True \(DGGS13 trace not local\)\. \(2\) The same baseline\-implementation caveat as the horse\-racing run applies\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.5Second\-order schedule on NYSE portfolio substitute \(§[C\.6\.1](https://arxiv.org/html/2607.08789#A3.SS6.SSS1)\)

##### Headline\.

Second\-order intrinsic at mean regret 16, comparable to AdaHedge \( 16\) and CoinBetting \( 18\)\.

##### Caveats\.

\(1\) synthetic\_substitute=True \(Cover 91 NYSE trace not redistributable\)\. \(2\) The same baseline\-implementation caveat as the horse\-racing run applies\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.6Bandit decomposition and EXP3\-IX tightness

The bandit decomposition of Section[6](https://arxiv.org/html/2607.08789#S6)extends the prefix identity from the full\-information regime to importance\-weighted estimates\. The empirical question is whether the residual between measured regret and the predicted bandit decomposition is at floating\-point precision \(the algebra is exact\), and whether the algorithm\-specific intrinsic\-time termQ^t\\hat\{Q\}\_\{t\}recovers the expected scaling factor for each common bandit algorithm\.

##### Hypothesis\.

The bandit decompositionRTbandit=MTplay\+ΞT−MTest\+BiasT\+D^T\+B^T\+∑tηt​Q^tR\_\{T\}^\{\\mathrm\{bandit\}\}=M\_\{T\}^\{\\mathrm\{play\}\}\+\\Xi\_\{T\}\-M\_\{T\}^\{\\mathrm\{est\}\}\+\\mathrm\{Bias\}\_\{T\}\+\\hat\{D\}\_\{T\}\+\\hat\{B\}\_\{T\}\+\\sum\_\{t\}\\eta\_\{t\}\\hat\{Q\}\_\{t\}is pathwise\-exact: the residual between the realized regretct​\(At\)−⟨u,ct⟩c\_\{t\}\(A\_\{t\}\)\-\\langle u,c\_\{t\}\\ranglesummed overttand the right\-hand decomposition is zero up to floating\-point error\. EXP3\-IX’s meanQ^t\\hat\{Q\}\_\{t\}atγt=log⁡K/\(K​T\)\\gamma\_\{t\}=\\sqrt\{\\log K/\(KT\)\}is within1\.05×1\.05\\timesof its oracle prediction on every seed; Tsallis\-INF shows a different worm\-plot signature \(largerωinfo\\omega^\{\\mathrm\{info\}\}\) consistent with its entropy regularizer\.

##### Protocol\.

Standard Bernoulli bandit,K∈\{8,16,32\}K\\in\\\{8,16,32\\\}, arm means drawn uniformly from\[0,1\]\[0,1\],T∈\{103,104,105\}T\\in\\\{10^\{3\},10^\{4\},10^\{5\}\\\}\. Algorithms: EXP3\[[5](https://arxiv.org/html/2607.08789#bib.bib5)\], EXP3\-IX\[[44](https://arxiv.org/html/2607.08789#bib.bib44)\], Tsallis\-INF\[[75](https://arxiv.org/html/2607.08789#bib.bib75)\], KL\-UCB\[[32](https://arxiv.org/html/2607.08789#bib.bib32)\]as a stochastic\-only baseline\. Per\-configuration residual against floating\-point envelope, decomposition shares, mean and BCa 95% CIs over5050seeds\. The battery comprises a synthetic identity check and a head\-to\-head comparison with KL\-UCB stochastic calibration\.

#### C\.6\.7Synthetic bandit decomposition residual \(§[C\.6\.6](https://arxiv.org/html/2607.08789#A3.SS6.SSS6)\)

##### Headline\.

log10 max \|Delta\_T\| = \-9\.91 across 360 cells \(EXP3 \+ EXP3\-IX\)\.

##### Caveats\.

\(1\) Approximation: only EXP3 and EXP3\-IX \(not Tsallis\-INF\) are verified at machine precision in this run\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.8Bernoulli bandit head\-to\-head: EXP3 / EXP3\-IX / Tsallis\-INF / KL\-UCB \(§[C\.6\.6](https://arxiv.org/html/2607.08789#A3.SS6.SSS6)\)

##### Headline\.

EXP3 and EXP3\-IX have similar terminal regret; KL\-UCB matches stochastic regime as expected\.

##### Caveats\.

\(1\) Tsallis\-INF is approximated by a scaled EXP3\-IX, not its canonical entropy\-regularized update; comparison is qualitative\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.9Pressure\-target boosting on UCI datasets

The pressure\-target recurrence of Section[3\.3](https://arxiv.org/html/2607.08789#S3.SS3)reproduces the classical AdaBoost coefficientαt∗=12​log⁡\{\(1−εt\)/εt\}\\alpha\_\{t\}^\{\*\}=\\tfrac\{1\}\{2\}\\log\\\{\(1\-\\varepsilon\_\{t\}\)/\\varepsilon\_\{t\}\\\}when the weak hypothesis has binary outputs, and the exponential loss decays by factor2​εt​\(1−εt\)2\\sqrt\{\\varepsilon\_\{t\}\(1\-\\varepsilon\_\{t\}\)\}per round\. The empirical content is twofold: a synthetic coefficient verification and three real\-data runs on classification datasets from the University of California, Irvine \(UCI\) Machine Learning Repository\[[4](https://arxiv.org/html/2607.08789#bib.bib4)\]\.

##### Hypothesis\.

On UCIletter,adult, andcovtypewith decision\-stump weak learners, the pressure\-recurrence coefficient agrees with the closed\-form AdaBoost value to floating\-point precision; the training and test exponential loss tracks the theoretical∏t2​εt​\(1−εt\)\\prod\_\{t\}2\\sqrt\{\\varepsilon\_\{t\}\(1\-\\varepsilon\_\{t\}\)\}envelope on every seed; and the worm\-plot decomposition on the signed\-margin state showsωdrift≡0\\omega^\{\\mathrm\{drift\}\}\\equiv 0because the pressure game is Nature\-first and the one\-step CGF constraint is active, so retempering drift does not appear\.

##### Protocol\.

Replicate the AdaBoost setup of\[[27](https://arxiv.org/html/2607.08789#bib.bib27),[66](https://arxiv.org/html/2607.08789#bib.bib66)\]on each of the three datasets\. Report \(i\) coefficient comparisonαt∗\\alpha\_\{t\}^\{\*\}vs\. closed\-form per round; \(ii\) train/test exponential loss with the theoretical envelope overlaid; \(iii\) the worm\-plot decomposition on the signed\-margin state with all three shares plotted as a stacked share againsttt\. Mean and BCa 95% CIs over1010seeds\.

#### C\.6\.10Synthetic AdaBoost coefficient verification \(§[C\.6\.9](https://arxiv.org/html/2607.08789#A3.SS6.SSS9)\)

##### Headline\.

log10 max coef residual = \-300 \(essentially zero\); log10 max envelope residual = \-14\.27\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.11AdaBoost pressure\-target verification on UCI letter substitute \(§[C\.6\.9](https://arxiv.org/html/2607.08789#A3.SS6.SSS9)\)

##### Headline\.

Coef residual at FP zero; envelope residual at FP precision \( 1e\-14\)\.

##### Caveats\.

\(1\) synthetic\_substitute=True \(UCI letter not local\)\. 26\-class substitute uses Gaussian mixture with d=16\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.12AdaBoost pressure\-target verification on UCI adult substitute \(§[C\.6\.9](https://arxiv.org/html/2607.08789#A3.SS6.SSS9)\)

##### Headline\.

Coef residual at FP zero; envelope residual at FP precision\.

##### Caveats\.

\(1\) synthetic\_substitute=True \(UCI adult not local\)\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.13AdaBoost pressure\-target verification on UCI covtype substitute \(§[C\.6\.9](https://arxiv.org/html/2607.08789#A3.SS6.SSS9)\)

##### Headline\.

Coef residual at FP zero across 7\-class subproblems; envelope residual at FP precision\.

##### Caveats\.

\(1\) synthetic\_substitute=True \(UCI covtype not local\)\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.14Self\-play on structured matrix games: Shapley RPS, Colonel\-Blotto

The matrix\-game side\-information recipe of §[C\.1\.2](https://arxiv.org/html/2607.08789#A3.SS1.SSS2)and the forecast ablation of §[C\.2\.7](https://arxiv.org/html/2607.08789#A3.SS2.SSS7)cover the random\-payoff regime\. The structured\-game subsection reported here adds two canonical hand\-crafted games where the equilibrium structure is non\-trivial and the duality\-gap\-to\-regret ratio is the diagnostic of interest\.

##### Hypothesis\.

When two intrinsic\-time regret\-matching players play a finite zero\-sum game against each other, the duality gapmaxq⁡p¯T⊤​M​q−minp⁡p⊤​M​q¯T\\max\_\{q\}\\bar\{p\}\_\{T\}^\{\\top\}Mq\-\\min\_\{p\}p^\{\\top\}M\\bar\{q\}\_\{T\}decays atO​\(1/T\)O\(1/\\sqrt\{T\}\)and equals the sum of individual regrets divided byTTup to floating\-point precision\. On structured games \(Shapley RPS, Colonel\-Blotto\), intrinsic\-time regret matching matches Optimistic Hedge\[[71](https://arxiv.org/html/2607.08789#bib.bib71)\]on average and outperforms vanilla Hedge on each of the structured matrices\.

##### Protocol\.

RandomK×KK\\times Kmatrix games withK∈\{5,10,20\}K\\in\\\{5,10,20\\\}and entries in\[−1,1\]\[\-1,1\]as a calibration baseline; the Shapley rock–paper–scissors matrix; the Colonel\-Blotto matrix of\[[34](https://arxiv.org/html/2607.08789#bib.bib34)\]at small action sets\. Comparators: Optimistic Hedge, vanilla Hedge\. Report duality gap and the per\-player regret sum, with the equality of the gap and the regret sum verified pathwise\. Mean and BCa 95% CIs over2525seeds\. The battery comprises a synthetic random\-matrix duality\-gap identity check and the structured\-game runs\.

#### C\.6\.15Synthetic self\-play duality\-gap identity \(§[C\.6\.14](https://arxiv.org/html/2607.08789#A3.SS6.SSS14)\)

##### Headline\.

log10 max \|Delta\_T\| = \-13\.30; convergence slopes per K: \-0\.49, \-0\.52, \-0\.53 \(target \-0\.5\)\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.16Self\-play on Shapley RPS and Colonel\-Blotto \(§[C\.6\.14](https://arxiv.org/html/2607.08789#A3.SS6.SSS14)\)

##### Headline\.

Duality gap 0 on Shapley RPS, modest on Blotto\-K3, K5; identity residual at FP precision\.

##### Caveats\.

None\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.17Thompson sampling and the PPR martingale

The prior\-posterior\-ratio \(PPR\)Rt​\(θ\):=π0​\(θ\)/πt​\(θ\)R\_\{t\}\(\\theta\):=\\pi\_\{0\}\(\\theta\)/\\pi\_\{t\}\(\\theta\)between the initial priorπ0\\pi\_\{0\}and the round\-ttposteriorπt\\pi\_\{t\}on a candidate parameterθ\\thetais a nonnegative martingale, and Thompson sampling’s intrinsic\-time profile coincides with Hedge’s on full\-information paths up to a sampling fluctuation whose amplitude is controlled by Freedman’s bound applied to the play martingale\. The empirical content is to verify Ville’s bound on the realized PPR martingale and to check the intrinsic\-time coupling at finite horizon\.

##### Hypothesis\.

OnKK\-armed Gaussian bandits with posterior Thompson sampling\[[63](https://arxiv.org/html/2607.08789#bib.bib63),[65](https://arxiv.org/html/2607.08789#bib.bib65)\], the measured intrinsic timeVTTSV\_\{T\}^\{\\mathrm\{TS\}\}approaches the parallel\-Hedge intrinsic timeVTHedgeV\_\{T\}^\{\\mathrm\{Hedge\}\}at rateT−1/2T^\{\-1/2\}, i\.e\.,\|VTTS−VTHedge\|/VTHedge→0\|V\_\{T\}^\{\\mathrm\{TS\}\}\-V\_\{T\}^\{\\mathrm\{Hedge\}\}\|/V\_\{T\}^\{\\mathrm\{Hedge\}\}\\to 0\. The PPR martingale satisfies Ville’s boundPr⁡\(suptRt≥1/α\)≤α\\Pr\(\\sup\_\{t\}R\_\{t\}\\geq 1/\\alpha\)\\leq\\alphaforα∈\{0\.1,0\.05,0\.01\}\\alpha\\in\\\{0\.1,0\.05,0\.01\\\}on every seed\.

##### Protocol\.

K∈\{8,16\}K\\in\\\{8,16\\\}arms, posterior Thompson sampling and a parallel Hedge run on the realized posterior means\. Report \(i\) the relative intrinsic\-time gap\|VTTS−VTHedge\|/VTHedge\|V\_\{T\}^\{\\mathrm\{TS\}\}\-V\_\{T\}^\{\\mathrm\{Hedge\}\}\|/V\_\{T\}^\{\\mathrm\{Hedge\}\}as a function ofT∈\{102,103,104,105\}T\\in\\\{10^\{2\},10^\{3\},10^\{4\},10^\{5\}\\\}on log\-log axes; \(ii\) the empirical crossing fractionPr⁡\(suptRt≥1/α\)\\Pr\(\\sup\_\{t\}R\_\{t\}\\geq 1/\\alpha\)versus the nominal levelα\\alphaforα∈\{0\.1,0\.05,0\.01\}\\alpha\\in\\\{0\.1,0\.05,0\.01\\\}\. Mean and BCa 95% CIs over200200seeds \(the Ville check is sensitive to tail mass and benefits from many seeds\)\. The battery comprises the PPR Ville identity check and the intrinsic\-time coupling run\.

#### C\.6\.18Synthetic Ville\-bound verification on the prior\-posterior\-ratio martingale \(§[C\.6\.17](https://arxiv.org/html/2607.08789#A3.SS6.SSS17)\)

##### Headline\.

Ville coverage holds across alpha grid; Wilson 95% upper bound is conservative due to MC noise at small alpha\.

##### Caveats\.

\(1\) At alpha=0\.05 and alpha=0\.02 with 500 seeds, Wilson 95

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

#### C\.6\.19Thompson\-vs\-Hedge intrinsic\-time coupling \(§[C\.6\.17](https://arxiv.org/html/2607.08789#A3.SS6.SSS17)\)

##### Headline\.

Per\-K log\-log slope of \|V\_TS \- V\_Hedge\|/V\_Hedge: target \-0\.5; observed slopes are negative but vary by K\.

##### Caveats\.

\(1\) TS posterior\-mean choice for the parallel Hedge is one specific implementation choice; other proxies could give different slopes\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

### C\.7Comparison with prior methods

#### C\.7\.1Comparison with current adaptive online learning

The decomposition view does not call for a regret horse race, but a benchmark against current adaptive online learning baselines is informative because it places the paper’s schedules on the same axis the field has been using\. We compare the retempered schedule, the pressure\-target schedule, and a hybrid*gap\-retempered*schedule \(prior\-retempered posteriors with the AdaHedge gap\-implied rate\) against AdaHedge\[[20](https://arxiv.org/html/2607.08789#bib.bib20)\], NormalHedge\[[13](https://arxiv.org/html/2607.08789#bib.bib13)\], AdaNormalHedge\[[50](https://arxiv.org/html/2607.08789#bib.bib50)\], the follow\-the\-regularized\-leader \(FTRL\) algorithm with1/21/2\-Tsallis entropy\[[75](https://arxiv.org/html/2607.08789#bib.bib75)\], and Squint\[[45](https://arxiv.org/html/2607.08789#bib.bib45)\]on the four families with1212random seeds; mean and inter\-quartile range \(IQR\) error bars appear in Figure[18](https://arxiv.org/html/2607.08789#A3.F18)\. The state\-of\-the\-art \(SOTA\) baselines have very different dynamic ranges—AdaHedge and AdaNormalHedge sit at≈0\.5\\approx 0\.5regret on stochastic paths, while Squint sits at≈31\\approx 31—so the figure uses a brokenyy\-axis on the panels where the outliers compress the paper algorithms; the diagonal break markers indicate where the axis discontinues\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig9_sota_comparison.png)Figure 18:Terminal regret to the best fixed expert \(T=2000T=2000,K=8K=8,1212seeds, mean with IQR\)\. Blue bars: paper schedules; orange bars: SOTA adaptive baselines\. Brokenyy\-axis where the FTRL\-Tsallis / Squint outliers would otherwise compress the paper algorithms\.Three observations from Figure[18](https://arxiv.org/html/2607.08789#A3.F18)match the framework’s predictions\.

First, on the predictable families \(i\.i\.d\., martingale, mixed\) the gap\-retempered schedule, AdaHedge, AdaNormalHedge, and NormalHedge sit in a tight cluster at terminal regret0\.40\.4–0\.650\.65\. The plain retempered schedule sits at≈2\.3\\approx 2\.3, the difference being exactly theC2​Γ\+Q∗T​\(c\)C^\{2\}\\Gamma\+Q\_\{\*\}^\{T\}\(c\)initial tax that the second\-order envelope of Theorem[3\.2](https://arxiv.org/html/2607.08789#S3.Thmtheorem2)predicts in the cap\-binding regime: withΓ=log⁡8≈2\.08\\Gamma=\\log 8\\approx 2\.08andC=1/2C=1/\\sqrt\{2\}, one hasC2​Γ≈1\.04C^\{2\}\\Gamma\\approx 1\.04andQ∗T​\(c\)≈1\.2Q\_\{\*\}^\{T\}\(c\)\\approx 1\.2on these paths, summing to roughly the observed2\.32\.3gap\. The retempered schedule is paying the upfront tax in exchange for the worst\-case envelope; on these easy paths the tax is wasted but bounded\.

Second, on the cycling\-adversarial family the pressure\-target schedule attains substantially negative regret to the best fixed expert \(−28\.9±3\.6\-28\.9\\pm 3\.6\)\. The reason is geometric, not lucky: the local update with the line\-search rule chases the rotating leader as soon as a switch happens, and the cumulative loss of the resulting weight sequence is much smaller than the cumulative loss of any single fixed expert \(which has been the leader for only a quarter of the rounds\)\. AdaHedge and AdaNormalHedge cannot do this—their gap\-implied rate ramps up only after the cumulative gap accumulates, so they enter each new segment at a rate that lags the change\-point by hundreds of rounds\.

Third, Squint pays≈31\\approx 31in regret on the i\.i\.d\., martingale, and mixed paths\. Squint’s analysis requiresη≤1/2\\eta\\leq 1/2to control the second\-order moment, and on well\-separated stochastic paths the optimal fixed temperature is well above1/21/2\. The paper’s intrinsic\-time schedule sidesteps this constraint at the cost of an explicitC2​ΓC^\{2\}\\Gammainitialization tax: under the capηt≤1\\eta\_\{t\}\\leq 1, one pays a known additive term and recovers the high\-temperature regime that Squint’s design forbids\. This is the empirical content of the cap design choice, and seeing it in a head\-to\-head comparison clarifies what the cap is buying\.

##### Baseline\-implementation caveat\.

A subset of the baselines reported in Figure[18](https://arxiv.org/html/2607.08789#A3.F18)are run from our re\-implementations rather than the original authors’ code\. Where a baseline implementation is unavailable in the realized cell, we substitute a different parameter\-free algorithm from the same family as a stand\-in — for example, NormalHedge\[[13](https://arxiv.org/html/2607.08789#bib.bib13)\]is run in place of AdaHedge\[[20](https://arxiv.org/html/2607.08789#bib.bib20)\]when the published code is not at hand\. These are distinct algorithms with different learning\-rate mechanisms, not interchangeable implementations of one method; we therefore label each such cell as a same\-family stand\-in in the companion fragment header rather than as the named baseline\. The qualitative ordering of*our*schedules against this baseline set is what we report; a sharper head\-to\-head against the original authors’ AdaHedge implementation is a clean follow\-up, and we do not claim a state\-of\-the\-art ranking against the exact published baselines on the strength of the stand\-ins\.

##### Cap binding andΓ\\Gammasensitivity\.

The capηt≤1\\eta\_\{t\}\\leq 1in Algorithm[1](https://arxiv.org/html/2607.08789#alg1)is functional rather than cosmetic\. Figure[19](https://arxiv.org/html/2607.08789#A3.F19)\(left\) compares the capped Algorithm 1 schedule with the uncapped second\-order ruleηt=C​Γ/Vt−1\\eta\_\{t\}=C\\sqrt\{\\Gamma/V\_\{t\-1\}\}on i\.i\.d\. stochastic and on cycling\-adversarial paths\. On the stochastic family the uncapped rule grows unboundedly \(we plot it on a symmetric\-log axis\); on the adversarial family both rules eventually decay below11, with the capped version delayed because it sits at the cap untilVtV\_\{t\}accumulates past the threshold\. Figure[19](https://arxiv.org/html/2607.08789#A3.F19)\(right\) sweepsΓ\\Gammaover two orders of magnitude on three families: the defaultΓ=log⁡K\\Gamma=\\log K\(vertical dotted line\) lies in the flat regime on every family, confirming the schedule degrades gracefully across two orders of magnitude inΓ\\Gamma\.

![Refer to caption](https://arxiv.org/html/2607.08789v1/figures/fig10_diagnostics.png)Figure 19:Left:cap\-binding diagnostic\. Solid lines are the capped Algorithm 1 ruleηt=min⁡\{1,C​Γ/Vt−1\}\\eta\_\{t\}=\\min\\\{1,C\\sqrt\{\\Gamma/V\_\{t\-1\}\}\\\}; dashed are the uncapped rule\. Symmetric\-logyy\-axis to keep the uncapped i\.i\.d\. trajectory legible\.Right:sensitivity of the retempered schedule toΓ\\Gammaon three families\. The defaultΓ=log⁡K\\Gamma=\\log Kis robust over two decades\.

#### C\.7\.2Baselines comparison on horse\-racing/electricity/NYSE substitutes \(§[C\.7\.1](https://arxiv.org/html/2607.08789#A3.SS7.SSS1)\)

##### Headline\.

Per\-dataset summaries in CSV; second\-order intrinsic competitive with AdaHedge across all three substitutes\.

##### Caveats\.

\(1\) synthetic\_substitute=True for all three datasets\. \(2\) Baseline implementations approximate; canonical reference implementations of NormalHedge / Squint / Coin\-Betting / MetaGrad / iProd are not used\.

##### Reproducibility\.

Per\-seed numbers and headline summaries are provided with the paper’s accompanying code release\.

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