适用于所有可混合适当损失及更广泛损失的全预测

# Omniprediction for All Mixable Proper Losses and Beyond
Source: [https://arxiv.org/html/2610.41023](https://arxiv.org/html/2610.41023)
mnlargesymbols’164 mnlargesymbols’171

## Omniprediction for All Mixable Proper Losses and BeyondThanks:Authors are listed in alphabetical order\.

Haipeng Luo††thanks:University of Southern California\. Email:haipengl@usc\.edu\.Lillian J\. Ratliff††thanks:University of Washington\. Email:ratliffl@uw\.edu\.

###### Abstract

Omniprediction concerns the task of making predictions that are simultaneously accurate with respect to different losses\. Earlier results by[Chernov and Vovk \(2009\)](https://arxiv.org/html/2610.41023#bib.bib7)show that in an online learning setting,𝒪⁡\(log⁡\(\|ℱ\|​\|ℒ\|\)\)\\mathcal\{O\}\(\\log\(\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|\)\)regret is achievable simultaneously against every predictor in a finite predictor classℱ\{\\mathcal\{F\}\}and for every loss in a finite classℒ\{\\mathcal\{L\}\}of mixable proper losses, while recent advances\([Okoroafor et al\., 2025](https://arxiv.org/html/2610.41023#bib.bib3);[Farina and Perdomo, 2026](https://arxiv.org/html/2610.41023#bib.bib1)\)show that𝒪~​\(T​log⁡\(\|ℱ\|​\|ℒ\|\)\)\\widetilde\{\\mathcal\{O\}\}\(\\sqrt\{T\\log\(\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|\)\}\)regret is achievable simultaneously for every loss in a finite classℒ\{\\mathcal\{L\}\}of bounded losses\.

In this work, we ask whether sublinear regret remains possible whenℒ\{\\mathcal\{L\}\}is aninfiniteclass of proper losses\. Our first result gives a negative answer in general: even whenℱ\{\\mathcal\{F\}\}contains only two linear predictors, it is impossible to achieve sublinear regret simultaneously for all bounded proper losses\. We then show that𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3\)\\widetilde\{\\mathcal\{O\}\}\(T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\)regret is achievable simultaneously forallmixable proper losses and that this bound is optimal up to logarithmic factors\. This optimal rate stands in stark contrast to the constant \(inTT\) regret achievable for a finite class of mixable proper losses\. Finally, we extend our algorithm to obtain a best\-of\-all\-worlds result: in addition to the new regret guarantee for all mixable proper losses, the resulting algorithm simultaneously achieves the existing guarantees for finite loss classes\. With appropriate choices of these finite classes, our algorithm simultaneously achieves constant regret for squared loss and log loss,𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3\)\\widetilde\{\\mathcal\{O\}\}\(T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\)regret for all mixable proper losses, and𝒪~​\(T​log⁡\|ℱ\|\)\\widetilde\{\\mathcal\{O\}\}\(\\sqrt\{T\\log\|\{\\mathcal\{F\}\}\|\}\)regret for00\-11loss, all Lipschitz losses, all GLM losses, and other structured families\.

###### Contents

1. [1Introduction](https://arxiv.org/html/2610.41023#S1)1. [1\.1Our Results](https://arxiv.org/html/2610.41023#S1.SS1) 2. [1\.2Technical Overview](https://arxiv.org/html/2610.41023#S1.SS2)1. [1\.2\.1Key ideas behind algorithms and upper bounds](https://arxiv.org/html/2610.41023#S1.SS2.SSS1) 3. [1\.3Related Work](https://arxiv.org/html/2610.41023#S1.SS3)
2. [2Preliminaries](https://arxiv.org/html/2610.41023#S2)
3. [3Algorithms and Results](https://arxiv.org/html/2610.41023#S3)1. [3\.1Warm\-up: all Lipschitz proper losses](https://arxiv.org/html/2610.41023#S3.SS1) 2. [3\.2All mixable proper losses](https://arxiv.org/html/2610.41023#S3.SS2) 3. [3\.3A best\-of\-all\-worlds omnipredictor](https://arxiv.org/html/2610.41023#S3.SS3)
4. [4Conclusion](https://arxiv.org/html/2610.41023#S4)
5. [AProofs of Main Results](https://arxiv.org/html/2610.41023#A1)1. [A\.1Impossibility for all bounded proper losses](https://arxiv.org/html/2610.41023#A1.SS1) 2. [A\.2Lipschitz and mixable lower bounds for every horizon](https://arxiv.org/html/2610.41023#A1.SS2) 3. [A\.3The universal mixable upper bound](https://arxiv.org/html/2610.41023#A1.SS3) 4. [A\.4The best\-of\-all\-worlds guarantee](https://arxiv.org/html/2610.41023#A1.SS4) 5. [A\.5Consequences for the five stated loss families](https://arxiv.org/html/2610.41023#A1.SS5)
6. [BAuxiliary Analysis](https://arxiv.org/html/2610.41023#A2)
7. [CPreliminary Facts](https://arxiv.org/html/2610.41023#A3)
8. [DApproximate Bases for GLM Losses](https://arxiv.org/html/2610.41023#A4)
9. [References](https://arxiv.org/html/2610.41023#bib)

## 1Introduction

A standard machine learning approach to making predictions is to choose a loss function and train a predictor to minimize it\. In many applications, however, the same predictions inform multiple downstream decisions, each with a different loss function, and these losses may not be known at training time\.*Omniprediction*addresses this mismatch by seeking a single predictor whose outputs can be post\-processed to compete with a benchmark classℱ\{\\mathcal\{F\}\}simultaneously for every loss in a classℒ\{\\mathcal\{L\}\}\. The post\-processing treats the predictor’s output as the true outcome probability and chooses a Bayes\-optimal action for the desired loss\. Thus, rather than training a separate model for each objective, omniprediction separates the task of forecasting from the choice of how to act on a forecast\. This paradigm was introduced in the batch setting by[Gopalan et al\. \(2022\)](https://arxiv.org/html/2610.41023#bib.bib8)and subsequently extended to the online adversarial setting by[Garg et al\. \(2024\)](https://arxiv.org/html/2610.41023#bib.bib9)\.

In the online setting, the learner must produce forecasts sequentially, without knowing future outcomes, and the goal is to achieve sublinear cumulative regret against the best predictor inℱ\{\\mathcal\{F\}\}for every loss of interest\. More specifically, we consider the following contextual online setting for forecasting binary outcomes: for each roundt=1,…,Tt=1,\\ldots,T, the nature reveals a contextxtx\_\{t\}from an arbitrary context space𝒳\{\\mathcal\{X\}\}, the learner then makes a predictionPt∈\[0,1\]P\_\{t\}\\in\[0,1\], and finally the nature reveals the true outcomeyt∈\{0,1\}y\_\{t\}\\in\\left\\\{0,1\\right\\\}\. Here,PtP\_\{t\}represents the learner’s forecast on the probability ofyty\_\{t\}being11and is generally random; we letDtD\_\{t\}be the distribution over\[0,1\]\[0,1\]from whichPtP\_\{t\}is drawn\. For a benchmark classℱ⊆\[0,1\]𝒳\{\\mathcal\{F\}\}\\subseteq\[0,1\]^\{\\mathcal\{X\}\}and a loss functionℓ:\[0,1\]×\{0,1\}→ℝ∪\{\+∞\}\\ell:\[0,1\]\\times\\left\\\{0,1\\right\\\}\\to\\mathbb\{R\}\\cup\\\{\+\\infty\\\}, the regret of the learner is defined as:

Regℓ,ℱ​\(T\)=∑t=1T𝔼Pt∼Dt​\[ℓ⁡\(πℓ​\(Pt\),yt\)\]−inff∈ℱ∑t=1Tℓ⁡\(f⁡\(xt\),yt\),\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\_\{t\}\\sim D\_\{t\}\}\\left\[\\ell\\left\(\\pi\_\{\\ell\}\(P\_\{t\}\),y\_\{t\}\\right\)\\right\]\-\\inf\_\{f\\in\{\\mathcal\{F\}\}\}\\sum\_\{t=1\}^\{T\}\\ell\\left\(f\(x\_\{t\}\),y\_\{t\}\\right\),whereπℓ​\(p\)=arg⁡minq∈\[0,1\]⁡𝔼Y∼Ber​\(p\)​\[ℓ⁡\(q,Y\)\]\\pi\_\{\\ell\}\(p\)=\\argmin\_\{q\\in\[0,1\]\}\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[\\ell\(q,Y\)\\right\]\(withBer​\(p\)\\texttt\{Ber\}\(p\)being the Bernoulli distribution with meanppand tie broken in an arbitrary, fixed way\) is the post\-processing function that returns one of the best actions based on the learner’s forecast on the outcome and the loss function\. Therefore, regretRegℓ,ℱ​\(T\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)can be interpreted as the difference between the total loss of a downstream agent who consumes the learner’s prediction and acts optimally and that of the best fixed benchmark from the classℱ\{\\mathcal\{F\}\}\. Since different agents might care about different loss functions, the goal of omniprediction is to make sureRegℓ,ℱ​\(T\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)is small simultaneously for allℓ∈ℒ\\ell\\in\{\\mathcal\{L\}\}whereℒ\{\\mathcal\{L\}\}is a certain broad class of loss functions\.

Although the terminology of omniprediction is recent, earlier work in fact already established strong simultaneous regret guarantees for certain loss families\. In particular,[Chernov and Vovk \(2009\)](https://arxiv.org/html/2610.41023#bib.bib7)considered a finite classℒ\{\\mathcal\{L\}\}ofproperandmixablelosses:

- •\(Proper Losses\) A lossℓ\\ellis proper if𝔼Y∼Ber​\(p\)​\[ℓ⁡\(p,Y\)\]≤𝔼Y∼Ber​\(p\)​\[ℓ⁡\(q,Y\)\],∀p,q∈\[0,1\],\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[\\ell\(p,Y\)\\right\]\\leq\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[\\ell\(q,Y\)\\right\],\\forall p,q\\in\[0,1\],that is, predictingppis optimal when the outcome is indeed drawn fromBer​\(p\)\\texttt\{Ber\}\(p\)\. In this case we can simply letπℓ​\(p\)=p\\pi\_\{\\ell\}\(p\)=p\.
- •\(Mixable Losses\) A lossℓ\\ellisβ\\beta\-mixable if, for every distributionDDover\[0,1\]\[0,1\], there existsp∈\[0,1\]p\\in\[0,1\]such thatexp⁡\(−β​ℓ​\(p,y\)\)≥𝔼P∼D​exp⁡\(−β​ℓ​\(P,y\)\),∀y∈\{0,1\}\.\\exp\\left\(\-\\beta\\ell\(p,y\)\\right\)\\geq\\mathbb\{E\}\_\{P\\sim D\}\\exp\\left\(\-\\beta\\ell\(P,y\)\\right\),\\forall y\\in\\left\\\{0,1\\right\\\}\.Denote byβℓ⋆\\beta\_\{\\ell\}^\{\\star\}the largest constantβ\\betasuch thatℓ\\ellisβ\\beta\-mixable\.

It is well\-known that standard losses such as squared loss and log loss belong to this class; in particular, squared lossℓ⁡\(p,y\)=\(p−y\)2\\ell\(p,y\)=\(p\-y\)^\{2\}is22\-mixable while log lossℓ⁡\(p,y\)=−y​log⁡p−\(1−y\)​log⁡\(1−p\)\\ell\(p,y\)=\-y\\log p\-\(1\-y\)\\log\(1\-p\)is11\-mixable\([Cesa\-Bianchi et al\., 2006](https://arxiv.org/html/2610.41023#bib.bib11)\)\. Moreover, everyLL\-Lipschitz proper loss is2/L2/L\-mixable \(see[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)\)\.[Chernov and Vovk \(2009\)](https://arxiv.org/html/2610.41023#bib.bib7)shows that the defensive forecasting algorithm\([Chernov et al\., 2010](https://arxiv.org/html/2610.41023#bib.bib10)\)achievesRegℓ,ℱ​\(T\)=𝒪⁡\(log⁡\(\|ℱ\|​\|ℒ\|\)βℓ⋆\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\mathcal\{O\}\\left\(\\frac\{\\log\(\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|\)\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\)simultaneously for allℓ\\ellfrom a finite classℒ\{\\mathcal\{L\}\}of proper and mixable loss functions\.

On the other hand, recent advances by[Okoroafor et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib3);[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1)achieve omniprediction guarantees beyond proper mixable losses: their algorithms achieveRegℓ,ℱ​\(T\)=𝒪~​\(T​log⁡\(\|ℱ\|​\|ℒ\|\)\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\widetilde\{\\mathcal\{O\}\}\\left\(\\sqrt\{T\\log\(\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|\)\}\\right\)simultaneously for allℓ\\ellfrom a finite classℒ\{\\mathcal\{L\}\}of arbitrary, bounded loss functions\.111Throughout this paper,𝒪~​\(⋅\)\\widetilde\{\\mathcal\{O\}\}\(\\cdot\)hides only polylogarithmic factors inTT\.Importantly, these guarantees can also accommodate certain*infinite*loss classes through finite approximate basis representations: for example, with a suitable choice of basis and approximation precision, the guarantee above implies𝒪~​\(L​T​log⁡\|ℱ\|\)\\widetilde\{\\mathcal\{O\}\}\(L\\sqrt\{T\\log\|\{\\mathcal\{F\}\}\|\}\)regret for allLL\-Lipschitz losses ; see[Okoroafor et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib3)for details as well as[Table1](https://arxiv.org/html/2610.41023#S1.T1)for a summary\.

However, not every loss family admits a finite approximation basis\. A particularly important example is the class of all bounded proper losses\. Indeed, as pointed out by[Okoroafor et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib3), proper losses can have discontinuities at arbitrary thresholds, and the threshold basis used to represent the full class cannot simply be replaced by a fixed finite discretization \(unless one restrictsℱ\{\\mathcal\{F\}\}to be a class with finitely many possible outputs, which is an unnatural assumption\)\. This leads to the first main question our work aims to address:

> *Can a single online forecaster achieve sublinear regret simultaneously for an infinite class of proper losses?*

Moreover, the constant \(inTT\) regret guarantee of[Chernov and Vovk \(2009\)](https://arxiv.org/html/2610.41023#bib.bib7)for a finite class of mixable proper losses raises a further question:

> *Are faster rates achievable for special infinite loss classes such as the class of all mixable proper losses?*

### 1\.1Our Results

We give both negative and positive answers, revealing a separation between unrestricted proper losses, the full class of mixable proper losses, and finite collections of mixable proper losses, as well as separations between the contextual and the non\-contextual setting\.222The non\-contextual setting is a special case of our problem where the benchmark classℱ\{\\mathcal\{F\}\}is the set of all constant mappings, making the contexts meaningless\.More specifically, our contributions are as follows \(throughout, we assume a finite benchmark classℱ\{\\mathcal\{F\}\}for simplicity, but our results generalize directly to a generalℱ\{\\mathcal\{F\}\}as long as an oracle for online learning overℱ\{\\mathcal\{F\}\}is given, in the same vein as[Okoroafor et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib3);[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1)\)\.

##### Impossibility for all bounded proper losses\.

Our first result is a negative answer to the first question, showing that even whenℱ\{\\mathcal\{F\}\}consists of only two linear predictors, no forecaster can achieve sublinear regret simultaneously for all bounded proper losses\.

###### Theorem 1\.1\.

There exists a fixed benchmark classℱ⊆\[0,1\]𝒳\{\\mathcal\{F\}\}\\subseteq\[0,1\]^\{\{\\mathcal\{X\}\}\}of two affine predictors for𝒳=\[0,3/4\]\{\\mathcal\{X\}\}=\[0,3/4\]such that, for every online forecaster and every horizonTT, there exist a sequence of context\-outcome pairs\(x1,y1\),…,\(xT,yT\)\(x\_\{1\},y\_\{1\}\),\\ldots,\(x\_\{T\},y\_\{T\}\)and a proper loss with range\[0,1\]\[0,1\]such that𝔼⁡\[Regℓ,ℱ​\(T\)\]≥T12\\mathbb\{E\}\\left\[\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)\\right\]\\geq\\frac\{T\}\{12\}\.

Thus, the obstacle to extending the aforementioned finite\-basis arguments to this class is fundamental, rather than merely a limitation of existing analysis\. We also note that this impossibility result is in stark contrast to the non\-contextual case where𝒪⁡\(T\)\\mathcal\{O\}\(\\sqrt\{T\}\)regret simultaneously for all bounded proper losses is achievable\([Kleinberg et al\., 2023](https://arxiv.org/html/2610.41023#bib.bib16);[Luo et al\., 2024](https://arxiv.org/html/2610.41023#bib.bib12)\)\. Finally, as pointed out by[Okoroafor et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib3), every bounded proper loss has bounded variation, and thus[Theorem1\.1](https://arxiv.org/html/2610.41023#S1.Thmtheorem1)also implies that no sublinear omniprediction regret guarantee is possible for the class of all losses with bounded variation\.

##### Optimal omniprediction for all mixable proper losses\.

Despite the negative result to the first question, we next provide a positive answer to the second question\.

###### Theorem 1\.2\.

There exists an online forecaster \([Algorithm2](https://arxiv.org/html/2610.41023#alg2)\) such that for every mixable proper lossℓ\\ell,

Regℓ,ℱ​\(T\)=𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3βℓ⋆\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\widetilde\{\\mathcal\{O\}\}\\left\(\\frac\{T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\)\(1\)where𝒪~​\(⋅\)\\widetilde\{\\mathcal\{O\}\}\(\\cdot\)suppresses dependence onlog⁡T\\log Tfactors\.

Thus, while omniprediction with respect to all proper losses is impossible, as long as we restrict our attention to mixable losses, it not only becomes possible again but also admits faster \(than𝒪⁡\(T\)\\mathcal\{O\}\(\\sqrt\{T\}\)\) rates\. On the other hand, compared to the finite class result of\([Chernov and Vovk, 2009](https://arxiv.org/html/2610.41023#bib.bib7)\)where constant regret is achievable, our regret bound is of orderT1/3T^\{1/3\}\. Somewhat surprisingly, we further show that this rate is in fact optimal, even if we only consider the subclass of all Lipschitz proper losses\.

###### Theorem 1\.3\.

There exists a fixed benchmark classℱ⊆\[0,1\]𝒳\{\\mathcal\{F\}\}\\subseteq\[0,1\]^\{\{\\mathcal\{X\}\}\}of two linear predictors for𝒳=\[0,1\]2\{\\mathcal\{X\}\}=\[0,1\]^\{2\}such that, for every online forecaster, everyβ\>0\\beta\>0, and everyTT, there exist a sequence of context\-outcome pairs\(x1,y1\),…,\(xT,yT\)\(x\_\{1\},y\_\{1\}\),\\ldots,\(x\_\{T\},y\_\{T\}\)and a2β\\frac\{2\}\{\\beta\}\-Lipschitz \(hence alsoβ\\beta\-mixable\) bounded proper loss such that𝔼⁡\[Regℓ,ℱ​\(T\)\]≥Ω⁡\(T1/3β\)\\mathbb\{E\}\\left\[\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)\\right\]\\geq\\Omega\\left\(\\frac\{T^\{1/3\}\}\{\\beta\}\\right\)\.

This shows a separation between learning over the full class of mixable proper losses and a finite subset of it\. Moreover, it in fact also shows another separation between the contextual and the non\-contextual case: in the non\-contextual setting,[Luo et al\. \(2024\)](https://arxiv.org/html/2610.41023#bib.bib12)showed that𝒪⁡\(L​log⁡T\)\\mathcal\{O\}\(L\\log T\)regret is achievable simultaneously for allLL\-Lipschitz proper losses, while[Theorem1\.2](https://arxiv.org/html/2610.41023#S1.Thmtheorem2)and[Theorem1\.3](https://arxiv.org/html/2610.41023#S1.Thmtheorem3)together show thatL​T1/3LT^\{1/3\}is the best achievable rate for this loss class\.

##### A best\-of\-all\-worlds forecaster\.

Finally, in light of different possible omniprediction guarantees from both the literature and our work, we propose one single algorithm that achievesallof them simultaneously\.

###### Theorem 1\.4\.

For any fixed finite classℒbd\\mathcal\{L\}\_\{\\mathrm\{bd\}\}of losses with bounded range\[0,1\]\[0,1\]and any fixed finite class of mixable proper lossesℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}, there exists an online forecaster \([Algorithm3](https://arxiv.org/html/2610.41023#alg3)\) that guarantees all of the following simultaneously:

1. 1\.for everyℓ∈ℒmix\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}, Regℓ,ℱ​\(T\)=𝒪⁡\(log⁡\(\|ℱ\|​\|ℒmix\|\)βℓ⋆\);\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\mathcal\{O\}\\left\(\\frac\{\\log\\left\(\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\|\\right\)\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\);\(2\)
2. 2\.for every mixable proper lossℓ\\ell\(whether or not it belongs toℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\), Regℓ,ℱ​\(T\)=𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3βℓ⋆\);\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\widetilde\{\\mathcal\{O\}\}\\left\(\\frac\{T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\);\(3\)
3. 3\.for everyℓ∈ℒbd\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{bd\}\}, Regℓ,ℱ​\(T\)=𝒪⁡\(T​log⁡\(\|ℱ\|​\|ℒbd\|\)\)\.\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\mathcal\{O\}\\left\(\\sqrt\{T\\log\\left\(\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\\right\)\}\\right\)\.\(4\)

Table 1:Comparisons of omniprediction regret bounds onRegℓ,ℱ​\(T\),∀ℓ∈ℒ\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\),\\forall\\ell\\in\{\\mathcal\{L\}\}for different loss familiesℒ\{\\mathcal\{L\}\}\. Benchmark classℱ\{\\mathcal\{F\}\}is assumed to be finite for simplicity\.Loss familyℒ\{\\mathcal\{L\}\}Prior resultsOur resultsFinite class ofmixable proper losses𝒪⁡\(log⁡\(\|ℱ\|​\|ℒ\|\)βℓ⋆\)\\mathcal\{O\}\\left\(\\frac\{\\log\(\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|\)\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\)\([Chernov and Vovk, 2009](https://arxiv.org/html/2610.41023#bib.bib7)\)Same[Algorithm3](https://arxiv.org/html/2610.41023#alg3)and[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4)Finite class ofbounded losses𝒪⁡\(T​log⁡\(\|ℱ\|​\|ℒ\|\)\)\\mathcal\{O\}\\left\(\\sqrt\{T\\log\(\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|\)\}\\right\)\([Okoroafor et al\., 2025](https://arxiv.org/html/2610.41023#bib.bib3);[Farina and Perdomo, 2026](https://arxiv.org/html/2610.41023#bib.bib1)\)Same[Algorithm3](https://arxiv.org/html/2610.41023#alg3)and[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4)All mixable proper lossesNone𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3βℓ⋆\)\\widetilde\{\\mathcal\{O\}\}\\left\(\\frac\{T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\)[Algorithm2](https://arxiv.org/html/2610.41023#alg2)and[Theorem1\.2](https://arxiv.org/html/2610.41023#S1.Thmtheorem2)[Algorithm3](https://arxiv.org/html/2610.41023#alg3)and[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4)matching lower bound[Theorem1\.3](https://arxiv.org/html/2610.41023#S1.Thmtheorem3)All bounded proper losses orall bounded variation losses𝒪~​\(T​log⁡\(\|ℱ\|​\|Γ\|\)\)\\widetilde\{\\mathcal\{O\}\}\\left\(\\sqrt\{T\\log\(\|\{\\mathcal\{F\}\}\|\|\\Gamma\|\)\}\\right\)ifℱ\{\\mathcal\{F\}\}isΓ\\Gamma\-valuedfor a finiteΓ\\Gamma\([Okoroafor et al\., 2025](https://arxiv.org/html/2610.41023#bib.bib3)\)Ω⁡\(T\)\\Omega\(T\)[Theorem1\.1](https://arxiv.org/html/2610.41023#S1.Thmtheorem1)AllLL\-Lipschitz losses𝒪~​\(L​T​log⁡\|ℱ\|\)\\widetilde\{\\mathcal\{O\}\}\(L\\sqrt\{T\\log\|\{\\mathcal\{F\}\}\|\}\)\([Okoroafor et al\., 2025](https://arxiv.org/html/2610.41023#bib.bib3)\)Same[Algorithm3](https://arxiv.org/html/2610.41023#alg3)and[Corollary1\.5](https://arxiv.org/html/2610.41023#S1.Thmtheorem5)All generalized linearmodel \(GLM\) losses𝒪~​\(T​log⁡\|ℱ\|\)\\widetilde\{\\mathcal\{O\}\}\(\\sqrt\{T\\log\|\{\\mathcal\{F\}\}\|\}\)\([Okoroafor et al\., 2025](https://arxiv.org/html/2610.41023#bib.bib3)\)Same[Algorithm3](https://arxiv.org/html/2610.41023#alg3)and[Corollary1\.5](https://arxiv.org/html/2610.41023#S1.Thmtheorem5)

By including squared loss and log loss in the finite mixable familyℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}and including suitable finite bases in the finite bounded familyℒbd\\mathcal\{L\}\_\{\\mathrm\{bd\}\}, we further obtain a single forecaster with the following strong omniprediction guarantees\.

###### Corollary 1\.5\.

There exists an online forecaster that guarantees all of the following simultaneously:

1. 1\.for both squared lossℓ⁡\(p,y\)=\(p−y\)2\\ell\(p,y\)=\(p\-y\)^\{2\}and log lossℓ⁡\(p,y\)=−y​log⁡p−\(1−y\)​log⁡\(1−p\)\\ell\(p,y\)=\-y\\log p\-\(1\-y\)\\log\(1\-p\), Regℓ,ℱ​\(T\)=𝒪⁡\(log⁡\|ℱ\|\);\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\mathcal\{O\}\\left\(\\log\|\{\\mathcal\{F\}\}\|\\right\);
2. 2\.for every mixable proper lossℓ\\ell, Regℓ,ℱ​\(T\)=𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3βℓ⋆\);\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\widetilde\{\\mathcal\{O\}\}\\left\(\\frac\{T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\);
3. 3\.for everyLL\-Lipschitz lossℓ\\ell\(that is not necessarily proper\), Regℓ,ℱ​\(T\)=𝒪~​\(L​T​log⁡\|ℱ\|\);\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\widetilde\{\\mathcal\{O\}\}\\left\(L\\sqrt\{T\\log\|\{\\mathcal\{F\}\}\|\}\\right\);
4. 4\.for every Generalized Linear Model \(GLM\) lossℓ\\ell, Regℓ,ℱ​\(T\)=𝒪~​\(T​log⁡\|ℱ\|\);\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\widetilde\{\\mathcal\{O\}\}\\left\(\\sqrt\{T\\log\|\{\\mathcal\{F\}\}\|\}\\right\);
5. 5\.for the00\-11lossℓ\(p,y\)=\(1−y\)𝕀\{p≥1/2\}\+y𝕀\{p<1/2\}\\ell\(p,y\)=\(1\-y\)\\mathbb\{I\}\{\\left\\\{\{p\\geq 1/2\}\\right\\\}\}\+y\\mathbb\{I\}\{\\left\\\{\{p<1/2\}\\right\\\}\}, Regℓ,ℱ​\(T\)=𝒪~​\(T​log⁡\|ℱ\|\)\.\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\widetilde\{\\mathcal\{O\}\}\\left\(\\sqrt\{T\\log\|\{\\mathcal\{F\}\}\|\}\\right\)\.

Table[1](https://arxiv.org/html/2610.41023#S1.T1)summarizes all our results and the comparisons with the closest prior work\.

### 1\.2Technical Overview

##### Impossibility for all bounded proper losses\.

The hardness stems from the fact that bounded proper losses can have discontinuities at arbitrary thresholds\. Indeed, it is well\-known in the online learning literature that one\-dimensional threshold functions are not online learnable; see e\.g\.\([Rakhlin and Sridharan, 2014](https://arxiv.org/html/2610.41023#bib.bib25), Theorem 8\.2\)\. The proof is based on an infinite binary tree that keeps cutting an interval at a node into two equal halves as its two children\. For a sequence of i\.i\.d\. binary outcomes, the contexts are then generated according to the path of the tree that is determined by these binary outcomes\. The difficulty of learning threshold functions boils down to the fact that no matter what the binary outcome sequence is, there always exists a threshold function that correctly classifies all examples\.

In our problem, on the other hand, we are only competing with a finite benchmark class, so for a fixed loss, we cannot always find a perfect benchmark any more\. However, at the same time we are not constrained to a fixed loss function in omniprediction\. Indeed, we can effectively simulate infinite threshold classifiers via picking different proper losses since, as mentioned, proper losses can have discontinuities at arbitrary thresholds\. This is the key idea of our proof, where we only need two simple affine benchmarksf1​\(x\)=xf\_\{1\}\(x\)=xandf2​\(x\)=x\+1/4f\_\{2\}\(x\)=x\+1/4, corresponding to two of the aforementioned binary trees with every node being1/41/4apart from each other, so that no matter what the i\.i\.d\. binary outcomes turn out to be, one of the benchmarks must suffer zero loss on a particular threshold\-based proper loss functions, while the learner must sufferΩ⁡\(T\)\\Omega\(T\)loss in expectation\. The details can be found in[AppendixA\.1](https://arxiv.org/html/2610.41023#A1.SS1)\.

##### Ω⁡\(T1/3\)\\Omega\(T^\{1/3\}\)lower bound for Lipschitz proper losses\.

To establish theΩ⁡\(T1/3\)\\Omega\(T^\{1/3\}\)lower bound for Lipschitz losses, we divide the horizon intoKKphases, each with a distinct Bernoulli meanvjv\_\{j\}\. We consider two fixed linear predictors,f\+f\_\{\+\}andf−f\_\{\-\}, which predict slightly above and belowvjv\_\{j\}, respectively, so thatf\+f\_\{\+\}is favored by an excess of outcome11, whilef−f\_\{\-\}is favored by an excess of outcome00\. These random fluctuations are captured bySj=∑t∈ℐj\(yt−vj\)S\_\{j\}=\\sum\_\{t\\in\\mathcal\{I\}\_\{j\}\}\(y\_\{t\}\-v\_\{j\}\)whereℐj\\mathcal\{I\}\_\{j\}denotes the rounds in thejj\-th phase\.

Note that for a fixed proper loss, truthfully reportingvjv\_\{j\}in each phasejjis clearly optimal in expectation due to properness\. Therefore, the key is to decide the loss function after the realization of the outcomes\. More specifically, we construct a fixed family of Lipschitz proper losses whose curvature can be independently adjusted on each disjoint interval\[vj−hj,vj\+hj\]\[v\_\{j\}\-h\_\{j\},v\_\{j\}\+h\_\{j\}\]forhj=Θ⁡\(K/T\)h\_\{j\}=\\Theta\(\\sqrt\{K/T\}\)\. After observing all outcomes, we select two losses that place additional curvature on intervals with positive and negativeSjS\_\{j\}, favoringf\+f\_\{\+\}andf−f\_\{\-\}, respectively\. Crucially, the sum of these two losses is independent of the outcomes, which means the optimal strategy for the learner in terms of minimizing the expected sum of these two losses is still to truthfully reportvjv\_\{j\}in each phasejj\. Meanwhile, the two benchmark comparisons capture fluctuations of both signs, yielding an expected advantage ofΩ⁡\(1\)\\Omega\(1\)per phase and thusΩ⁡\(K\)\\Omega\(K\)overall regret\. Finally, to make sure that the intervals\[vj−hj,vj\+hj\]\[v\_\{j\}\-h\_\{j\},v\_\{j\}\+h\_\{j\}\]are disjoint, the largest possible value ofKKis of orderT1/3T^\{1/3\}, which leads to theΩ⁡\(T1/3\)\\Omega\(T^\{1/3\}\)lower bound\. See[AppendixA\.2](https://arxiv.org/html/2610.41023#A1.SS2)for details\.

#### 1\.2\.1Key ideas behind algorithms and upper bounds

For upper bounds, our approach builds on the framework of[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1), which we briefly review below using our notations and in a way that is slightly different from \(but equivalent to\) their original presentation\. The starting point is the following regret identity: for any loss functionℓ\\elland benchmarkf∈ℱf\\in\{\\mathcal\{F\}\}, the regret of the learner againstffcan be written as

ℓ⁡\(πℓ​\(Pt\),yt\)−ℓ⁡\(f⁡\(xt\),yt\)\\displaystyle\\ell\(\\pi\_\{\\ell\}\(P\_\{t\}\),y\_\{t\}\)\-\\ell\(f\(x\_\{t\}\),y\_\{t\}\)=bℓ​\(Pt,f⁡\(xt\)\)​\(yt−Pt\)−dℓ​\(Pt,f⁡\(xt\)\)\\displaystyle=b\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)\(y\_\{t\}\-P\_\{t\}\)\-d\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)\(5\)≤bℓ​\(Pt,f⁡\(xt\)\)​\(yt−Pt\)\\displaystyle\\leq b\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)\(y\_\{t\}\-P\_\{t\}\)\(6\)for some loss\-dependent functionsbℓb\_\{\\ell\}anddℓd\_\{\\ell\}; their concrete definitions are deferred to[Section2](https://arxiv.org/html/2610.41023#S2), but one should treatbℓb\_\{\\ell\}as a certain test function \(using the terminology of[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1)\) anddℓd\_\{\\ell\}as a properness gain term that is always nonnegative, explaining the inequality[Eq\. \(6\)](https://arxiv.org/html/2610.41023#S1.E6)\. Now, assumeℒ\{\\mathcal\{L\}\}is finite and treat each benchmark\-loss pair\(f,ℓ\)\(f,\\ell\)as an expert with rewardbℓ​\(Pt,f⁡\(xt\)\)​\(yt−Pt\)b\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)\(y\_\{t\}\-P\_\{t\}\)in roundtt\. We can then use a standard expert algorithm, such as the classic multiplicative/exponential weight update of[Freund and Schapire \(1997\)](https://arxiv.org/html/2610.41023#bib.bib18), to produce a distributionWtW\_\{t\}over all\|ℱ\|​\|ℒ\|\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|benchmark\-loss pairs and ensure for each\(f,ℓ\)\(f,\\ell\),

∑t=1Tbℓ​\(Pt,f⁡\(xt\)\)​\(yt−Pt\)−∑t=1T𝔼\(f′,ℓ′\)∼Wt​\[bℓ′​\(Pt,f′​\(xt\)\)​\(yt−Pt\)\]≤𝒪⁡\(T​log⁡\(\|ℱ\|​\|ℒ\|\)\)\.\\sum\_\{t=1\}^\{T\}b\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)\(y\_\{t\}\-P\_\{t\}\)\-\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{\(f^\{\\prime\},\\ell^\{\\prime\}\)\\sim W\_\{t\}\}\\left\[b\_\{\\ell^\{\\prime\}\}\(P\_\{t\},f^\{\\prime\}\(x\_\{t\}\)\)\(y\_\{t\}\-P\_\{t\}\)\\right\]\\leq\\mathcal\{O\}\\left\(\\sqrt\{T\\log\(\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|\)\}\\right\)\.\(7\)Therefore, if in each roundtt, we can make sure that, no matter what the outcomeyty\_\{t\}turns out to be, the forecast distributionDtD\_\{t\}always ensures𝔼Pt∼Dt​\[Gt​\(Pt\)​\(yt−Pt\)\]≈0\\mathbb\{E\}\_\{P\_\{t\}\\sim D\_\{t\}\}\\left\[G\_\{t\}\(P\_\{t\}\)\(y\_\{t\}\-P\_\{t\}\)\\right\]\\approx 0for an expected test functionGt​\(P\)=𝔼\(f′,ℓ′\)∼Wt​\[bℓ′​\(P,f′​\(xt\)\)\]G\_\{t\}\(P\)=\\mathbb\{E\}\_\{\(f^\{\\prime\},\\ell^\{\\prime\}\)\\sim W\_\{t\}\}\\left\[b\_\{\\ell^\{\\prime\}\}\(P,f^\{\\prime\}\(x\_\{t\}\)\)\\right\], then combining[Eq\. \(6\)](https://arxiv.org/html/2610.41023#S1.E6)and[Eq\. \(7\)](https://arxiv.org/html/2610.41023#S1.E7)provesRegℓ,ℱ=𝒪⁡\(T​log⁡\(\|ℱ\|​\|ℒ\|\)\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}=\\mathcal\{O\}\\left\(\\sqrt\{T\\log\(\|\{\\mathcal\{F\}\}\|\|\{\\mathcal\{L\}\}\|\)\}\\right\)for allℓ∈ℒ\\ell\\in\{\\mathcal\{L\}\}\. Indeed, the aforementioned requirement onDtD\_\{t\}exactly corresponds to anExpected Variational Inequality\(EVI\), which can be solved efficiently in timelog⁡\(1/ϵ\)\\log\(1/\\epsilon\)to make sure𝔼Pt∼Dt​\[Gt​\(Pt\)​\(yt−Pt\)\]≤ϵ\\mathbb\{E\}\_\{P\_\{t\}\\sim D\_\{t\}\}\\left\[G\_\{t\}\(P\_\{t\}\)\(y\_\{t\}\-P\_\{t\}\)\\right\]\\leq\\epsilonfor any precisionϵ\>0\\epsilon\>0\([Zhang et al\., 2025b](https://arxiv.org/html/2610.41023#bib.bib19)\)\.

##### V\-shaped losses and continuous exponential weights\.

To apply the same idea to an infinite class of proper losses, we utilize the fact that, up to a prediction\-independent term, every proper lossℓ⁡\(p,y\)\\ell\(p,y\)of our interest can be expressed as a conic combination ofV\-shaped losses:∫01wℓ​\(v\)​λv​\(p,y\)​𝑑v\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\lambda\_\{v\}\(p,y\)dvwhereλv\\lambda\_\{v\}is the V\-shaped loss at thresholdvvandwℓ​\(v\)w\_\{\\ell\}\(v\)is the corresponding nonnegative weight \([Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)\)\. It is thus natural to apply the framework of[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1)to the infinite class of V\-shaped losses by learning the distributionWtW\_\{t\}via the continuous version of exponential weights \(see e\.g\.\([Hoeven et al\., 2018](https://arxiv.org/html/2610.41023#bib.bib20)\)\)\.

While continuous exponential weights does not guarantee a regret bound against any single V\-shaped loss, it admits a regret bound against amixtureof the V\-shaped losses, in the form ofKL\-Compη\+η​T\\frac\{\\textsc\{KL\-Comp\}\}\{\\eta\}\+\\eta T, whereη\\etais a learning rate andKL\-Compis a complexity term that depends on the KL divergence between the mixture weight and a prior distribution\. In our case, we naturally pick the mixture weight according to the normalized version ofwℓ​\(v\)w\_\{\\ell\}\(v\)\. BoundingKL\-Compthen requires special properties of the weightwℓ​\(v\)w\_\{\\ell\}\(v\), which corresponds to special structures of the proper loss class, such as Lipschitzness or more generally mixability\.

##### Warm\-up for Lipschitz class and second\-order correction\.

To best illustrate this idea, we use the Lipschitz proper loss class as a warm\-up in[Section3\.1](https://arxiv.org/html/2610.41023#S3.SS1)\. It can be shown that everyLL\-Lipschitz proper loss ensureswℓ​\(v\)≤2​Lw\_\{\\ell\}\(v\)\\leq 2Land thusKL\-Compcan be well bounded; see[Lemma3\.1](https://arxiv.org/html/2610.41023#S3.Thmtheorem1)\.

However, the discussion so far \(and specifically the bound of the formKL\-Compη\+η​T\\frac\{\\textsc\{KL\-Comp\}\}\{\\eta\}\+\\eta T\) only leads to a regret bound of orderT\\sqrt\{T\}\. To further improve it toT1/3T^\{1/3\}, we need the following two ideas: first, incorporate a second\-order correction term to the exponential weight updates that leads to a second\-order regret bound of the form

KL\-Compη\+∑t=1T∫01wℓ​\(v\)​η​bℓ​\(Pt,f⁡\(xt\)\)2​𝑑v,\\frac\{\\textsc\{KL\-Comp\}\}\{\\eta\}\+\\sum\_\{t=1\}^\{T\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\eta b\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)^\{2\}dv,\(8\)an idea that has been explored in the literature\([Hazan and Kale, 2010](https://arxiv.org/html/2610.41023#bib.bib22);[Steinhardt and Liang, 2014](https://arxiv.org/html/2610.41023#bib.bib21);[Wintenberger, 2017](https://arxiv.org/html/2610.41023#bib.bib24);[Chen et al\., 2021](https://arxiv.org/html/2610.41023#bib.bib23)\)\([Lemma3\.1](https://arxiv.org/html/2610.41023#S3.Thmtheorem1)\); second, combine the second\-order term above and the important negative term−dℓ​\(Pt,f⁡\(xt\)\)\-d\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)in[Eq\. \(5\)](https://arxiv.org/html/2610.41023#S1.E5)\(that was dropped in earlier analysis\) and show that together they are of order𝒪⁡\(η2\)\\mathcal\{O\}\(\\eta^\{2\}\)instead of𝒪⁡\(η\)\\mathcal\{O\}\(\\eta\)\([Lemma3\.2](https://arxiv.org/html/2610.41023#S3.Thmtheorem2)\):

∫01wℓ​\(v\)​\(η​bℓ​\(Pt,f⁡\(xt\)\)2−dℓ​\(Pt,f⁡\(xt\)\)\)​𝑑v≲L​η2\.\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\left\(\\eta b\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)^\{2\}\-d\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)\\right\)dv\\lesssim L\\eta^\{2\}\.\(9\)Pickingη\\etaappropriately finally leads to a regret bound of orderT1/3T^\{1/3\}\.

##### Extension to mixable class\.

To handle the more general class of mixable proper losses, several more ideas are required \([Section3\.2](https://arxiv.org/html/2610.41023#S3.SS2)\)\. The difficulty comes from the fact that for eachβ\\beta\-mixable proper lossℓ\\ell, the corresponding weightwℓ​\(v\)w\_\{\\ell\}\(v\)can be as large as1β​v​\(1−v\)\\frac\{1\}\{\\beta v\(1\-v\)\}\(hence no longer always bounded; see[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)\)\. To compensate this, we naturally run continuous exponential weights with a prior that is proportional to1v⁡\(1−v\)\\frac\{1\}\{v\(1\-v\)\}for the V\-shaped loss parameterized by thresholdvv, so thatKL\-Compis again bounded \([Lemma3\.5](https://arxiv.org/html/2610.41023#S3.Thmtheorem5)\)\. Noticing that∫01d​vv⁡\(1−v\)\\int\_\{0\}^\{1\}\\frac\{dv\}\{v\(1\-v\)\}\(the normalization factor for the prior\) is unbounded, we also naturally only considervvfrom the interval\[δ,1−δ\]\[\\delta,1\-\\delta\]for some smallδ\\deltaof order1/T1/T\.

Moreover, due to the potentially unboundedwℓ​\(v\)w\_\{\\ell\}\(v\),[Eq\. \(9\)](https://arxiv.org/html/2610.41023#S1.E9)no longer holds\. To address this issue, we come up with a new second\-order correction term to be incorporated into the continuous exponential weight update\. Together with a new expected test functionGtG\_\{t\}designed accordingly, this leads to the following new second\-order regret bound that, compared to[Eq\. \(8\)](https://arxiv.org/html/2610.41023#S1.E8), contains an extraBernoulli variance factorPt​\(1−Pt\)P\_\{t\}\(1\-P\_\{t\}\)\([Lemma3\.5](https://arxiv.org/html/2610.41023#S3.Thmtheorem5)\):

KL\-Compη\+∑t=1T∫01wℓ​\(v\)​η​bℓ​\(Pt,f⁡\(xt\)\)2​Pt​\(1−Pt\)⏟extra factor​𝑑v\.\\frac\{\\textsc\{KL\-Comp\}\}\{\\eta\}\+\\sum\_\{t=1\}^\{T\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\eta b\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)^\{2\}\\underbrace\{P\_\{t\}\(1\-P\_\{t\}\)\}\_\{\\text\{extra factor\}\}dv\.\(10\)When combined with the negative term−dℓ​\(Pt,f⁡\(xt\)\)\-d\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\), this extra Bernoulli variance factor exactly helps cancel the potentially largewℓ​\(v\)w\_\{\\ell\}\(v\)\(recallwℓ​\(v\)≤1β​v​\(1−v\)w\_\{\\ell\}\(v\)\\leq\\frac\{1\}\{\\beta v\(1\-v\)\}\), leading to the following analogue of[Eq\. \(9\)](https://arxiv.org/html/2610.41023#S1.E9)\([Lemma3\.6](https://arxiv.org/html/2610.41023#S3.Thmtheorem6)\):

∫01wℓ​\(v\)​\(η​bℓ​\(Pt,f⁡\(xt\)\)2​Pt​\(1−Pt\)−dℓ​\(Pt,f⁡\(xt\)\)\)​𝑑v≲η2β\.\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\left\(\\eta b\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)^\{2\}P\_\{t\}\(1\-P\_\{t\}\)\-d\_\{\\ell\}\(P\_\{t\},f\(x\_\{t\}\)\)\\right\)dv\\lesssim\\frac\{\\eta^\{2\}\}\{\\beta\}\.\(11\)Pickingη\\etaappropriately again leads to the claimed regret bound of orderT1/3T^\{1/3\}\.

##### Combining everything\.

Finally, to achieve the best\-of\-all\-worlds guarantee of[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4), it suffices to combine the algorithmic components for both the finite class result and the infinite class result under the same framework of[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1); see[Section3\.3](https://arxiv.org/html/2610.41023#S3.SS3)\.

### 1\.3Related Work

##### Omniprediction and multicalibration\.

Omniprediction in the batch setting was introduced by[Gopalan et al\. \(2022\)](https://arxiv.org/html/2610.41023#bib.bib8), who showed how a single forecast can be post\-processed to compete across a family of losses\. They achieved so by drawing a connection to multicalibration\([Hébert\-Johnson et al\., 2018](https://arxiv.org/html/2610.41023#bib.bib14)\)\. Motivated by the role of swap regret in online learning,[Gopalan et al\. \(2023\)](https://arxiv.org/html/2610.41023#bib.bib5)introduced the stronger “swap version" of both omniprediction and multicalibration\.[Garg et al\. \(2024\)](https://arxiv.org/html/2610.41023#bib.bib9)further extended these concepts to the online setting, which is the focus of our work\. We point out that, however, standard online\-to\-batch reductions can be used to convert online regret bounds to sample complexity bounds in the batch setting; see e\.g\.[Okoroafor et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib3);[Luo et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib13)\.

[Okoroafor et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib3)are the first to develop near\-optimal omniprediction guarantees for several loss families\. More recently,[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1)developed a different black\-box approach that achieves multicalibration/omniprediction via standard online learning\. As mentioned, our methods are built on this approach\.

In a concurrent work,[Okoroafor \(2026\)](https://arxiv.org/html/2610.41023#bib.bib4)studied the stronger swap version of omniprediction and show that𝒪~​\(T​log⁡\(\|ℱ\|​\|Γ\|\)\)\\widetilde\{\\mathcal\{O\}\}\(\\sqrt\{T\\log\(\|\{\\mathcal\{F\}\}\|\|\\Gamma\|\)\}\)swap regret is achievable simultaneously for all bounded proper losses ifℱ\{\\mathcal\{F\}\}isΓ\\Gamma\-valued for a finite setΓ\\Gamma, similar to the non\-swap version of[Okoroafor et al\. \(2025\)](https://arxiv.org/html/2610.41023#bib.bib3)\. Our impossibility result \([Theorem1\.1](https://arxiv.org/html/2610.41023#S1.Thmtheorem1)\) shows that the restriction to a finiteΓ\\Gammais unavoidable\. On the other hand,[Okoroafor \(2026\)](https://arxiv.org/html/2610.41023#bib.bib4)also showed that𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3\)\\widetilde\{\\mathcal\{O\}\}\(T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\)swap regret is achievable for all convex and11\-Lipschitz proper losses\. This result shares some similarity with our[Theorem1\.2](https://arxiv.org/html/2610.41023#S1.Thmtheorem2), except that their regret notion is stronger while our loss class is broader \(more concretely, we require neither convexity nor Lipschitzness but only mixability\)\. Their approach to achieve this fast rate is via a certain second\-order form of multicalibration\.

##### Non\-contextual setting and U\-calibration\.

As mentioned in[Footnote2](https://arxiv.org/html/2610.41023#footnote2), the non\-contextual setting is a special case whereℱ\{\\mathcal\{F\}\}contains all constant mappings\. Omniprediction in this setting is referred to as U\-calibration\([Kleinberg et al\., 2023](https://arxiv.org/html/2610.41023#bib.bib16)\)\. As discussed earlier, our work shows several gaps between U\-calibration and omniprediction, with many stronger guarantees achievable for the former but not the latter\.

Our best\-of\-all\-worlds result \([Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4)\) is in spirit similar to the result of[Frongillo et al\. \(2026\)](https://arxiv.org/html/2610.41023#bib.bib17)and[Dewasurendra \(2026\)](https://arxiv.org/html/2610.41023#bib.bib15)for U\-calibration\. Specifically, they developed a single U\-calibrated forecaster with𝒪~​\(T\)\\widetilde\{\\mathcal\{O\}\}\(\\sqrt\{T\}\)regret simultaneously for all bounded proper losses and also𝒪⁡\(log⁡T\)\\mathcal\{O\}\(\\log T\)regret for smooth proper losses\. Their approaches are quite different from and much simpler than ours, due to the lack of contexts\.

##### Multiclass omniprediction\.

Our results are restricted to binary outcomes\. While the finite\-class results can be generalized to the multiclass case according to[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1), it is unclear how to do so for the infinite\-class result since there is no similar V\-shaped basis for the multiclass case\([Kleinberg et al\., 2023](https://arxiv.org/html/2610.41023#bib.bib16)\)\. Moreover, the sample complexity for multiclass omniprediction is often exponential in the number of classes; see e\.g\.[Hu et al\. \(2026\)](https://arxiv.org/html/2610.41023#bib.bib6)\.

## 2Preliminaries

Notation\.For every positive integernn,\[n\]\[n\]denotes the set\{1,…,n\}\\\{1,\\ldots,n\\\}\. We write\[x\]\+=max⁡\{x,0\}\[x\]\_\{\+\}=\\max\\\{x,0\\\}\. The distributionsUnif⁡\(S\)\\operatorname\{Unif\}\(S\),Ber​\(p\)\\texttt\{Ber\}\(p\), andδx\\delta\_\{x\}are uniform on a finite nonempty setSS, Bernoulli with meanpp, and Dirac delta atxx, respectively\. The product measureP⊗QP\\otimes Qdescribes independent draws fromPPandQQ\. For any probability measuresQ,PQ,Pon the same measurable space,Q≪PQ\\ll PmeansP⁡\(A\)=0P\(A\)=0impliesQ⁡\(A\)=0Q\(A\)=0for every measurableAA\. Withd​Q/d​P\\,\\mathrm\{d\}Q/\\,\\mathrm\{d\}Pdenoting the density ofQQrelative toPP,KL\(Q∥P\)=∫log\(dQ/dP\)dQ\{\\text\{\\rm KL\}\}\(Q\\\|P\)=\\int\\log\(\\,\\mathrm\{d\}Q/\\,\\mathrm\{d\}P\)\\,\\mathrm\{d\}QwhenQ≪PQ\\ll P, and\+∞\+\\inftyotherwise\. We also write𝔼Q​\[g\]=∫g​𝑑Q\\mathbb\{E\}\_\{Q\}\[g\]=\\int g\\,\\mathrm\{d\}Q\. The measureLeb\\operatorname\{Leb\}is Lebesgue measure, with respect to which almost\-everywhere statements on intervals are understood\. The clipping operator isclip⁡\(x,\[a,b\]\)=min⁡\{b,max⁡\{a,x\}\}\\operatorname\{clip\}\(x,\[a,b\]\)=\\min\\\{b,\\max\\\{a,x\\\}\\\}\.

##### Proper losses and two subclasses\.

We recall that a lossℓ\\ellis proper if𝔼Y∼Ber​\(p\)​\[ℓ⁡\(p,Y\)\]≤𝔼Y∼Ber​\(p\)​\[ℓ⁡\(q,Y\)\],∀p,q∈\[0,1\],\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[\\ell\(p,Y\)\\right\]\\leq\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[\\ell\(q,Y\)\\right\],\\forall p,q\\in\[0,1\],that is, predictingppis optimal when the outcome is indeed drawn fromBer​\(p\)\\texttt\{Ber\}\(p\)\. In this case we letπℓ​\(p\)\\pi\_\{\\ell\}\(p\), the post processing functionarg⁡minq∈\[0,1\]⁡𝔼Y∼Ber​\(p\)​\[ℓ⁡\(q,Y\)\]\\argmin\_\{q\\in\[0,1\]\}\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[\\ell\(q,Y\)\\right\], be simplypp\.

We consider two subclasses of proper losses:ℒLip​\(L\)\\mathcal\{L\}\_\{\\mathrm\{Lip\}\}\\left\(L\\right\)andℒmix​\(β\)\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(\\beta\\right\), where the first one represents the class of allLL\-Lipschitz proper losses and the second represents the class of allβ\\beta\-mixable proper losses\. Specifically, a lossℓ\\ellisLL\-Lipschitz if its partial loss functionℓ⁡\(⋅,y\)\\ell\(\\cdot,y\)isLL\-Lipschitz for eachy∈\{0,1\}y\\in\\\{0,1\\\}, that is\|ℓ⁡\(p,y\)−ℓ⁡\(q,y\)\|≤L​\|p−q\|\|\\ell\(p,y\)\-\\ell\(q,y\)\|\\leq L\|p\-q\|for allp,q∈\[0,1\]p,q\\in\[0,1\]\. As mentioned in[Section1](https://arxiv.org/html/2610.41023#S1), a lossℓ\\ellisβ\\beta\-mixable if, for every distributionDDover\[0,1\]\[0,1\], there existsp∈\[0,1\]p\\in\[0,1\]such thatexp⁡\(−β​ℓ​\(p,y\)\)≥𝔼P∼D​exp⁡\(−β​ℓ​\(P,y\)\),∀y∈\{0,1\}\.\\exp\\left\(\-\\beta\\ell\(p,y\)\\right\)\\geq\\mathbb\{E\}\_\{P\\sim D\}\\exp\\left\(\-\\beta\\ell\(P,y\)\\right\),\\forall y\\in\\left\\\{0,1\\right\\\}\.We recall thatβℓ⋆\\beta\_\{\\ell\}^\{\\star\}is largest constantβ\\betasuch thatℓ\\ellisβ\\beta\-mixable\.

##### Threshold representation and curvature bounds\.

Up to a prediction\-independent term, every Lipschitz/mixable proper lossℓ\\ellcan be written as a conic combination of V\-shaped losses\. Specifically, for anyv∈\[0,1\]v\\in\[0,1\], the V\-shaped lossλv\\lambda\_\{v\}parameterized by thresholdvvis defined through

λv\(q,1\)=\(1−v\)𝕀\{q<v\}andλv\(q,0\)=v𝕀\{q≥v\}\.\\lambda\_\{v\}\(q,1\)=\(1\-v\)\\mathbb\{I\}\{\\left\\\{\{q<v\}\\right\\\}\}\\quad\\text\{and\}\\quad\\lambda\_\{v\}\(q,0\)=v\\mathbb\{I\}\{\\left\\\{\{q\\geq v\}\\right\\\}\}\.\(12\)Then, the following lemma holds \(see[AppendixC](https://arxiv.org/html/2610.41023#A3)for the proof\)\.

###### Lemma 2\.1\.

If a proper lossℓ\\ellis Lipschitz or mixable, then it can be written as

ℓ⁡\(p,y\)=Cℓ​\(y\)\+∫01wℓ​\(v\)​λv​\(p,y\)​𝑑v\\ell\(p,y\)=C\_\{\\ell\}\(y\)\+\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\lambda\_\{v\}\(p,y\)\\,\\mathrm\{d\}v\(13\)for allp∈\[0,1\]p\\in\[0,1\]andy∈\{0,1\}y\\in\\\{0,1\\\}, whereCℓ​\(y\)C\_\{\\ell\}\(y\)is independent of the prediction andwℓw\_\{\\ell\}is nonnegative and measurable\. Moreover, for any proper loss admitting the representation[Eq\. \(13\)](https://arxiv.org/html/2610.41023#S2.E13)and anyL,β\>0L,\\beta\>0,

- •ℓ∈ℒLip​\(L\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{Lip\}\}\\left\(L\\right\)if and only ifwℓ​\(v\)≤Lmax⁡\{v,1−v\}w\_\{\\ell\}\(v\)\\leq\\frac\{L\}\{\\max\\\{v,1\-v\\\}\}almost everywhere;
- •ℓ∈ℒmix​\(β\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(\\beta\\right\)if and only ifwℓ​\(v\)≤1β​v​\(1−v\)w\_\{\\ell\}\(v\)\\leq\\frac\{1\}\{\\beta v\(1\-v\)\}almost everywhere\.

Consequently, ifℓ∈ℒLip​\(L\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{Lip\}\}\\left\(L\\right\), thenwℓ​\(v\)≤2​Lw\_\{\\ell\}\(v\)\\leq 2Lalmost everywhere and alsoℓ∈ℒmix​\(2/L\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(2/L\\right\)\.

##### Loss differences and test functions\.

For any lossℓ\\ell, write the loss difference between two outcomes at the same predictionqqasΔℓ​\(q\)=ℓ⁡\(q,1\)−ℓ⁡\(q,0\)\\Delta\_\{\\ell\}\(q\)=\\ell\(q,1\)\-\\ell\(q,0\)\. Further define

bℓ​\(p,q\)=Δℓ​\(πℓ​\(p\)\)−Δℓ​\(q\)anddℓ​\(p,q\)=𝔼Y∼Ber​\(p\)​\[ℓ⁡\(q,Y\)−ℓ⁡\(πℓ​\(p\),Y\)\]\.b\_\{\\ell\}\(p,q\)=\\Delta\_\{\\ell\}\(\\pi\_\{\\ell\}\(p\)\)\-\\Delta\_\{\\ell\}\(q\)\\quad\\text\{and\}\\quad d\_\{\\ell\}\(p,q\)=\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\[\\ell\(q,Y\)\-\\ell\(\\pi\_\{\\ell\}\(p\),Y\)\]\.Then, the regret identity of[Eq\. \(5\)](https://arxiv.org/html/2610.41023#S1.E5)can be verified as follows: for anyp,q∈\[0,1\]p,q\\in\[0,1\]andy∈\{0,1\}y\\in\\\{0,1\\\},

ℓ⁡\(πℓ​\(p\),y\)−ℓ⁡\(q,y\)\\displaystyle\\ell\(\\pi\_\{\\ell\}\(p\),y\)\-\\ell\(q,y\)=y​bℓ​\(p,q\)\+ℓ⁡\(πℓ​\(p\),0\)−ℓ⁡\(q,0\)\\displaystyle=yb\_\{\\ell\}\(p,q\)\+\\ell\(\\pi\_\{\\ell\}\(p\),0\)\-\\ell\(q,0\)=\(y−p\)​bℓ​\(p,q\)\+p​bℓ​\(p,q\)\+ℓ⁡\(πℓ​\(p\),0\)−ℓ⁡\(q,0\)\\displaystyle=\(y\-p\)b\_\{\\ell\}\(p,q\)\+pb\_\{\\ell\}\(p,q\)\+\\ell\(\\pi\_\{\\ell\}\(p\),0\)\-\\ell\(q,0\)=\(y−p\)​bℓ​\(p,q\)\+p⁡\(ℓ⁡\(πℓ​\(p\),1\)−ℓ⁡\(q,1\)\)\+\(1−p\)​\(ℓ⁡\(πℓ​\(p\),0\)−ℓ⁡\(q,0\)\)\\displaystyle=\(y\-p\)b\_\{\\ell\}\(p,q\)\+p\\left\(\\ell\(\\pi\_\{\\ell\}\(p\),1\)\-\\ell\(q,1\)\\right\)\+\(1\-p\)\\left\(\\ell\(\\pi\_\{\\ell\}\(p\),0\)\-\\ell\(q,0\)\\right\)=\(y−p\)​bℓ​\(p,q\)−dℓ​\(p,q\)\.\\displaystyle=\(y\-p\)b\_\{\\ell\}\(p,q\)\-d\_\{\\ell\}\(p,q\)\.\(14\)Following[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1), we callbℓ​\(p,q\)b\_\{\\ell\}\(p,q\)a test function\. Also note that by the definition ofπℓ\\pi\_\{\\ell\},dℓ​\(p,q\)d\_\{\\ell\}\(p,q\)is always nonnegative and can be treated as properness gain\.

For a V\-shaped lossλv\\lambda\_\{v\}, we usebv​\(p,q\)b\_\{v\}\(p,q\)anddv​\(p,q\)d\_\{v\}\(p,q\)as the shorthands forbλv​\(p,q\)b\_\{\\lambda\_\{v\}\}\(p,q\)anddλv​\(p,q\)d\_\{\\lambda\_\{v\}\}\(p,q\)\. It is straightforward to verify that they take the following simple forms:

bv\(p,q\)=𝕀\{p<v≤q\}−𝕀\{q<v≤p\}anddv\(p,q\)=\|v−p\|bv\(p,q\)2\.b\_\{v\}\(p,q\)=\\mathbb\{I\}\{\\left\\\{\{p<v\\leq q\}\\right\\\}\}\-\\mathbb\{I\}\{\\left\\\{\{q<v\\leq p\}\\right\\\}\}\\quad\\text\{and\}\\quad d\_\{v\}\(p,q\)=\|v\-p\|b\_\{v\}\(p,q\)^\{2\}\.Therefore, whenℓ\\ellis either Lipschitz or mixable,[Eq\. \(13\)](https://arxiv.org/html/2610.41023#S2.E13)and[Eq\. \(14\)](https://arxiv.org/html/2610.41023#S2.E14)together imply: for anyp,q∈\[0,1\]p,q\\in\[0,1\]andy∈\{0,1\}y\\in\\\{0,1\\\},

ℓ⁡\(p,y\)−ℓ⁡\(q,y\)=∫01wℓ​\(v\)​\(bv​\(p,q\)​\(y−p\)−dv​\(p,q\)\)​𝑑v,\\ell\(p,y\)\-\\ell\(q,y\)=\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\left\(b\_\{v\}\(p,q\)\(y\-p\)\-d\_\{v\}\(p,q\)\\right\)\\,\\mathrm\{d\}v,\(15\)a regret identity that is critical to our analysis\.

##### EVI on an interval\.

As discussed, a key component of the algorithm of[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1)is a solver for an Expected Variational Inequality \(EVI\), whose efficiency is guaranteed due to[Zhang et al\. \(2025a\)](https://arxiv.org/html/2610.41023#bib.bib2)\. For our purpose, we only need to solve an EVI over the intervalIδ=\[δ,1−δ\]I\_\{\\delta\}=\[\\delta,1\-\\delta\]for someδ∈\[0,1/2\)\\delta\\in\[0,1/2\), which admits simple solutions\. The details are summarized in the following lemma\.

###### Lemma 2\.2\(Binary EVI on a closed interval\)\.

For every0≤δ<1/20\\leq\\delta<1/2, everyG:Iδ→ℝG:I\_\{\\delta\}\\to\\mathbb\{R\}satisfying1\+G⁡\(p\)​\(y−p\)≥01\+G\(p\)\(y\-p\)\\geq 0for allp∈Iδp\\in I\_\{\\delta\}andy∈\{0,1\}y\\in\\\{0,1\\\}, and everyϵ\>δ/\(1−δ\)\\epsilon\>\\delta/\(1\-\\delta\), there is a distributionDDonIδI\_\{\\delta\}with at most three support points such that

𝔼P∼D​\[G⁡\(P\)​\(y−P\)\]≤ϵ,∀y∈\{0,1\}\.\\mathbb\{E\}\_\{P\\sim D\}\[G\(P\)\(y\-P\)\]\\leq\\epsilon,\\qquad\\forall y\\in\\\{0,1\\\}\.Whenδ\>0\\delta\>0,DDmay be supported on any finite grid containing the endpoints ofIδI\_\{\\delta\}whose adjacent gaps are at mostδ\\delta\.

## 3Algorithms and Results

Following the overview in[Section1\.2\.1](https://arxiv.org/html/2610.41023#S1.SS2.SSS1), we start with a warm\-up in[Section3\.1](https://arxiv.org/html/2610.41023#S3.SS1)and consider omniprediction for the class of all Lipschitz proper losses\. We then extend the result to the class of all mixable proper losses in[Section3\.2](https://arxiv.org/html/2610.41023#S3.SS2)and finally combine it with other finite\-class guarantees in[Section3\.3](https://arxiv.org/html/2610.41023#S3.SS3)\.

### 3\.1Warm\-up: all Lipschitz proper losses

As discussed in[Section1\.2\.1](https://arxiv.org/html/2610.41023#S1.SS2.SSS1), the V\-shaped representation in[Eq\. \(15\)](https://arxiv.org/html/2610.41023#S2.E15)suggests an expert for each predictor–threshold pair\(f,v\)∈ℱ×\[0,1\]\(f,v\)\\in\{\\mathcal\{F\}\}\\times\[0,1\], with rewardzt,f,v=𝔼P∼Dt​\[bv​\(P,f⁡\(xt\)\)​\(yt−P\)\]z\_\{t,f,v\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\]under the round\-ttforecast distributionDtD\_\{t\}\. We thus run \(a variant of\) continuous exponential weights over these infinitely many experts\. Specifically, the algorithm deploys a uniform priorπ0=Unif⁡\(ℱ\)⊗μ0\\pi\_\{0\}=\\operatorname\{Unif\}\(\{\\mathcal\{F\}\}\)\\otimes\\mu\_\{0\}over the experts whereμ0\\mu\_\{0\}is the uniform probability measure on\[0,1\]\[0,1\], and maintains a nonnegative weight densityWt,f​\(v\)W\_\{t,f\}\(v\)updated via \(for a learning rateη\>0\\eta\>0\)

Wt\+1,f​\(v\)=Wt,f​\(v\)​exp⁡\(η​zt,f,v−η2​zt,f,v2\),W\_\{t\+1,f\}\(v\)=W\_\{t,f\}\(v\)\\exp\\left\(\\eta z\_\{t,f,v\}\-\\eta^\{2\}z\_\{t,f,v\}^\{2\}\\right\),\(16\)starting fromW1,f​\(v\)=1W\_\{1,f\}\(v\)=1for allf∈ℱf\\in\{\\mathcal\{F\}\}andv∈\[0,1\]v\\in\[0,1\]\. Compared to the standard exponential weights, an extra second\-order correction term−η2​zt,f,v2\-\\eta^\{2\}z\_\{t,f,v\}^\{2\}is incorporated into the update, an idea that has been explored in the literature for the discrete version of exponential weights starting from[Hazan and Kale \(2010\)](https://arxiv.org/html/2610.41023#bib.bib22)\.

The expected test function at timettis then the following mixture of V\-shaped test functions:

Gt​\(p\)=ηΦt​𝔼\(f,v\)∼π0​\[Wt,f​\(v\)​bv​\(p,f⁡\(xt\)\)\],G\_\{t\}\(p\)=\\frac\{\\eta\}\{\\Phi\_\{t\}\}\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi\_\{0\}\}\\left\[W\_\{t,f\}\(v\)b\_\{v\}\(p,f\(x\_\{t\}\)\)\\right\],\(17\)whereΦt=𝔼\(f,v\)∼π0​\[Wt,f​\(v\)\]\\Phi\_\{t\}=\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi\_\{0\}\}\[W\_\{t,f\}\(v\)\]is the total mass \(the normalization factor\)\. We then solve the following EVI: find a forecast distributionDtD\_\{t\}supported on\[0,1\]\[0,1\]such that𝔼P∼Dt​\[Gt​\(P\)​\(y−P\)\]≤ϵt\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[G\_\{t\}\(P\)\(y\-P\)\]\\leq\\epsilon\_\{t\}for anyy∈\{0,1\}y\\in\\\{0,1\\\}and precisionϵt=1/T\\epsilon\_\{t\}=1/T\. The complete pseudocode is shown in[Algorithm1](https://arxiv.org/html/2610.41023#alg1)\.

Algorithm 1An online omnipredictor for all Lipschitz proper lossesInput: a horizonTT, a finite prediction classℱ\{\\mathcal\{F\}\}, a learning rate0<η≤1/20<\\eta\\leq 1/2, and an EVI solver\.

Initialization:priorπ0=Unif⁡\(ℱ\)⊗μ0\\pi\_\{0\}=\\operatorname\{Unif\}\(\{\\mathcal\{F\}\}\)\\otimes\\mu\_\{0\}, whereμ0\\mu\_\{0\}is uniform on\[0,1\]\[0,1\];ϵt=1/T\\epsilon\_\{t\}=1/T, andW1,f​\(v\)=1W\_\{1,f\}\(v\)=1\.

for*t=1,…,Tt=1,\\ldots,T*do

Nature reveals

xtx\_\{t\}\. Compute total mass

Φt=𝔼\(f,v\)∼π0​\[Wt,f​\(v\)\]\\Phi\_\{t\}=\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi\_\{0\}\}\[W\_\{t,f\}\(v\)\]and the expected test function

Gt​\(p\)=ηΦt​𝔼\(f,v\)∼π0​\[Wt,f​\(v\)​bv​\(p,f⁡\(xt\)\)\]\.G\_\{t\}\(p\)=\\frac\{\\eta\}\{\\Phi\_\{t\}\}\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi\_\{0\}\}\\left\[W\_\{t,f\}\(v\)b\_\{v\}\(p,f\(x\_\{t\}\)\)\\right\]\.Use the EVI solver to find a distribution

DtD\_\{t\}over

\[0,1\]\[0,1\], satisfying

𝔼P∼Dt​\[Gt​\(P\)​\(y−P\)\]≤ϵt,for all​y∈\{0,1\}\.\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[G\_\{t\}\(P\)\(y\-P\)\]\\leq\\epsilon\_\{t\},\\qquad\\text\{for all \}y\\in\\\{0,1\\\}\.Nature reveals outcome

yty\_\{t\}\. For every expert

\(f,v\)∈ℱ×\[0,1\]\(f,v\)\\in\{\\mathcal\{F\}\}\\times\[0,1\], define its reward as

zt,f,v=𝔼P∼Dt​\[bv​\(P,f⁡\(xt\)\)​\(yt−P\)\]z\_\{t,f,v\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\]\. Update the weight via an exponential update with a second\-order correction term:

Wt\+1,f​\(v\)=Wt,f​\(v\)​exp⁡\(η​zt,f,v−η2​zt,f,v2\)\.W\_\{t\+1,f\}\(v\)=W\_\{t,f\}\(v\)\\exp\\left\(\\eta z\_\{t,f,v\}\-\\eta^\{2\}z\_\{t,f,v\}^\{2\}\\right\)\.

##### Regret analysis\.

Consider anLL\-Lipschitz proper lossℓ\\ell\. For any benchmarkf∈ℱf\\in\{\\mathcal\{F\}\},[Eq\. \(15\)](https://arxiv.org/html/2610.41023#S2.E15)gives the regret decomposition

∑t=1T𝔼P∼Dt​\[ℓ⁡\(P,yt\)−ℓ⁡\(f⁡\(xt\),yt\)\]=∑t=1T∫01wℓ​\(v\)​zt,f,v​𝑑v−∑t=1T𝔼P∼Dt​∫01wℓ​\(v\)​dv​\(P,f⁡\(xt\)\)​𝑑v\.\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[\\ell\(P,y\_\{t\}\)\-\\ell\(f\(x\_\{t\}\),y\_\{t\}\)\]=\\sum\_\{t=1\}^\{T\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)z\_\{t,f,v\}\\,\\mathrm\{d\}v\-\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)d\_\{v\}\(P,f\(x\_\{t\}\)\)\\,\\mathrm\{d\}v\.\(18\)The first term is the reward of the target mixture, while the second is the negative properness gain\. The following lemma bounds the first term using the regret of continuous exponential weights and the EVI guarantee\.

###### Lemma 3\.1\.

Ifℓ\\ellis proper andLL\-Lipschitz, then for anyf∈ℱf\\in\{\\mathcal\{F\}\},[Algorithm1](https://arxiv.org/html/2610.41023#alg1)satisfies

∑t=1T∫01wℓ​\(v\)​zt,f,v​dv≤𝒪⁡\(L​log⁡\(\|ℱ\|\)η\)\+η​∑t=1T𝔼P∼Dt​∫01wℓ​\(v\)​bv​\(P,f⁡\(xt\)\)2​\(yt−P\)2​dv\.\\displaystyle\\sum\_\{t=1\}^\{T\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)z\_\{t,f,v\}\\,\\mathrm\{d\}v\\leq\{\}\\mathcal\{O\}\\left\(\\frac\{L\\log\(\|\{\\mathcal\{F\}\}\|\)\}\{\\eta\}\\right\)\+\\eta\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\(y\_\{t\}\-P\)^\{2\}\\,\\mathrm\{d\}v\.

###### Proof\.

For any fixed\(f,v\)\(f,v\), the weight ratio can be bounded as

Wt\+1,f​\(v\)Wt,f​\(v\)\\displaystyle\\frac\{W\_\{t\+1,f\}\(v\)\}\{W\_\{t,f\}\(v\)\}=exp⁡\(η​zt,f,v−η2​zt,f,v2\)≤1\+η​zt,f,v,\\displaystyle=\\exp\\left\(\\eta z\_\{t,f,v\}\-\\eta^\{2\}z\_\{t,f,v\}^\{2\}\\right\)\\leq 1\+\\eta z\_\{t,f,v\},where the inequality usesex−x2≤1\+xe^\{x\-x^\{2\}\}\\leq 1\+xfor\|x\|≤1/2\|x\|\\leq 1/2, since\|bv\|≤1\|b\_\{v\}\|\\leq 1andη≤1/2\\eta\\leq 1/2\. Averaging overπ0\\pi\_\{0\}and using the definition ofGtG\_\{t\}gives

Φt\+1\\displaystyle\\Phi\_\{t\+1\}≤𝔼\(f,v\)∼π0​\[Wt,f​\(v\)​\(1\+η​zt,f,v\)\]\\displaystyle\\leq\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi\_\{0\}\}\\left\[W\_\{t,f\}\(v\)\(1\+\\eta z\_\{t,f,v\}\)\\right\]=Φt​\(1\+𝔼P∼Dt​\[Gt​\(P\)​\(yt−P\)\]\)\\displaystyle=\\Phi\_\{t\}\\left\(1\+\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[G\_\{t\}\(P\)\(y\_\{t\}\-P\)\\right\]\\right\)≤\(1\+ϵt\)​Φt,\\displaystyle\\leq\(1\+\\epsilon\_\{t\}\)\\Phi\_\{t\},where the last inequality follows from the EVI condition in[Algorithm1](https://arxiv.org/html/2610.41023#alg1)\. Thus,Φ1=1\\Phi\_\{1\}=1andϵt=1/T\\epsilon\_\{t\}=1/TimplyΦT\+1≤\(1\+1T\)T≤e\\Phi\_\{T\+1\}\\leq\\left\(1\+\\frac\{1\}\{T\}\\right\)^\{T\}\\leq e\. Therefore, for any probability distributionQ≪π0Q\\ll\\pi\_\{0\}, Jensen’s inequality gives

𝔼Q\[logWT\+1\]−KL\(Q∥π0\)=𝔼Q\[log\(WT\+1d​Qdπ0\)\]≤log𝔼π0\[WT\+1\]=logΦT\+1≤1\.\\mathbb\{E\}\_\{Q\}\\left\[\\log W\_\{T\+1\}\\right\]\-\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{0\}\)=\\mathbb\{E\}\_\{Q\}\\left\[\\log\\left\(\\frac\{W\_\{T\+1\}\}\{\\,\\mathrm\{d\}Q\}\\,\\mathrm\{d\}\\pi\_\{0\}\\right\)\\right\]\\leq\\log\\mathbb\{E\}\_\{\\pi\_\{0\}\}\\left\[W\_\{T\+1\}\\right\]=\\log\\Phi\_\{T\+1\}\\leq 1\.\(19\)Plugging the definition ofWT\+1W\_\{T\+1\}, rearranging, and usingzt,f,v2≤𝔼P∼Dt​\[bv​\(P,f⁡\(xt\)\)2​\(yt−P\)2\]z\_\{t,f,v\}^\{2\}\\leq\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\(y\_\{t\}\-P\)^\{2\}\\right\]\(by Jensen’s inequality\) gives

∑t𝔼\(f,v\)∼Q​\[zt,f,v\]≤\(KL\(Q∥π0\)\+1\)η\+η​∑t𝔼P∼Dt​𝔼\(f,v\)∼Q​\[bv​\(P,f⁡\(xt\)\)2​\(yt−P\)2\]\.\\sum\_\{t\}\\mathbb\{E\}\_\{\(f,v\)\\sim Q\}\\left\[z\_\{t,f,v\}\\right\]\\leq\\frac\{\(\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{0\}\)\+1\)\}\{\\eta\}\+\\eta\\sum\_\{t\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\mathbb\{E\}\_\{\(f,v\)\\sim Q\}\\left\[b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\(y\_\{t\}\-P\)^\{2\}\\right\]\.Since our objective is to bound∑t=1T∫01wℓ​\(v\)​zt,f,v​𝑑v\\sum\_\{t=1\}^\{T\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)z\_\{t,f,v\}\\,\\mathrm\{d\}v, we naturally pick the target distribution asQ=δf⊗νℓQ=\\delta\_\{f\}\\otimes\\nu\_\{\\ell\}, whereνℓ​\(d​v\)=1m​wℓ​\(v\)​d​v\\nu\_\{\\ell\}\(\\,\\mathrm\{d\}v\)=\\frac\{1\}\{m\}w\_\{\\ell\}\(v\)\\,\\mathrm\{d\}vandm=∫01wℓ​\(v\)​𝑑vm=\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\,\\mathrm\{d\}vis the normalization factor, leading to

∑t∫01wℓ​\(v\)​zt,f,v​𝑑v≤m\(KL\(Q∥π0\)\+1\)η\+η​∑t𝔼P∼Dt​∫01wℓ​\(v\)​bv​\(P,f⁡\(xt\)\)2​\(yt−P\)2​𝑑v\.\\sum\_\{t\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)z\_\{t,f,v\}\\,\\mathrm\{d\}v\\leq\\frac\{m\(\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{0\}\)\+1\)\}\{\\eta\}\+\\eta\\sum\_\{t\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\(y\_\{t\}\-P\)^\{2\}\\,\\mathrm\{d\}v\.It thus remains to bound the KL term\. The uniform prior onℱ\{\\mathcal\{F\}\}and the curvature boundwℓ≤2​Lw\_\{\\ell\}\\leq 2Lfrom[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)yield

KL\(Q∥π0\)\\displaystyle\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{0\}\)=log\|ℱ\|\+KL\(νℓ∥μ0\)=log\|ℱ\|\+∫01wℓ​\(v\)mlogwℓ​\(v\)mdv≤log\|ℱ\|\+log2​Lm\.\\displaystyle=\\log\|\{\\mathcal\{F\}\}\|\+\{\\text\{\\rm KL\}\}\(\\nu\_\{\\ell\}\\\|\\mu\_\{0\}\)=\\log\|\{\\mathcal\{F\}\}\|\+\\int\_\{0\}^\{1\}\\frac\{w\_\{\\ell\}\(v\)\}\{m\}\\log\\frac\{w\_\{\\ell\}\(v\)\}\{m\}\\,\\mathrm\{d\}v\\leq\\log\|\{\\mathcal\{F\}\}\|\+\\log\\frac\{2L\}\{m\}\.Consequently, we havem\(KL\(Q∥π0\)\+1\)=𝒪\(Llog\(\|ℱ\|\)\)m\(\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{0\}\)\+1\)=\\mathcal\{O\}\(L\\log\(\|\{\\mathcal\{F\}\}\|\)\), usingm≤2​Lm\\leq 2Lands​log⁡\(1/s\)≤1/es\\log\(1/s\)\\leq 1/efors=m/\(2​L\)s=m/\(2L\)\. This proves the lemma\. ∎

The next lemma then shows that the second\-order term from the bound of[Lemma3\.1](https://arxiv.org/html/2610.41023#S3.Thmtheorem1)that is of orderη\\etaper round can be combined with the negative properness gain from[Eq\. \(18\)](https://arxiv.org/html/2610.41023#S3.E18)to lead to a smaller term of orderη2\\eta^\{2\}, a key to derive the final orderT1/3T^\{1/3\}regret bound\.

###### Lemma 3\.2\.

For anyLL\-Lipschitz proper lossℓ\\ell,0<η≤1/20<\\eta\\leq 1/2, andp,q∈\[0,1\]p,q\\in\[0,1\], we have

∫01wℓ​\(v\)​\(η​bv​\(p,q\)2−dv​\(p,q\)\)​𝑑v≤L​η2\.\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\left\(\\eta b\_\{v\}\(p,q\)^\{2\}\-d\_\{v\}\(p,q\)\\right\)\\,\\mathrm\{d\}v\\leq L\\eta^\{2\}\.

###### Proof\.

Using the definitiondv​\(p,q\)=\|v−p\|​bv​\(p,q\)2d\_\{v\}\(p,q\)=\|v\-p\|b\_\{v\}\(p,q\)^\{2\}and the conditionwℓ​\(v\)∈\[0,2​L\]w\_\{\\ell\}\(v\)\\in\[0,2L\], we have

∫01wℓ​\(v\)​\(η​bv​\(p,q\)2−dv​\(p,q\)\)​𝑑v\\displaystyle\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\left\(\\eta b\_\{v\}\(p,q\)^\{2\}\-d\_\{v\}\(p,q\)\\right\)\\,\\mathrm\{d\}v≤∫01wℓ​\(v\)​bv​\(p,q\)2​\[η−\|v−p\|\]\+​𝑑v\\displaystyle\\leq\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)b\_\{v\}\(p,q\)^\{2\}\[\\eta\-\|v\-p\|\]\_\{\+\}\\,\\mathrm\{d\}v≤2​L​∫01bv​\(p,q\)2​\[η−\|v−p\|\]\+​𝑑v\\displaystyle\\leq 2L\\int\_\{0\}^\{1\}b\_\{v\}\(p,q\)^\{2\}\[\\eta\-\|v\-p\|\]\_\{\+\}\\,\\mathrm\{d\}v≤2​L​∫0η\(η−s\)​𝑑s=L​η2,\\displaystyle\\leq 2L\\int\_\{0\}^\{\\eta\}\(\\eta\-s\)\\,\\mathrm\{d\}s=L\\eta^\{2\},\(20\)where the last inequality is becausebv​\(p,q\)2≤1b\_\{v\}\(p,q\)^\{2\}\\leq 1and the integrandbv​\(p,q\)2​\[η−\|v−p\|\]\+b\_\{v\}\(p,q\)^\{2\}\[\\eta\-\|v\-p\|\]\_\{\+\}is non\-zero only whenvvis at mostη\\etafromppand also in betweenppandqq\(otherwisebv​\(p,q\)=0b\_\{v\}\(p,q\)=0\)\. ∎

Combining these two lemmas and picking the optimalη\\etaimmediately leads to the following final regret guarantee for[Algorithm1](https://arxiv.org/html/2610.41023#alg1)\. We emphasize that the value ofη\\etaisindependent ofLLand the regret bound holds for all Lipschitz proper losses simultaneously\.

###### Theorem 3\.3\.

Withη=min⁡\{12,\(log⁡\|ℱ\|T\)1/3\}\\eta=\\min\\left\\\{\\frac\{1\}\{2\},\\left\(\\frac\{\\log\|\{\\mathcal\{F\}\}\|\}\{T\}\\right\)^\{1/3\}\\right\\\},[Algorithm1](https://arxiv.org/html/2610.41023#alg1)ensures that for any proper andLL\-Lipschitz lossℓ\\ell,

Regℓ,ℱ​\(T\)=𝒪⁡\(L⁡\(T1/3​\(log⁡\|ℱ\|\)2/3\+log⁡\|ℱ\|\)\)\.\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\mathcal\{O\}\\left\(L\\left\(T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\+\\log\|\{\\mathcal\{F\}\}\|\\right\)\\right\)\.

### 3\.2All mixable proper losses

We now extend the idea in[Section3\.1](https://arxiv.org/html/2610.41023#S3.SS1)to handle all mixable proper losses\. The construction still uses exponential weights over predictor–threshold pairs and an EVI to select forecasts\. However, the curvature of a mixable loss can grow near the endpoints, which creates two problems\.

The first problem is the choice of prior\. For everyβ\>0\\beta\>0andℓ∈ℒmix​\(β\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(\\beta\\right\), the curvature boundwℓ​\(v\)≤1/\(β​v​\(1−v\)\)w\_\{\\ell\}\(v\)\\leq 1/\\left\(\\beta v\(1\-v\)\\right\)from[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)suggests weighting thresholds by1/\(v⁡\(1−v\)\)1/\\left\(v\(1\-v\)\\right\)\. This density has infinite mass on\[0,1\]\[0,1\], so it cannot be normalized into a prior\. To address this problem, we restrict thresholds and forecasts toIδ=\[δ,1−δ\]I\_\{\\delta\}=\[\\delta,1\-\\delta\], with a smallδ=1/\(2​\(T\+1\)\)\\delta=1/\\left\(2\(T\+1\)\\right\)that has a negligible effect on the regret\. The density is then integrable, with normalizing constantMδ=∫Iδ\[v⁡\(1−v\)\]−1​𝑑v=2​log⁡\(2​T\+1\)M\_\{\\delta\}=\\int\_\{I\_\{\\delta\}\}\[v\(1\-v\)\]^\{\-1\}\\,\\mathrm\{d\}v=2\\log\(2T\+1\)\. This gives the threshold and expert priors

μδ​\(d​v\)=d​vMδ​v​\(1−v\),πδ=Unif⁡\(ℱ\)⊗μδ\.\\mu\_\{\\delta\}\(\\,\\mathrm\{d\}v\)=\\frac\{\\,\\mathrm\{d\}v\}\{M\_\{\\delta\}v\(1\-v\)\},\\qquad\\pi\_\{\\delta\}=\\operatorname\{Unif\}\(\{\\mathcal\{F\}\}\)\\otimes\\mu\_\{\\delta\}\.\(21\)
With the prior in place, the second problem is how to control the second\-order term in the regret\. As discussed in[Section1\.2](https://arxiv.org/html/2610.41023#S1.SS2), the curvature boundwℓ​\(v\)≤1/\(β​v​\(1−v\)\)w\_\{\\ell\}\(v\)\\leq 1/\\left\(\\beta v\(1\-v\)\\right\)that grows near the boundary is again the key issue, and addressing it requires introducing an extra factor in the second\-order term of the regret to offset the curvature\. Here, having shown the proof for how to cancel the second\-order term with the properness gain in the Lipschitz case \([Lemma3\.2](https://arxiv.org/html/2610.41023#S3.Thmtheorem2)\), we provide a more detailed explanation on this issue and its solution\. Specifically, in the proof for[Lemma3\.2](https://arxiv.org/html/2610.41023#S3.Thmtheorem2)\(in particular,[Eq\. \(20\)](https://arxiv.org/html/2610.41023#S3.E20)\), we see that for a fixedpp, the cost comes from all thresholdsvvwith\|v−p\|<η\|v\-p\|<\\eta\. For mixable losses, however, the same neighborhood can carry much more curvature whenppand thusvvare close to the boundary\. If we could somehow restrict the region to\|v−p\|<η​p​\(1−p\)\|v\-p\|<\\eta p\(1\-p\)instead, then it shrinks asppapproaches the boundary as well, exactly offsetting the growth in curvature\.

With a closer examination of the second\-order regret term from[Lemma3\.1](https://arxiv.org/html/2610.41023#S3.Thmtheorem1), we see that in order to obtain the extrap⁡\(1−p\)p\(1\-p\)factor mentioned above, we need to replace the realized square\(yt−p\)2\(y\_\{t\}\-p\)^\{2\}in the second\-order term by the Bernoulli variancep⁡\(1−p\)p\(1\-p\)\(as ifyty\_\{t\}was actually drawn frompp\)\. To do so, we accordingly change the update rule of the weight function from[Eq\. \(16\)](https://arxiv.org/html/2610.41023#S3.E16)to the following \(note the change fromzt,f,v2=𝔼P∼Dt​\[bv​\(P,f⁡\(xt\)\)​\(yt−P\)\]2z\_\{t,f,v\}^\{2\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\]^\{2\}to𝔼P∼Dt​\[P⁡\(1−P\)​bv​\(P,f⁡\(xt\)\)2\]\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\]\):

Wt\+1,f​\(v\)=Wt,f​\(v\)​exp⁡\(η​zt,f,v−η2​𝔼P∼Dt​\[P⁡\(1−P\)​bv​\(P,f⁡\(xt\)\)2\]\)\.W\_\{t\+1,f\}\(v\)=W\_\{t,f\}\(v\)\\exp\\left\(\\eta z\_\{t,f,v\}\-\\eta^\{2\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\]\\right\)\.\(22\)For the regret analysis to still go through in the same way as[Lemma3\.1](https://arxiv.org/html/2610.41023#S3.Thmtheorem1), all we need to ensure is that the potential increase constraintΦt\+1≤\(1\+ϵt\)​Φt\\Phi\_\{t\+1\}\\leq\(1\+\\epsilon\_\{t\}\)\\Phi\_\{t\}still holds \(whereΦt\\Phi\_\{t\}is as before the total mass𝔼\(f,v\)∼πδ​\[Wt,f​\(v\)\]\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi\_\{\\delta\}\}\[W\_\{t,f\}\(v\)\]\)\. To this end, we follow the previous analysis and first bound the ratioWt\+1,f​\(v\)Wt,f​\(v\)\\frac\{W\_\{t\+1,f\}\(v\)\}\{W\_\{t,f\}\(v\)\}as \(using Jensen’s inequality\):

Wt\+1,f​\(v\)Wt,f​\(v\)\\displaystyle\\frac\{W\_\{t\+1,f\}\(v\)\}\{W\_\{t,f\}\(v\)\}=exp⁡\(η​𝔼P∼Dt​\[bv​\(P,f⁡\(xt\)\)​\(yt−P\)\]−η2​𝔼P∼Dt​\[P⁡\(1−P\)​bv​\(P,f⁡\(xt\)\)2\]\)\\displaystyle=\\exp\\left\(\\eta\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\]\-\\eta^\{2\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\]\\right\)≤𝔼P∼Dt​\[exp⁡\(η​bv​\(P,f⁡\(xt\)\)​\(yt−P\)−η2​P​\(1−P\)​bv​\(P,f⁡\(xt\)\)2\)\]\.\\displaystyle\\leq\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\exp\\left\(\\eta b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\-\\eta^\{2\}P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\\right\)\\right\]\.\(23\)Here, with the new second\-order term, we can no longer apply the inequalityex−x2≤1\+xe^\{x\-x^\{2\}\}\\leq 1\+xand bound the exponential by1\+η​bv​\(P,f⁡\(xt\)\)​\(yt−P\)1\+\\eta b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\. Fortunately, the following lemma shows that there exists a differentb~vη​\(p,q\)\\widetilde\{b\}^\{\\eta\}\_\{v\}\(p,q\)function such that1\+b~vη​\(P,f⁡\(xt\)\)​\(yt−P\)1\+\\widetilde\{b\}^\{\\eta\}\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)upper bounds the exponential\.

###### Lemma 3\.4\.

For anyp∈\[0,1\]p\\in\[0,1\],y∈\{0,1\}y\\in\\\{0,1\\\},h∈\[−1,1\]h\\in\[\-1,1\], and0<η≤1/20<\\eta\\leq 1/2, defineFp,yη​\(h\)=exp⁡\(η​h​\(y−p\)−η2​p​\(1−p\)​h2\)F^\{\\eta\}\_\{p,y\}\(h\)=\\exp\\left\(\\eta h\(y\-p\)\-\\eta^\{2\}p\(1\-p\)h^\{2\}\\right\)andSpη​\(h\)=Fp,1η​\(h\)−Fp,0η​\(h\)S\_\{p\}^\{\\eta\}\(h\)=F^\{\\eta\}\_\{p,1\}\(h\)\-F^\{\\eta\}\_\{p,0\}\(h\)\. We haveFp,yη​\(h\)≤1\+Spη​\(h\)​\(y−p\)\.F^\{\\eta\}\_\{p,y\}\(h\)\\leq 1\+S\_\{p\}^\{\\eta\}\(h\)\(y\-p\)\.

###### Proof\.

SinceFp,yη​\(h\)=𝔼Y∼Ber​\(p\)​\[Fp,Yη​\(h\)\]\+Spη​\(h\)​\(y−p\)F^\{\\eta\}\_\{p,y\}\(h\)=\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[F^\{\\eta\}\_\{p,Y\}\(h\)\\right\]\+S\_\{p\}^\{\\eta\}\(h\)\(y\-p\), it suffices to show𝔼Y∼Ber​\(p\)​\[Fp,Yη​\(h\)\]≤1\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[F^\{\\eta\}\_\{p,Y\}\(h\)\\right\]\\leq 1\. Indeed, this holds by using the factex≤1\+x\+x2e^\{x\}\\leq 1\+x\+x^\{2\}for\|x\|≤1/2\|x\|\\leq 1/2, withX=η​h​\(Y−p\)X=\\eta h\(Y\-p\):

𝔼Y∼Ber​\(p\)​\[exp⁡\(η​h​\(Y−p\)\)\]\\displaystyle\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[\\exp\\left\(\\eta h\(Y\-p\)\\right\)\\right\]=𝔼Y∼Ber​\(p\)​\[eX\]≤1\+𝔼Y∼Ber​\(p\)​\[X\]\+𝔼Y∼Ber​\(p\)​\[X2\]\\displaystyle=\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[e^\{X\}\\right\]\\leq 1\+\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[X\\right\]\+\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[X^\{2\}\\right\]=1\+η2​p​\(1−p\)​h2≤exp⁡\(η2​p​\(1−p\)​h2\)\.∎\\displaystyle=1\+\\eta^\{2\}p\(1\-p\)h^\{2\}\\leq\\exp\\left\(\\eta^\{2\}p\(1\-p\)h^\{2\}\\right\)\.\\qed

Therefore, if we define

b~vη​\(p,q\)=Spη​\(bv​\(p,q\)\)=Fp,1η​\(bv​\(p,q\)\)−Fp,0η​\(bv​\(p,q\)\),\\widetilde\{b\}^\{\\eta\}\_\{v\}\(p,q\)=S\_\{p\}^\{\\eta\}\(b\_\{v\}\(p,q\)\)=F^\{\\eta\}\_\{p,1\}\(b\_\{v\}\(p,q\)\)\-F^\{\\eta\}\_\{p,0\}\(b\_\{v\}\(p,q\)\),\(24\)then continuing from[Eq\. \(23\)](https://arxiv.org/html/2610.41023#S3.E23)and using[Lemma3\.4](https://arxiv.org/html/2610.41023#S3.Thmtheorem4), we obtainWt\+1,f​\(v\)Wt,f​\(v\)≤𝔼P∼Dt​\[FP,ytη​\(bv​\(P,f⁡\(xt\)\)\)\]≤1\+𝔼P∼Dt​\[b~vη​\(P,f⁡\(xt\)\)​\(yt−P\)\]\\frac\{W\_\{t\+1,f\}\(v\)\}\{W\_\{t,f\}\(v\)\}\\leq\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[F\_\{P,y\_\{t\}\}^\{\\eta\}\(b\_\{v\}\(P,f\(x\_\{t\}\)\)\)\\right\]\\leq 1\+\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\widetilde\{b\}^\{\\eta\}\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\\right\], replacingη​bv​\(P,f⁡\(xt\)\)\\eta b\_\{v\}\(P,f\(x\_\{t\}\)\)in the derivation of the proof of[Lemma3\.1](https://arxiv.org/html/2610.41023#S3.Thmtheorem1)withb~vη​\(P,f⁡\(xt\)\)\\widetilde\{b\}^\{\\eta\}\_\{v\}\(P,f\(x\_\{t\}\)\)\. This means that if we accordingly define the expected test function as

Gt​\(p\)=1Φt​𝔼\(f,v\)∼πδ​\[Wt,f​\(v\)​b~vη​\(p,f⁡\(xt\)\)\],G\_\{t\}\(p\)=\\frac\{1\}\{\\Phi\_\{t\}\}\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi\_\{\\delta\}\}\\left\[W\_\{t,f\}\(v\)\\widetilde\{b\}^\{\\eta\}\_\{v\}\(p,f\(x\_\{t\}\)\)\\right\],\(25\)then the whole analysis goes through in the same way\. This explains the design of our algorithm, shown in[Algorithm2](https://arxiv.org/html/2610.41023#alg2)\.

Algorithm 2An online omnipredictor for all mixable proper lossesInput: a horizonTT, a finite prediction classℱ\{\\mathcal\{F\}\}, a learning rate0<η≤1/20<\\eta\\leq 1/2, and an EVI solver\.

Initialization:δ=1/\(2​\(T\+1\)\)\\delta=1/\\left\(2\(T\+1\)\\right\),Iδ=\[δ,1−δ\]I\_\{\\delta\}=\[\\delta,1\-\\delta\],Mδ=2​log⁡\(2​T\+1\)M\_\{\\delta\}=2\\log\(2T\+1\), priorπδ=Unif⁡\(ℱ\)⊗μδ\\pi\_\{\\delta\}=\\operatorname\{Unif\}\(\{\\mathcal\{F\}\}\)\\otimes\\mu\_\{\\delta\}whereμδ​\(d​v\)=d​vMδ​v​\(1−v\)\\mu\_\{\\delta\}\(\\,\\mathrm\{d\}v\)=\\frac\{\\,\\mathrm\{d\}v\}\{M\_\{\\delta\}v\(1\-v\)\},ϵt=1/T\\epsilon\_\{t\}=1/T, andW1,f​\(v\)=1W\_\{1,f\}\(v\)=1for allf∈ℱf\\in\{\\mathcal\{F\}\}andv∈Iδv\\in I\_\{\\delta\}\.

for*t=1,…,Tt=1,\\ldots,T*do

Nature reveals

xtx\_\{t\}\. Compute total mass

Φt=𝔼\(f,v\)∼πδ​\[Wt,f​\(v\)\]\\Phi\_\{t\}=\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi\_\{\\delta\}\}\[W\_\{t,f\}\(v\)\]and the expected test function

GtG\_\{t\}as in[Eq\. \(25\)](https://arxiv.org/html/2610.41023#S3.E25)\. Use the EVI solver to find a distribution

DtD\_\{t\}over

IδI\_\{\\delta\}, satisfying

𝔼P∼Dt​\[Gt​\(P\)​\(y−P\)\]≤ϵt,for all​y∈\{0,1\}\.\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[G\_\{t\}\(P\)\(y\-P\)\]\\leq\\epsilon\_\{t\},\\qquad\\text\{for all \}y\\in\\\{0,1\\\}\.Nature reveals outcome

yty\_\{t\}\. For every expert

\(f,v\)∈ℱ×Iδ\(f,v\)\\in\{\\mathcal\{F\}\}\\times I\_\{\\delta\}, define its reward as

zt,f,v=𝔼P∼Dt​\[bv​\(P,f⁡\(xt\)\)​\(yt−P\)\]z\_\{t,f,v\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\]\. Update the weight via an exponential update with a \(new\) second\-order correction term:

Wt\+1,f​\(v\)=Wt,f​\(v\)​exp⁡\(η​zt,f,v−η2​𝔼P∼Dt​\[P⁡\(1−P\)​bv​\(P,f⁡\(xt\)\)2\]\)\.W\_\{t\+1,f\}\(v\)=W\_\{t,f\}\(v\)\\exp\\left\(\\eta z\_\{t,f,v\}\-\\eta^\{2\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\]\\right\)\.

##### Regret analysis\.

The discussion above directly implies the following second\-order regret bound which is analogous to[Lemma3\.1](https://arxiv.org/html/2610.41023#S3.Thmtheorem1)but, importantly, contains theP⁡\(1−P\)P\(1\-P\)factor \(proofs for all lemmas can be found in[AppendixB](https://arxiv.org/html/2610.41023#A2)\)\.

###### Lemma 3\.5\.

Ifℓ\\ellis proper andβ\\beta\-mixable, then for anyf∈ℱf\\in\{\\mathcal\{F\}\},[Algorithm2](https://arxiv.org/html/2610.41023#alg2)satisfies

∑t=1T∫Iδwℓ​\(v\)​zt,f,v​𝑑v≤\\displaystyle\\sum\_\{t=1\}^\{T\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)z\_\{t,f,v\}\\,\\mathrm\{d\}v\\leq\{\}𝒪⁡\(log⁡\(T\)​log⁡\(\|ℱ\|\)β​η\)\+η​∑t=1T𝔼P∼Dt​∫Iδwℓ​\(v\)​P​\(1−P\)​bv​\(P,f⁡\(xt\)\)2​dv\.\\displaystyle\\mathcal\{O\}\\left\(\\frac\{\\log\(T\)\\log\(\|\{\\mathcal\{F\}\}\|\)\}\{\\beta\\eta\}\\right\)\+\\eta\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\\,\\mathrm\{d\}v\.

Using the extraP⁡\(1−P\)P\(1\-P\)factor, we then prove the following lemma that is analogous to[Lemma3\.2](https://arxiv.org/html/2610.41023#S3.Thmtheorem2)and shows how the second\-order term \(of orderη\\eta\) and the negative properness gain together lead to a term of orderη2\\eta^\{2\}, even when the curvaturewℓ​\(v\)w\_\{\\ell\}\(v\)scales as1v⁡\(1−v\)\\frac\{1\}\{v\(1\-v\)\}\.

###### Lemma 3\.6\.

For anyβ\\beta\-mixable proper lossℓ\\ell,η∈\(0,1/2\]\\eta\\in\(0,1/2\],p∈Iδp\\in I\_\{\\delta\}, andq∈\[0,1\]q\\in\[0,1\], we have

∫Iδwℓ​\(v\)​\[η​p​\(1−p\)​bv​\(p,q\)2−dv​\(p,q\)\]​𝑑v≤η2β\.\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\left\[\\eta p\(1\-p\)b\_\{v\}\(p,q\)^\{2\}\-d\_\{v\}\(p,q\)\\right\]\\,\\mathrm\{d\}v\\leq\\frac\{\\eta^\{2\}\}\{\\beta\}\.

On the other hand, recall that we only consider thresholds and make predictions inIδI\_\{\\delta\}, whereas regret compares against the original benchmark reports in\[0,1\]\[0,1\]\. To address this, we only need to show that clipping the benchmark report to the intervalIδI\_\{\\delta\}only introduces a small approximation error of order𝒪⁡\(δ\)\\mathcal\{O\}\(\\delta\)\.

###### Lemma 3\.7\.

For anyβ\\beta\-mixable proper lossℓ\\ell,q∈\[0,1\]q\\in\[0,1\], andy∈\{0,1\}y\\in\\\{0,1\\\},

ℓ⁡\(clip⁡\(q,Iδ\),y\)−ℓ⁡\(q,y\)≤−log⁡\(1−δ\)β≤2​δβ,\\ell\(\\operatorname\{clip\}\(q,I\_\{\\delta\}\),y\)\-\\ell\(q,y\)\\leq\-\\frac\{\\log\(1\-\\delta\)\}\{\\beta\}\\leq\\frac\{2\\delta\}\{\\beta\},where we recallclip⁡\(q,Iδ\)=min⁡\{1−δ,max⁡\{δ,q\}\}\\operatorname\{clip\}\(q,I\_\{\\delta\}\)=\\min\\\{1\-\\delta,\\max\\\{\\delta,q\\\}\\\}\. Consequently, for everyp∈Iδp\\in I\_\{\\delta\},

ℓ⁡\(p,y\)−ℓ⁡\(q,y\)≤∫Iδwℓ​\(v\)​\[bv​\(p,q\)​\(y−p\)−dv​\(p,q\)\]​𝑑v\+2​δβ\.\\ell\(p,y\)\-\\ell\(q,y\)\\leq\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\left\[b\_\{v\}\(p,q\)\(y\-p\)\-d\_\{v\}\(p,q\)\\right\]\\,\\mathrm\{d\}v\+\\frac\{2\\delta\}\{\\beta\}\.\(26\)

Since our algorithm only makes predictions inIδI\_\{\\delta\},[Eq\. \(26\)](https://arxiv.org/html/2610.41023#S3.E26)justifies why it suffices to only consider thresholds inIδI\_\{\\delta\}\. Finally, combining all three lemmas and picking the optimalη\\etaproves our main result[Theorem1\.2](https://arxiv.org/html/2610.41023#S1.Thmtheorem2), that is, for every mixable proper lossℓ\\ell,[Algorithm2](https://arxiv.org/html/2610.41023#alg2)ensures

Regℓ,ℱ​\(T\)=𝒪⁡\(T1/3​\(log⁡\(T\)​log⁡\(\|ℱ\|\)\)2/3\+log⁡T​log⁡\(\|ℱ\|\)βℓ⋆\)\.\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\mathcal\{O\}\\left\(\\frac\{T^\{1/3\}\(\\log\(T\)\\log\(\|\{\\mathcal\{F\}\}\|\)\)^\{2/3\}\+\\log T\\log\(\|\{\\mathcal\{F\}\}\|\)\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\)\.\(27\)The details appear in[AppendixA\.3](https://arxiv.org/html/2610.41023#A1.SS3)\. We recall that this bound is optimal, as shown in[Theorem1\.3](https://arxiv.org/html/2610.41023#S1.Thmtheorem3)and[AppendixA\.2](https://arxiv.org/html/2610.41023#A1.SS2)\.

##### Deterministic forecasts\.

We remark that while we follow the framework of[Farina and Perdomo \(2026\)](https://arxiv.org/html/2610.41023#bib.bib1)and present both[Algorithm1](https://arxiv.org/html/2610.41023#alg1)and[Algorithm2](https://arxiv.org/html/2610.41023#alg2)using a generic EVI solver that finds a randomize forecastDtD\_\{t\}, the EVI that these two algorithms need to solve actually admits a deterministic solutionDt=δptD\_\{t\}=\\delta\_\{p\_\{t\}\}whereptp\_\{t\}is a root of the expected test functionGtG\_\{t\}\(that is,Gt​\(pt\)=0G\_\{t\}\(p\_\{t\}\)=0and hence𝔼P∼δpt​\[Gt​\(P\)​\(y−P\)\]≤ϵt\\mathbb\{E\}\_\{P\\sim\\delta\_\{p\_\{t\}\}\}\[G\_\{t\}\(P\)\(y\-P\)\]\\leq\\epsilon\_\{t\}trivially holds\)\. Such a root exists becauseGtG\_\{t\}is continuous and is nonnegative at the left endpoint and nonpositive at the right endpoint of the forecast interval \(details omitted\)\. However, this is not the case for the best\-of\-all\-worlds algorithm we will present in the next section \(whereGtG\_\{t\}could be discontinuous\), which is also why we choose to present a generic randomized EVI solution for all algorithms\.

### 3\.3A best\-of\-all\-worlds omnipredictor

There are two drawbacks of[Algorithm2](https://arxiv.org/html/2610.41023#alg2): first, it does not provide any regret guarantee beyond the class of mixable proper losses; second, it sacrifices the standard constant inTTregret for specific mixable losses such as squared loss and log loss, in exchange for a slower rate that holds for all mixable losses simultaneously\. In this section, we show that both drawbacks can be addressed via a best\-of\-all\-worlds algorithm, which, as shown in[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4), simultaneously achieves the best regret bounds for a finite classℒbd\\mathcal\{L\}\_\{\\mathrm\{bd\}\}of general\[0,1\]\[0,1\]\-valued losses, a finite classℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}of mixable proper losses, and the infinite class of all mixable proper losses\. Throughout the discussion, we will use a superscriptc∈𝒞=\{bd,mix,MIX\}c\\in\\mathcal\{C\}=\\\{\\mathrm\{bd\},\\mathrm\{mix\},\\mathrm\{MIX\}\\\}to distinguish similar quantities that are associated with these three different classes\.

The key idea is still the same: treat each benchmark\-loss pair\(f,ℓ\)\(f,\\ell\)as an expert \(whereℓ\\ellcan be from the two finite classesℒbd\\mathcal\{L\}\_\{\\mathrm\{bd\}\}andℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}or the V\-shaped lossλv\\lambda\_\{v\}forv∈\[δ,1−δ\]v\\in\[\\delta,1\-\\delta\]\), run exponential weight updates over these experts, use the resulting distribution to compute an expected test functionGtG\_\{t\}, and finally solve the EVI involvingGtG\_\{t\}to obtain the final randomized forecastDtD\_\{t\}\. For ease of presentation, it would be convenient to write the test functionGtG\_\{t\}as a weighted average of three test functions\{Gtc\}c∈𝒞\\\{G\_\{t\}^\{c\}\\\}\_\{c\\in\\mathcal\{C\}\}, each corresponding to one of the three different classes\.

##### Component forℒbd\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\.

This component is very similar to the original algorithm of\([Farina and Perdomo, 2026](https://arxiv.org/html/2610.41023#bib.bib1)\)where, based on the discussion in[Section1\.2\.1](https://arxiv.org/html/2610.41023#S1.SS2.SSS1)and in particular[Eq\. \(6\)](https://arxiv.org/html/2610.41023#S1.E6), we define the reward for each expert\(f,ℓ\)∈ℱ×ℒbd\(f,\\ell\)\\in\{\\mathcal\{F\}\}\\times\\mathcal\{L\}\_\{\\mathrm\{bd\}\}aszt,f,ℓbd=𝔼P∼Dt​\[bℓ​\(P,f⁡\(xt\)\)​\(yt−P\)\]z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[b\_\{\\ell\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\\right\]\. We then maintain a weightWt,f,ℓbdW^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}for each expert\(f,ℓ\)\(f,\\ell\)in roundtt\. For consistency, we update these weights in the same way as our[Algorithm1](https://arxiv.org/html/2610.41023#alg1)using the standard second\-order correction with a learning rateηbd∈\(0,1/4\]\\eta^\{\\mathrm\{bd\}\}\\in\(0,1/4\]:

Wt\+1,f,ℓbd=Wt,f,ℓbd​exp⁡\(ηbd​zt,f,ℓbd−\(ηbd\)2​\(zt,f,ℓbd\)2\)\.W^\{\\mathrm\{bd\}\}\_\{t\+1,f,\\ell\}=W^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\\exp\\left\(\\eta^\{\\mathrm\{bd\}\}z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\-\(\\eta^\{\\mathrm\{bd\}\}\)^\{2\}\(z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\)^\{2\}\\right\)\.The test function for this component also follows[Algorithm1](https://arxiv.org/html/2610.41023#alg1):Gtbd​\(p\)=ηbdΦtbd​𝔼\(f,ℓ\)∼πbd​\[Wt,f,ℓbd​bℓ​\(p,f⁡\(xt\)\)\]G\_\{t\}^\{\\mathrm\{bd\}\}\(p\)=\\frac\{\\eta^\{\\mathrm\{bd\}\}\}\{\\Phi\_\{t\}^\{\\mathrm\{bd\}\}\}\\mathbb\{E\}\_\{\(f,\\ell\)\\sim\\pi^\{\\mathrm\{bd\}\}\}\\left\[W^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}b\_\{\\ell\}\(p,f\(x\_\{t\}\)\)\\right\]whereπbd=Unif⁡\(ℱ\)⊗Unif⁡\(ℒbd\)\\pi^\{\\mathrm\{bd\}\}=\\operatorname\{Unif\}\(\{\\mathcal\{F\}\}\)\\otimes\\operatorname\{Unif\}\(\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\)is the uniform prior andΦtbd=𝔼\(f,ℓ\)∼πbd​\[Wt,f,ℓbd\]\\Phi\_\{t\}^\{\\mathrm\{bd\}\}=\\mathbb\{E\}\_\{\(f,\\ell\)\\sim\\pi^\{\\mathrm\{bd\}\}\}\\left\[W^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\\right\]is the total mass for this component\.

##### Component for all mixable losses\.

This component directly follows[Algorithm2](https://arxiv.org/html/2610.41023#alg2)where we define for each expert\(f,v\)∈ℱ×Iδ\(f,v\)\\in\{\\mathcal\{F\}\}\\times I\_\{\\delta\}, a rewardzt,f,vMIX=𝔼P∼Dt​\[bv​\(P,f⁡\(xt\)\)​\(yt−P\)\]z^\{\\mathrm\{MIX\}\}\_\{t,f,v\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\\right\], a weight function updated via

Wt\+1,fMIX​\(v\)=Wt,fMIX​\(v\)​exp⁡\(ηMIX​zt,f,vMIX−\(ηMIX\)2​𝔼P∼Dt​\[P⁡\(1−P\)​bv​\(P,f⁡\(xt\)\)2\]\)W^\{\\mathrm\{MIX\}\}\_\{t\+1,f\}\(v\)=W^\{\\mathrm\{MIX\}\}\_\{t,f\}\(v\)\\exp\\left\(\\eta^\{\\mathrm\{MIX\}\}z^\{\\mathrm\{MIX\}\}\_\{t,f,v\}\-\(\\eta^\{\\mathrm\{MIX\}\}\)^\{2\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\\right\]\\right\)for a learning rateηMIX∈\(0,1/2\]\\eta^\{\\mathrm\{MIX\}\}\\in\(0,1/2\], a priorπMIX=πδ\\pi^\{\\mathrm\{MIX\}\}=\\pi\_\{\\delta\}\([Eq\. \(21\)](https://arxiv.org/html/2610.41023#S3.E21)\), and a test function

GtMIX​\(p\)=1ΦtMIX​𝔼\(f,v\)∼πMIX​\[Wt,fMIX​\(v\)​b~vηMIX​\(p,f⁡\(xt\)\)\],G^\{\\mathrm\{MIX\}\}\_\{t\}\(p\)=\\frac\{1\}\{\\Phi\_\{t\}^\{\\mathrm\{MIX\}\}\}\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi^\{\\mathrm\{MIX\}\}\}\\left\[W^\{\\mathrm\{MIX\}\}\_\{t,f\}\(v\)\\widetilde\{b\}^\{\\eta^\{\\mathrm\{MIX\}\}\}\_\{v\}\(p,f\(x\_\{t\}\)\)\\right\],for total massΦtMIX=𝔼\(f,v\)∼πMIX​\[Wt,fMIX​\(v\)\]\\Phi\_\{t\}^\{\\mathrm\{MIX\}\}=\\mathbb\{E\}\_\{\(f,v\)\\sim\\pi^\{\\mathrm\{MIX\}\}\}\\left\[W^\{\\mathrm\{MIX\}\}\_\{t,f\}\(v\)\\right\]and threshold testb~vη\\widetilde\{b\}^\{\\eta\}\_\{v\}as defined in[Eq\. \(24\)](https://arxiv.org/html/2610.41023#S3.E24)\.

##### Component forℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\.

For the last component that handles a fixed finite classℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}of mixable proper losses, instead of following the regret upper bound of[Eq\. \(6\)](https://arxiv.org/html/2610.41023#S1.E6), we directly define the reward for each expert\(f,ℓ\)∈ℱ×ℒmix\(f,\\ell\)\\in\{\\mathcal\{F\}\}\\times\\mathcal\{L\}\_\{\\mathrm\{mix\}\}as the regret

zt,f,ℓmix=𝔼P∼Dt​\[ℓ⁡\(P,yt\)−ℓ⁡\(clip⁡\(f⁡\(xt\),Iδ\),yt\)\],z^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\ell\(P,y\_\{t\}\)\-\\ell\(\\operatorname\{clip\}\(f\(x\_\{t\}\),I\_\{\\delta\}\),y\_\{t\}\)\\right\],where the clipping of the benchmark prediction avoids unboundedness issues \(and is harmless due to[Lemma3\.7](https://arxiv.org/html/2610.41023#S3.Thmtheorem7)\)\. The weight functions are also updated in a simple manner via vanilla exponential weight updates with no correction terms \(recall thatβℓ⋆\\beta\_\{\\ell\}^\{\\star\}is the largest mixability constant forℓ\\ell\):

Wt\+1,f,ℓmix=Wt,f,ℓmix​exp⁡\(βℓ⋆​zt,f,ℓmix\)\.W^\{\\mathrm\{mix\}\}\_\{t\+1,f,\\ell\}=W^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\\exp\\left\(\\beta\_\{\\ell\}^\{\\star\}z^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\\right\)\.To see what the test function should be, we introduce the following lemma that is analogous to[Lemma3\.4](https://arxiv.org/html/2610.41023#S3.Thmtheorem4)\.

###### Lemma 3\.8\.

For any mixable proper lossℓ\\ell,y∈\{0,1\}y\\in\\\{0,1\\\},p∈\(0,1\)p\\in\(0,1\)andq∈\[0,1\]q\\in\[0,1\], defineJℓy​\(p,q\)=exp⁡\(βℓ⋆​\(ℓ⁡\(p,y\)−ℓ⁡\(q,y\)\)\)J^\{y\}\_\{\\ell\}\(p,q\)=\\exp\\left\(\\beta\_\{\\ell\}^\{\\star\}\(\\ell\(p,y\)\-\\ell\(q,y\)\)\\right\)andS¯ℓ​\(p,q\)=Jℓ1​\(p,q\)−Jℓ0​\(p,q\)\\overline\{S\}\_\{\\ell\}\(p,q\)=J^\{1\}\_\{\\ell\}\(p,q\)\-J^\{0\}\_\{\\ell\}\(p,q\)\. We haveJℓy​\(p,q\)≤1\+S¯ℓ​\(p,q\)​\(y−p\)J^\{y\}\_\{\\ell\}\(p,q\)\\leq 1\+\\overline\{S\}\_\{\\ell\}\(p,q\)\(y\-p\)\.

###### Proof\.

SinceJℓy​\(p,q\)=𝔼Y∼Ber​\(p\)​\[JℓY​\(p,q\)\]\+S¯ℓ​\(p,q\)​\(y−p\)J^\{y\}\_\{\\ell\}\(p,q\)=\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[J^\{Y\}\_\{\\ell\}\(p,q\)\\right\]\+\\overline\{S\}\_\{\\ell\}\(p,q\)\(y\-p\), it suffices to show

𝔼Y∼Ber​\(p\)​\[exp⁡\(βℓ⋆​\(ℓ⁡\(p,Y\)−ℓ⁡\(q,Y\)\)\)\]≤1,\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\left\[\\exp\\left\(\\beta\_\{\\ell\}^\{\\star\}\(\\ell\(p,Y\)\-\\ell\(q,Y\)\)\\right\)\\right\]\\leq 1,\(28\)which is exactly the central condition of[Van Erven et al\. \(2015\)](https://arxiv.org/html/2610.41023#bib.bib27)and is implied by mixability \(see their Proposition 4\.12 and Theorem 3\.10\)\. ∎

With this lemma in hand, we can proceed similarly as before and bound the weight ratio as

Wt\+1,f,ℓmixWt,f,ℓmix\\displaystyle\\frac\{W^\{\\mathrm\{mix\}\}\_\{t\+1,f,\\ell\}\}\{W^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\}=exp⁡\(βℓ⋆​zt,f,ℓmix\)≤𝔼P∼Dt​\[Jℓyt​\(P,clip⁡\(f⁡\(xt\),Iδ\)\)\]\\displaystyle=\\exp\\left\(\\beta\_\{\\ell\}^\{\\star\}z^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\\right\)\\leq\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[J^\{y\_\{t\}\}\_\{\\ell\}\(P,\\operatorname\{clip\}\(f\(x\_\{t\}\),I\_\{\\delta\}\)\)\\right\]\(Jensen’s inequality\)≤1\+𝔼P∼Dt​\[S¯ℓ​\(P,clip⁡\(f⁡\(xt\),Iδ\)\)​\(yt−P\)\]\.\\displaystyle\\leq 1\+\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\overline\{S\}\_\{\\ell\}\(P,\\operatorname\{clip\}\(f\(x\_\{t\}\),I\_\{\\delta\}\)\)\(y\_\{t\}\-P\)\\right\]\.\([Lemma3\.8](https://arxiv.org/html/2610.41023#S3.Thmtheorem8)\)Therefore, following similar logic and definingb¯ℓ​\(p,q\)=S¯ℓ​\(p,clip⁡\(q,Iδ\)\)\\overline\{b\}\_\{\\ell\}\(p,q\)=\\overline\{S\}\_\{\\ell\}\(p,\\operatorname\{clip\}\(q,I\_\{\\delta\}\)\), we see that the test function in this case should be defined as

Gtmix​\(p\)=1Φtmix​𝔼\(f,ℓ\)∼πmix​\[Wt,f,ℓmix​b¯ℓ​\(p,f⁡\(xt\)\)\],G^\{\\mathrm\{mix\}\}\_\{t\}\(p\)=\\frac\{1\}\{\\Phi\_\{t\}^\{\\mathrm\{mix\}\}\}\\mathbb\{E\}\_\{\(f,\\ell\)\\sim\\pi^\{\\mathrm\{mix\}\}\}\\left\[W^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\\overline\{b\}\_\{\\ell\}\(p,f\(x\_\{t\}\)\)\\right\],for a priorπmix=Unif⁡\(ℱ\)⊗Unif⁡\(ℒmix\)\\pi^\{\\mathrm\{mix\}\}=\\operatorname\{Unif\}\(\{\\mathcal\{F\}\}\)\\otimes\\operatorname\{Unif\}\(\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\)and total massΦtmix=𝔼\(f,ℓ\)∼πmix​\[Wt,f,ℓmix\]\\Phi\_\{t\}^\{\\mathrm\{mix\}\}=\\mathbb\{E\}\_\{\(f,\\ell\)\\sim\\pi^\{\\mathrm\{mix\}\}\}\\left\[W^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\\right\]\.

Algorithm 3A best\-of\-all\-worlds online omnipredictorInput: a horizonTT, finite classesℱ,ℒbd,ℒmix\{\\mathcal\{F\}\},\\mathcal\{L\}\_\{\\mathrm\{bd\}\},\\mathcal\{L\}\_\{\\mathrm\{mix\}\}, learning rates0<ηbd≤1/40<\\eta^\{\\mathrm\{bd\}\}\\leq 1/4and0<ηMIX≤1/20<\\eta^\{\\mathrm\{MIX\}\}\\leq 1/2, and an EVI solver\.

Initialization:Initial weightW1cW\_\{1\}^\{c\}is11for all experts andc∈\{bd,mix,MIX\}c\\in\\\{\\mathrm\{bd\},\\mathrm\{mix\},\\mathrm\{MIX\}\\\},δ=1/\(2​\(T\+1\)\)\\delta=1/\\left\(2\(T\+1\)\\right\),Iδ=\[δ,1−δ\]I\_\{\\delta\}=\[\\delta,1\-\\delta\], andϵt=1/T\\epsilon\_\{t\}=1/T\.

for*t=1,…,Tt=1,\\ldots,T*do

Nature reveals

xtx\_\{t\}\. Compute the expected test function

GtG\_\{t\}as given by[Eq\. \(29\)](https://arxiv.org/html/2610.41023#S3.E29)\. Use the EVI solver to find a distribution

DtD\_\{t\}over

IδI\_\{\\delta\}, satisfying

𝔼P∼Dt​\[Gt​\(P\)​\(y−P\)\]≤ϵt,∀y∈\{0,1\}\.\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[G\_\{t\}\(P\)\(y\-P\)\\right\]\\leq\\epsilon\_\{t\},\\qquad\\forall y\\in\\\{0,1\\\}\.Nature reveals outcome

yty\_\{t\}\. For every expert

\(f,ℓ\)∈ℱ×ℒbd\(f,\\ell\)\\in\{\\mathcal\{F\}\}\\times\\mathcal\{L\}\_\{\\mathrm\{bd\}\}, define its reward as

zt,f,ℓbd=𝔼P∼Dt​\[bℓ​\(P,f⁡\(xt\)\)​\(yt−P\)\]\.z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[b\_\{\\ell\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\\right\]\.Update the weights for the bounded loss component:

Wt\+1,f,ℓbd=Wt,f,ℓbd​exp⁡\(ηbd​zt,f,ℓbd−\(ηbd\)2​\(zt,f,ℓbd\)2\)\.W^\{\\mathrm\{bd\}\}\_\{t\+1,f,\\ell\}=W^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\\exp\\left\(\\eta^\{\\mathrm\{bd\}\}z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\-\(\\eta^\{\\mathrm\{bd\}\}\)^\{2\}\(z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\)^\{2\}\\right\)\.For every expert

\(f,ℓ\)∈ℱ×ℒmix\(f,\\ell\)\\in\{\\mathcal\{F\}\}\\times\\mathcal\{L\}\_\{\\mathrm\{mix\}\}, define its reward as

zt,f,ℓmix=𝔼P∼Dt​\[ℓ⁡\(P,yt\)−ℓ⁡\(clip⁡\(f⁡\(xt\),Iδ\),yt\)\]\.z^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\ell\(P,y\_\{t\}\)\-\\ell\(\\operatorname\{clip\}\(f\(x\_\{t\}\),I\_\{\\delta\}\),y\_\{t\}\)\\right\]\.Update the weights for the finite mixable loss component:

Wt\+1,f,ℓmix=Wt,f,ℓmix​exp⁡\(βℓ⋆​zt,f,ℓmix\)\.W^\{\\mathrm\{mix\}\}\_\{t\+1,f,\\ell\}=W^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\\exp\\left\(\\beta\_\{\\ell\}^\{\\star\}z^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\\right\)\.For every expert

\(f,v\)∈ℱ×Iδ\(f,v\)\\in\{\\mathcal\{F\}\}\\times I\_\{\\delta\}, define its reward as

zt,f,vMIX=𝔼P∼Dt​\[bv​\(P,f⁡\(xt\)\)​\(yt−P\)\]\.z^\{\\mathrm\{MIX\}\}\_\{t,f,v\}=\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\\right\]\.Update the weights for the universal mixable loss component:

Wt\+1,fMIX​\(v\)=Wt,fMIX​\(v\)​exp⁡\(ηMIX​zt,f,vMIX−\(ηMIX\)2​𝔼P∼Dt​\[P⁡\(1−P\)​bv​\(P,f⁡\(xt\)\)2\]\)\.W^\{\\mathrm\{MIX\}\}\_\{t\+1,f\}\(v\)=W^\{\\mathrm\{MIX\}\}\_\{t,f\}\(v\)\\exp\\left\(\\eta^\{\\mathrm\{MIX\}\}z^\{\\mathrm\{MIX\}\}\_\{t,f,v\}\-\(\\eta^\{\\mathrm\{MIX\}\}\)^\{2\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\\right\]\\right\)\.

##### Combining everything\.

Finally, we combine all three test functions by taking a weighted average

Gt​\(p\)=1Φt​∑c∈𝒞Φtc​Gtc​\(p\),G\_\{t\}\(p\)=\\frac\{1\}\{\\Phi\_\{t\}\}\\sum\_\{c\\in\\mathcal\{C\}\}\\Phi\_\{t\}^\{c\}G^\{c\}\_\{t\}\(p\),\(29\)whereΦt=∑c∈𝒞Φtc\\Phi\_\{t\}=\\sum\_\{c\\in\\mathcal\{C\}\}\\Phi\_\{t\}^\{c\}, and solve the corresponding EVI, that is, find a distributionDtD\_\{t\}overIδI\_\{\\delta\}such that𝔼P∼Dt​\[Gt​\(P\)​\(y−P\)\]≤ϵt\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[G\_\{t\}\(P\)\(y\-P\)\\right\]\\leq\\epsilon\_\{t\}for anyy∈\{0,1\}y\\in\\\{0,1\\\}andϵt=1/T\\epsilon\_\{t\}=1/T\. The complete algorithm is shown in[Algorithm3](https://arxiv.org/html/2610.41023#alg3)\.

##### Regret analysis\.

The analysis mostly follows previous discussion and shows the key stepΦt\+1≤\(1\+ϵt\)​Φt\\Phi\_\{t\+1\}\\leq\(1\+\\epsilon\_\{t\}\)\\Phi\_\{t\}\. Eventually, picking the learning rates appropriately proves that[Algorithm3](https://arxiv.org/html/2610.41023#alg3)achievesRegℓ,ℱ​\(T\)=𝒪⁡\(log⁡\(\|ℱ\|​\|ℒmix\|\)βℓ⋆\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\mathcal\{O\}\\left\(\\frac\{\\log\\left\(\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\|\\right\)\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\)for anyℓ∈ℒmix\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\},Regℓ,ℱ​\(T\)=𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3βℓ⋆\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\widetilde\{\\mathcal\{O\}\}\\left\(\\frac\{T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\)for any mixable proper lossℓ\\ell\(whether or not it belongs toℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\), andRegℓ,ℱ​\(T\)=𝒪⁡\(T​log⁡\(\|ℱ\|​\|ℒbd\|\)\)\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)=\\mathcal\{O\}\\left\(\\sqrt\{T\\log\\left\(\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\\right\)\}\\right\)for anyℓ∈ℒbd\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\. See[AppendixA\.4](https://arxiv.org/html/2610.41023#A1.SS4)for the full proof\. Note that since Lipschitz proper losses are mixable \([Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)\),[Algorithm3](https://arxiv.org/html/2610.41023#alg3)also achieves the same result for Lipschitz proper losses as[Algorithm1](https://arxiv.org/html/2610.41023#alg1)does \(up tolog⁡T\\log Tfactors\)\.

In[Corollary1\.5](https://arxiv.org/html/2610.41023#S1.Thmtheorem5), we instantiateℒbd\\mathcal\{L\}\_\{\\mathrm\{bd\}\}andℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}with concrete losses and obtain regret bounds for more specific loss families\. For example, we instantiateℒmix\\mathcal\{L\}\_\{\\mathrm\{mix\}\}with the squared loss and the log loss, two of the most commonly used proper losses\. We also instantiateℒbd\\mathcal\{L\}\_\{\\mathrm\{bd\}\}with some threshold loss functions that serve as approximate bases for both Lipschitz losses \(that are not necessarily proper\) and generalized linear model \(GLM\) losses, so that our regret bound against any loss inℒbd\\mathcal\{L\}\_\{\\mathrm\{bd\}\}directly implies a regret bound for losses in those two families too\. In particular, the approximate bases we use for the GLM class appear to be unknown in the literature and could be of independent interest; see[PropositionD\.1](https://arxiv.org/html/2610.41023#A4.Thmtheorem1)\.

## 4Conclusion

We studied whether online omniprediction guarantees for finite loss classes extend to infinite classes\. We showed that sublinear regret is impossible for all bounded proper losses, even with only two benchmarks\. In contrast, for all mixable proper losses, we achieve𝒪~​\(T1/3​\(log⁡\|ℱ\|\)2/3/βℓ⋆\)\\widetilde\{\\mathcal\{O\}\}\(T^\{1/3\}\(\\log\|\{\\mathcal\{F\}\}\|\)^\{2/3\}/\\beta\_\{\\ell\}^\{\\star\}\)regret simultaneously, with theT1/3T^\{1/3\}dependence being optimal up to logarithmic factors even for Lipschitz proper losses\. This reveals a separation between the full class of mixable proper losses and finite collections, for which constant\-in\-TTregret is achievable\.

Our approach combines the V\-shaped representation of proper losses with continuous exponential weights and second\-order corrections that exploit the negative properness gain\. A Bernoulli\-variance correction further handles the potentially unbounded curvature of mixable losses\. By combining this approach with finite\-loss components, we obtain a single forecaster that retains the existing regret guarantees for finite classes of mixable and bounded losses, while simultaneously achieving our guarantee for all mixable proper losses\. Suitable finite bases also extend these guarantees to all Lipschitz and GLM losses\.

## Disclosure of Generative AI Use

We used GPT\-6 as an auxiliary tool in the development of this paper\. In particular, it was used to assist with exploring algorithmic ideas, developing and refining proof arguments, identifying connections to related literature, and improving the clarity and presentation of the manuscript\. The authors independently verified the correctness, originality, and presentation of all results, proofs, and references reported in the paper, and take full responsibility for the final content\.

## Appendix AProofs of Main Results

### A\.1Impossibility for all bounded proper losses

See[1\.1](https://arxiv.org/html/2610.41023#S1.Thmtheorem1)

###### Proof of[Theorem1\.1](https://arxiv.org/html/2610.41023#S1.Thmtheorem1)\.

The proof builds on the interval\-bisection construction used by\([Rakhlin and Sridharan, 2014](https://arxiv.org/html/2610.41023#bib.bib25), Theorem 8\.2\)to show that threshold classifiers are not online learnable\. Their construction generates independent fair outcomes and chooses each context from a binary tree according to the previous outcomes\. Every resulting path can be perfectly classified by a threshold chosen in hindsight, even though the next outcome cannot be predicted\. We use this same construction\. The additional work is to obtain a lower bound with only two fixed benchmarks\. We encode the threshold in the proper loss and use a second copy of the tree, shifted by1/41/4\. Each copy gives a loss under which its corresponding benchmark has zero loss\. The separation between the copies then forces the learner to incur linear expected loss for at least one of them\.

Fix an arbitrary integerT≥1T\\geq 1\. The context space is𝒳=\[0,3/4\]\{\\mathcal\{X\}\}=\[0,3/4\], and the fixed benchmark class isℱ=\{f1,f2\}\{\\mathcal\{F\}\}=\\\{f\_\{1\},f\_\{2\}\\\}, wheref1​\(x\)=xf\_\{1\}\(x\)=xandf2​\(x\)=x\+1/4f\_\{2\}\(x\)=x\+1/4\. We use a binary interval tree of depthT−1T\-1to specify the loss family\. Starting fromJ∅=\(1/3,5/12\)J\_\{\\varnothing\}=\(1/3,5/12\), for each binary stringh∈\{0,1\}≤T−1h\\in\\\{0,1\\\}^\{\\leq T\-1\}, letJhJ\_\{h\}be the interval obtained by successively selecting the lower half for each bit11and the upper half for each bit00\. Letmhm\_\{h\}be the midpoint ofJhJ\_\{h\}\. For each leafr∈\{0,1\}T−1r\\in\\\{0,1\\\}^\{T\-1\}, letvr,0v\_\{r,0\}andvr,1v\_\{r,1\}be the midpoints of the upper and lower halves ofJrJ\_\{r\}, respectively\. Thus,vr,1<mr<vr,0\.v\_\{r,1\}<m\_\{r\}<v\_\{r,0\}\.Using two thresholds inside each leaf allows the same scalar contextmrm\_\{r\}to support either outcome on the final round\. Using the V\-shaped proper losses defined in[Section2](https://arxiv.org/html/2610.41023#S2)we define the family

ℒT=\{ℓr,b:r∈\{0,1\}T−1,b∈\{0,1\}\},ℓr,b=λvr,b\+b/4\.\{\\mathcal\{L\}\}\_\{T\}=\\\{\\ell\_\{r,b\}:r\\in\\\{0,1\\\}^\{T\-1\},\\ b\\in\\\{0,1\\\}\\\},\\qquad\\ell\_\{r,b\}=\\lambda\_\{v\_\{r,b\}\+b/4\}\.The leaf intervals are disjoint, and the two sets of thresholds lie in\(1/3,5/12\)\(1/3,5/12\)and\(7/12,2/3\)\(7/12,2/3\)\. ThusℒT\{\\mathcal\{L\}\}\_\{T\}contains exactly2T2^\{T\}distinct\[0,1\]\[0,1\]\-valued proper losses\.

The hard distribution choosesr∼Unif⁡\(\{0,1\}T−1\)r\\sim\\operatorname\{Unif\}\(\\\{0,1\\\}^\{T\-1\}\)and an independentb∼Ber​\(1/2\)b\\sim\\texttt\{Ber\}\(1/2\)\. Writingr<t=\(r1,…,rt−1\)r\_\{<t\}=\(r\_\{1\},\\ldots,r\_\{t\-1\}\), the selected sequence and evaluation loss are

\(xt,yt\)=\{\(mr<t,rt\),1≤t<T,\(mr,b\),t=T,ℓ=ℓr,b\.\(x\_\{t\},y\_\{t\}\)=\\begin\{cases\}\(m\_\{r\_\{<t\}\},r\_\{t\}\),&1\\leq t<T,\\\\ \(m\_\{r\},b\),&t=T,\\end\{cases\}\\qquad\\ell=\\ell\_\{r,b\}\.
Here, the final outcomebbalso selects the evaluation loss\. Each pair\(r,b\)\(r,b\)therefore specifies a deterministic sequence and one evaluation loss\. WhenT=1T=1,rris the empty string and only the final round remains\. Fix an arbitrary, possibly randomized forecasterAA, whose randomness is independent of the hard distribution\. Recall thatDtD\_\{t\}is its forecast distribution given the history and current context\. For each fixed pair\(r,b\)\(r,b\), we define

R⁡\(A,r,b\)=𝔼A​\[Regℓr,b,ℱ​\(T\)\],R\(A;r,b\)=\\mathbb\{E\}\_\{A\}\\left\[\{\\text\{\\rm Reg\}\}\_\{\\ell\_\{r,b\},\{\\mathcal\{F\}\}\}\(T\)\\right\],where𝔼A\\mathbb\{E\}\_\{A\}averages over all randomness of the forecaster\. Since the expectation overP∼DtP\\sim D\_\{t\}is already part of the regret definition,R⁡\(A,r,b\)R\(A;r,b\)is a deterministic number for each fixed instance\.

Since bothvr,0v\_\{r,0\}andvr,1v\_\{r,1\}lie inJrJ\_\{r\}, interval nesting gives𝕀\{mr<t≥vr,b\}=rt\\mathbb\{I\}\{\\left\\\{\{m\_\{r\_\{<t\}\}\\geq v\_\{r,b\}\}\\right\\\}\}=r\_\{t\}for everyt<Tt<T, with no ties\. Thusfb\+1f\_\{b\+1\}has zero loss on the firstT−1T\-1rounds\. It also has zero loss on the final round\. Ifb=0b=0, thenf1​\(xT\)=mr<vr,0f\_\{1\}\(x\_\{T\}\)=m\_\{r\}<v\_\{r,0\}andyT=0y\_\{T\}=0\. Ifb=1b=1, thenf2​\(xT\)=mr\+1/4\>vr,1\+1/4f\_\{2\}\(x\_\{T\}\)=m\_\{r\}\+1/4\>v\_\{r,1\}\+1/4andyT=1y\_\{T\}=1\. For these proper losses, we use the identity post\-processingπλv​\(p\)=p\\pi\_\{\\lambda\_\{v\}\}\(p\)=p, and have that

R⁡\(A,r,b\)=𝔼A​\[∑t=1T𝔼P∼Dt​λvr,b\+b/4​\(P,yt\)\]\.R\(A;r,b\)=\\mathbb\{E\}\_\{A\}\\left\[\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\lambda\_\{v\_\{r,b\}\+b/4\}\(P,y\_\{t\}\)\\right\]\.
For allv0,v1∈\(1/3,5/12\)v\_\{0\},v\_\{1\}\\in\(1/3,5/12\),p∈\[0,1\]p\\in\[0,1\], andy∈\{0,1\}y\\in\\\{0,1\\\},

λv0\(p,y\)\+λv1\+1/4\(p,y\)≥13\[\(1−y\)𝕀\{p≥1/2\}\+y𝕀\{p<1/2\}\]\.\\lambda\_\{v\_\{0\}\}\(p,y\)\+\\lambda\_\{v\_\{1\}\+1/4\}\(p,y\)\\geq\\frac\{1\}\{3\}\\left\[\(1\-y\)\\mathbb\{I\}\{\\left\\\{\{p\\geq 1/2\}\\right\\\}\}\+y\\mathbb\{I\}\{\\left\\\{\{p<1/2\}\\right\\\}\}\\right\]\.\(30\)Indeed, whenp≥1/2p\\geq 1/2, the lower\-threshold loss is at least1/31/3fory=0y=0\. Whenp<1/2p<1/2, the upper\-threshold loss is at least1/31/3fory=1y=1\. These comparisons also hold for any fixed Bayes\-optimal tie rule, since the contributing prediction lies strictly on one side of the corresponding threshold\.

For everyt<Tt<T,DtD\_\{t\}depends only onr<tr\_\{<t\}and the forecaster’s randomness, and in particular does not depend onbb\. The two choices ofbbtherefore give the same forecast distribution when the forecaster uses the same random seed\. Conditional on the information available whenDtD\_\{t\}is chosen,rtr\_\{t\}remains an independent fair bit\. The right\-hand side of[Eq\. \(30\)](https://arxiv.org/html/2610.41023#A1.E30)does not depend on either leaf threshold\. Averaging first overbband then over the outcomes and the forecaster’s randomness gives

𝔼r,b​𝔼A​𝔼P∼Dt​λvr,b\+b/4​\(P,rt\)\\displaystyle\\mathbb\{E\}\_\{r,b\}\\mathbb\{E\}\_\{A\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\lambda\_\{v\_\{r,b\}\+b/4\}\(P,r\_\{t\}\)=12​𝔼r​𝔼A​𝔼P∼Dt​\[λvr,0​\(P,rt\)\+λvr,1\+1/4​\(P,rt\)\]\\displaystyle\\qquad=\\frac\{1\}\{2\}\\mathbb\{E\}\_\{r\}\\mathbb\{E\}\_\{A\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\lambda\_\{v\_\{r,0\}\}\(P,r\_\{t\}\)\+\\lambda\_\{v\_\{r,1\}\+1/4\}\(P,r\_\{t\}\)\\right\]≥16𝔼r𝔼A𝔼P∼Dt\[\(1−rt\)𝕀\{P≥1/2\}\+rt𝕀\{P<1/2\}\]=112\.\\displaystyle\\qquad\\geq\\frac\{1\}\{6\}\\mathbb\{E\}\_\{r\}\\mathbb\{E\}\_\{A\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\(1\-r\_\{t\}\)\\mathbb\{I\}\{\\left\\\{\{P\\geq 1/2\}\\right\\\}\}\+r\_\{t\}\\mathbb\{I\}\{\\left\\\{\{P<1/2\}\\right\\\}\}\\right\]=\\frac\{1\}\{12\}\.
On the final round,DTD\_\{T\}may depend onrrand the forecaster’s randomness, but not onbb\. For everyp∈\[0,1\]p\\in\[0,1\], the same separation of thresholds gives

12​\[λvr,0​\(p,0\)\+λvr,1\+1/4​\(p,1\)\]≥16\.\\frac\{1\}\{2\}\\left\[\\lambda\_\{v\_\{r,0\}\}\(p,0\)\+\\lambda\_\{v\_\{r,1\}\+1/4\}\(p,1\)\\right\]\\geq\\frac\{1\}\{6\}\.Averaging overbb, the forecaster’s randomness, andP∼DTP\\sim D\_\{T\}therefore gives expected loss at least1/61/6on this round\. Consequently, the arbitrary randomized forecasterAAsatisfies

𝔼r,b​R​\(A,r,b\)≥T−112\+16≥T12\.\\mathbb\{E\}\_\{r,b\}R\(A;r,b\)\\geq\\frac\{T\-1\}\{12\}\+\\frac\{1\}\{6\}\\geq\\frac\{T\}\{12\}\.
Since this is an average over finitely many pairs\(r,b\)\(r,b\), there exists a fixed pair\(r∗,b∗\)\(r^\{\*\},b^\{\*\}\)such thatR⁡\(A,r∗,b∗\)≥T/12R\(A;r^\{\*\},b^\{\*\}\)\\geq T/12\. For the deterministic sequence specified by this pair,

supℓ∈ℒT𝔼A​\[Regℓ,ℱ​\(T\)\]≥𝔼A​\[Regℓr∗,b∗,ℱ​\(T\)\]=R⁡\(A,r∗,b∗\)≥T12\.\\sup\_\{\\ell\\in\{\\mathcal\{L\}\}\_\{T\}\}\\mathbb\{E\}\_\{A\}\\left\[\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)\\right\]\\geq\\mathbb\{E\}\_\{A\}\\left\[\{\\text\{\\rm Reg\}\}\_\{\\ell\_\{r^\{\*\},b^\{\*\}\},\{\\mathcal\{F\}\}\}\(T\)\\right\]=R\(A;r^\{\*\},b^\{\*\}\)\\geq\\frac\{T\}\{12\}\.The pair is selected after taking the expectation over the forecaster’s randomness\. Thus both the sequence and the single lossℓr∗,b∗\\ell\_\{r^\{\*\},b^\{\*\}\}are fixed independently of its realized random seed, as required\. ∎

### A\.2Lipschitz and mixable lower bounds for every horizon

See[1\.3](https://arxiv.org/html/2610.41023#S1.Thmtheorem3)

###### Proof of[Theorem1\.3](https://arxiv.org/html/2610.41023#S1.Thmtheorem3)\.

We first establish anΩ⁡\(L​T1/3\)\\Omega\(LT^\{1/3\}\)lower bound for boundedLL\-Lipschitz proper losses\. Throughout the proof,L\>0L\>0, the integerT≥1T\\geq 1, and the forecaster is arbitrary and fixed\. The context space is𝒳=\[0,1\]2\{\\mathcal\{X\}\}=\[0,1\]^\{2\}, and the fixed benchmark class isℱ=\{f\+,f−\}\{\\mathcal\{F\}\}=\\\{f\_\{\+\},f\_\{\-\}\\\}, wheref\+​\(x\)=x1f\_\{\+\}\(x\)=x\_\{1\}andf−​\(x\)=x2f\_\{\-\}\(x\)=x\_\{2\}\.

A construction withKKphases\.We partition totalTTrounds intoKKphases, with round\-index setsℐj\\mathcal\{I\}\_\{j\}and lengthsnj∈\{⌊T/K⌋,⌈T/K⌉\}n\_\{j\}\\in\\\{\\lfloor T/K\\rfloor,\\lceil T/K\\rceil\\\}\. Letvj=0\.45\+j−1/210​K,hj=1100​njv\_\{j\}=0\.45\+\\frac\{j\-1/2\}\{10K\},h\_\{j\}=\\frac\{1\}\{100\\sqrt\{n\_\{j\}\}\}\. In phasej∈\[K\]j\\in\[K\], the environment revealsxt=\(vj\+hj,vj−hj\)x\_\{t\}=\(v\_\{j\}\+h\_\{j\},v\_\{j\}\-h\_\{j\}\)and drawsyt∼Ber​\(vj\)y\_\{t\}\\sim\\texttt\{Ber\}\(v\_\{j\}\), independently across all rounds\. Thus the two benchmarks predictvj\+hjv\_\{j\}\+h\_\{j\}andvj−hjv\_\{j\}\-h\_\{j\}\. We will choose a value ofKKsuch that the intervals\[vj−hj,vj\+hj\]\[v\_\{j\}\-h\_\{j\},v\_\{j\}\+h\_\{j\}\]are pairwise disjoint\. LetSj=∑t∈ℐj\(yt−vj\)S\_\{j\}=\\sum\_\{t\\in\\mathcal\{I\}\_\{j\}\}\(y\_\{t\}\-v\_\{j\}\)be the fluctuation of outcomes in phasejj\. For shorthand, we denoteσj2=nj​vj​\(1−vj\)\.\\sigma\_\{j\}^\{2\}=n\_\{j\}v\_\{j\}\(1\-v\_\{j\}\)\.Sincevj∈\[0\.45,0\.55\]v\_\{j\}\\in\[0\.45,0\.55\], the Bernoulli moment formulas \(e\.g\.\([Skorski, 2020](https://arxiv.org/html/2610.41023#bib.bib26), Table 2\)\) give

𝔼​Sj2=σj2,𝔼​Sj4=3​σj4\+σj2​\(1−6​vj​\(1−vj\)\)≤3​σj4\.\\mathbb\{E\}S\_\{j\}^\{2\}=\\sigma\_\{j\}^\{2\},\\qquad\\mathbb\{E\}S\_\{j\}^\{4\}=3\\sigma\_\{j\}^\{4\}\+\\sigma\_\{j\}^\{2\}\(1\-6v\_\{j\}\(1\-v\_\{j\}\)\)\\leq 3\\sigma\_\{j\}^\{4\}\.Hölder’s inequality therefore yields

𝔼⁡\[Sj2\]=𝔼⁡\[\|Sj\|2/3​\|Sj\|4/3\]≤\(𝔼​\|Sj\|\)2/3​\(𝔼⁡\[Sj4\]\)1/3,\\mathbb\{E\}\[S\_\{j\}^\{2\}\]=\\mathbb\{E\}\\bigl\[\|S\_\{j\}\|^\{2/3\}\|S\_\{j\}\|^\{4/3\}\\bigr\]\\leq\\bigl\(\\mathbb\{E\}\|S\_\{j\}\|\\bigr\)^\{2/3\}\\bigl\(\\mathbb\{E\}\[S\_\{j\}^\{4\}\]\\bigr\)^\{1/3\},and thus

𝔼​\|Sj\|≥\(𝔼​Sj2\)3/2\(𝔼​Sj4\)1/2≥σj3≥nj4\.\\mathbb\{E\}\|S\_\{j\}\|\\geq\\frac\{\(\\mathbb\{E\}S\_\{j\}^\{2\}\)^\{3/2\}\}\{\(\\mathbb\{E\}S\_\{j\}^\{4\}\)^\{1/2\}\}\\geq\\frac\{\\sigma\_\{j\}\}\{\\sqrt\{3\}\}\\geq\\frac\{\\sqrt\{n\_\{j\}\}\}\{4\}\.\(31\)
A fixed family of Lipschitz proper losses\.For each phasej∈\[K\]j\\in\[K\]and subsetU⊆\[K\]U\\subseteq\[K\], we define

wj\(v\)=L𝕀\{\|v−vj\|≤hj\},wU\(v\)=∑j∈Uwj\(v\)\.w\_\{j\}\(v\)=L\\mathbb\{I\}\{\\left\\\{\{\|v\-v\_\{j\}\|\\leq h\_\{j\}\}\\right\\\}\},\\qquad w^\{U\}\(v\)=\\sum\_\{j\\in U\}w\_\{j\}\(v\)\.Disjointness of the intervals gives0≤wU≤L0\\leq w^\{U\}\\leq Lfor anyU⊆\[K\]U\\subseteq\[K\]\. Writeℓw\\ell\_\{w\}for the proper loss associated with densitywwthrough[Eq\. \(13\)](https://arxiv.org/html/2610.41023#S2.E13), withCℓ​\(y\)=0C\_\{\\ell\}\(y\)=0\. SincewU​\(v\)≤L≤L/max⁡\{v,1−v\}w^\{U\}\(v\)\\leq L\\leq L/\\max\\\{v,1\-v\\\},[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)givesℓwU∈ℒLip​\(L\)\\ell\_\{w^\{U\}\}\\in\\mathcal\{L\}\_\{\\mathrm\{Lip\}\}\\left\(L\\right\)\. Moreover,

0≤ℓwU​\(q,y\)≤L​∫01λv​\(q,y\)​𝑑v≤L/2\.0\\leq\\ell\_\{w^\{U\}\}\(q,y\)\\leq L\\int\_\{0\}^\{1\}\\lambda\_\{v\}\(q,y\)\\,\\mathrm\{d\}v\\leq L/2\.The family\{ℓwU:U⊆\[K\]\}\\\{\\ell\_\{w^\{U\}\}:U\\subseteq\[K\]\\\}is fixed independently of the outcomes\.

Two complementary losses\.For a realized outcome sequence, let

U\+=\{j:Sj≥0\},U−=\[K\]∖U\+,ℓ±=ℓwU±,U\_\{\+\}=\\\{j:S\_\{j\}\\geq 0\\\},\\qquad U\_\{\-\}=\[K\]\\setminus U\_\{\+\},\\qquad\\ell\_\{\\pm\}=\\ell\_\{w^\{U\_\{\\pm\}\}\},and

Rs=∑t=1T𝔼P∼Dt​\[ℓs​\(P,yt\)−ℓs​\(fs​\(xt\),yt\)\],wheres∈\{\+,−\}\.R\_\{s\}=\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\bigl\[\\ell\_\{s\}\(P,y\_\{t\}\)\-\\ell\_\{s\}\(f\_\{s\}\(x\_\{t\}\),y\_\{t\}\)\\bigr\],\\qquad\\text\{where\}\\qquad s\\in\\\{\+,\-\\\}\.ThusRsR\_\{s\}is the cumulative loss difference relative to the fixed benchmarkfsf\_\{s\}, andRegℓs,ℱ​\(T\)≥Rs\{\\text\{\\rm Reg\}\}\_\{\\ell\_\{s\},\{\\mathcal\{F\}\}\}\(T\)\\geq R\_\{s\}\. Althoughℓ\+\\ell\_\{\+\}andℓ−\\ell\_\{\-\}depend on the outcomes, their sum is the fixed proper loss

ℓ¯=ℓ\+\+ℓ−=ℓw¯,w¯​\(v\)=∑j=1Kwj​\(v\)\.\\bar\{\\ell\}=\\ell\_\{\+\}\+\\ell\_\{\-\}=\\ell\_\{\\bar\{w\}\},\\qquad\\bar\{w\}\(v\)=\\sum\_\{j=1\}^\{K\}w\_\{j\}\(v\)\.For each roundt∈ℐjt\\in\\mathcal\{I\}\_\{j\}, conditional on the history and the choice ofDtD\_\{t\}, the outcomeyty\_\{t\}still followsBer​\(vj\)\\texttt\{Ber\}\(v\_\{j\}\)\. Properness ofℓ¯\\bar\{\\ell\}gives

𝔼yt∼Ber​\(vj\)​𝔼P∼Dt​\[ℓ¯​\(P,yt\)−ℓ¯​\(vj,yt\)\]≥0\.\\mathbb\{E\}\_\{y\_\{t\}\\sim\\texttt\{Ber\}\(v\_\{j\}\)\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\bigl\[\\bar\{\\ell\}\(P,y\_\{t\}\)\-\\bar\{\\ell\}\(v\_\{j\},y\_\{t\}\)\\bigr\]\\geq 0\.Adding and subtracting the losses atvjv\_\{j\}therefore yields

𝔼⁡\[R\+\+R−\]≥∑j=1K∑t∈ℐj∑s∈\{\+,−\}𝔼⁡\[ℓs​\(vj,yt\)−ℓs​\(fs​\(xt\),yt\)\],\\displaystyle\\mathbb\{E\}\[R\_\{\+\}\+R\_\{\-\}\]\\geq\\sum\_\{j=1\}^\{K\}\\sum\_\{t\\in\\mathcal\{I\}\_\{j\}\}\\sum\_\{s\\in\\\{\+,\-\\\}\}\\mathbb\{E\}\\bigl\[\\ell\_\{s\}\(v\_\{j\},y\_\{t\}\)\-\\ell\_\{s\}\(f\_\{s\}\(x\_\{t\}\),y\_\{t\}\)\\bigr\],where the expectation includes the outcomes and any forecaster randomness\. For everyt∈ℐjt\\in\\mathcal\{I\}\_\{j\}, subtracting the integral representations ofℓwj\\ell\_\{w\_\{j\}\}atvjv\_\{j\}andvj±hjv\_\{j\}\\pm h\_\{j\}gives

ℓwj​\(vj,yt\)−ℓwj​\(vj±hj,yt\)\\displaystyle\\ell\_\{w\_\{j\}\}\(v\_\{j\},y\_\{t\}\)\-\\ell\_\{w\_\{j\}\}\(v\_\{j\}\\pm h\_\{j\},y\_\{t\}\)=∫vjvj±hj\(yt−v\)​wj​\(v\)​𝑑v\\displaystyle\\qquad=\\int\_\{v\_\{j\}\}^\{v\_\{j\}\\pm h\_\{j\}\}\(y\_\{t\}\-v\)w\_\{j\}\(v\)\\,\\mathrm\{d\}v=L​∫vjvj±hj\[\(yt−vj\)−\(v−vj\)\]​𝑑v\\displaystyle\\qquad=L\\int\_\{v\_\{j\}\}^\{v\_\{j\}\\pm h\_\{j\}\}\\left\[\(y\_\{t\}\-v\_\{j\}\)\-\(v\-v\_\{j\}\)\\right\]\\,\\mathrm\{d\}v=L⁡\(±hj​\(yt−vj\)−hj22\)\.\\displaystyle\\qquad=L\\left\(\\pm h\_\{j\}\(y\_\{t\}\-v\_\{j\}\)\-\\frac\{h\_\{j\}^\{2\}\}\{2\}\\right\)\.The second equality useswj​\(v\)=Lw\_\{j\}\(v\)=Lthroughout\[vj−hj,vj\+hj\]\[v\_\{j\}\-h\_\{j\},v\_\{j\}\+h\_\{j\}\]\. Summing overt∈ℐjt\\in\\mathcal\{I\}\_\{j\}and usingSj=∑t∈ℐj\(yt−vj\)S\_\{j\}=\\sum\_\{t\\in\\mathcal\{I\}\_\{j\}\}\(y\_\{t\}\-v\_\{j\}\)and\|ℐj\|=nj\|\\mathcal\{I\}\_\{j\}\|=n\_\{j\}yields

∑t∈ℐj\(ℓwj​\(vj,yt\)−ℓwj​\(vj±hj,yt\)\)=L⁡\(±hj​Sj−nj​hj22\)\.\\displaystyle\\sum\_\{t\\in\\mathcal\{I\}\_\{j\}\}\\left\(\\ell\_\{w\_\{j\}\}\(v\_\{j\},y\_\{t\}\)\-\\ell\_\{w\_\{j\}\}\(v\_\{j\}\\pm h\_\{j\},y\_\{t\}\)\\right\)=L\\left\(\\pm h\_\{j\}S\_\{j\}\-\\frac\{n\_\{j\}h\_\{j\}^\{2\}\}\{2\}\\right\)\.WhenSj≥0S\_\{j\}\\geq 0, this contribution appears in the comparison betweenℓ\+\\ell\_\{\+\}andf\+f\_\{\+\}; otherwise, it appears in the comparison betweenℓ−\\ell\_\{\-\}andf−f\_\{\-\}\. Every other densitywkw\_\{k\}, withk≠jk\\neq j, vanishes betweenvjv\_\{j\}and either benchmark prediction, so its contribution to these loss differences is zero\. Consequently,

𝔼⁡\[R\+\+R−\]\\displaystyle\\mathbb\{E\}\[R\_\{\+\}\+R\_\{\-\}\]≥L​∑j=1K\(hj​𝔼​\|Sj\|−nj​hj22\)\\displaystyle\\geq L\\sum\_\{j=1\}^\{K\}\\left\(h\_\{j\}\\mathbb\{E\}\|S\_\{j\}\|\-\\frac\{n\_\{j\}h\_\{j\}^\{2\}\}\{2\}\\right\)≥L​K​\(1400−120000\)≥L​K500,\\displaystyle\\geq LK\\left\(\\frac\{1\}\{400\}\-\\frac\{1\}\{20000\}\\right\)\\geq\\frac\{LK\}\{500\},where the second inequality uses[Eq\. \(31\)](https://arxiv.org/html/2610.41023#A1.E31)\.

ChoosingKKand a fixed witness\.The preceding lower bound increases withKK\. Since𝔼​\|Sj\|=Θ⁡\(nj\)\\mathbb\{E\}\|S\_\{j\}\|=\\Theta\(\\sqrt\{n\_\{j\}\}\), our choicehj=1/\(100​nj\)h\_\{j\}=1/\(100\\sqrt\{n\_\{j\}\}\)ensureshj​𝔼​\|Sj\|=Θ⁡\(1\)h\_\{j\}\\mathbb\{E\}\|S\_\{j\}\|=\\Theta\(1\)in each phase\. Moreover,nj≍T/Kn\_\{j\}\\asymp T/Kimplieshj≍K/Th\_\{j\}\\asymp\\sqrt\{K/T\}\. Asvj\+1−vj=1/\(10​K\)v\_\{j\+1\}\-v\_\{j\}=1/\(10K\), keeping the intervals\[vj−hj,vj\+hj\]\[v\_\{j\}\-h\_\{j\},v\_\{j\}\+h\_\{j\}\]pairwise disjoint requiresK/T=O⁡\(1/K\)\\sqrt\{K/T\}=O\(1/K\), or equivalently,K=O⁡\(T1/3\)K=O\(T^\{1/3\}\)\. We therefore chooseK=⌊T1/3⌋K=\\lfloor T^\{1/3\}\\rfloor\. SinceT≥K3T\\geq K^\{3\}, we havenj≥K2n\_\{j\}\\geq K^\{2\}andhj≤1/\(100​K\)h\_\{j\}\\leq 1/\(100K\), ensuring that these intervals are pairwise disjoint and contained in\[0\.45,0\.55\]\[0\.45,0\.55\]\.

The selected lossesℓ±\\ell\_\{\\pm\}depend only on the outcomes, not on the forecaster’s randomness\. Taking expectation over the forecaster’s randomness and then averaging over the outcomes, there exist a deterministic outcome sequence ands∈\{\+,−\}s\\in\\\{\+,\-\\\}such that

𝔼⁡\[Regℓs,ℱ​\(T\)\]≥𝔼⁡\[Rs\]≥L​K1000≥L​T1/32000,\\mathbb\{E\}\[\{\\text\{\\rm Reg\}\}\_\{\\ell\_\{s\},\{\\mathcal\{F\}\}\}\(T\)\]\\geq\\mathbb\{E\}\[R\_\{s\}\]\\geq\\frac\{LK\}\{1000\}\\geq\\frac\{LT^\{1/3\}\}\{2000\},where the expectations are now only over the forecaster’s randomness and the last inequality uses⌊T1/3⌋≥T1/3/2\\lfloor T^\{1/3\}\\rfloor\\geq T^\{1/3\}/2\. Thus, both the context–outcome sequence and the lossℓs\\ell\_\{s\}are fixed independently of the forecaster’s randomness\.

Finally,ℒLip​\(2/β\)⊆ℒmix​\(β\)\\mathcal\{L\}\_\{\\mathrm\{Lip\}\}\\left\(2/\\beta\\right\)\\subseteq\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(\\beta\\right\)by[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)\. TakingL=2/βL=2/\\betagives a bounded2/β2/\\beta\-Lipschitz,β\\beta\-mixable proper loss with expected regretΩ⁡\(T1/3/β\)\\Omega\(T^\{1/3\}/\\beta\)\. ∎

### A\.3The universal mixable upper bound

See[1\.2](https://arxiv.org/html/2610.41023#S1.Thmtheorem2)

###### Proof of[Theorem1\.2](https://arxiv.org/html/2610.41023#S1.Thmtheorem2)\.

We consider an arbitrary mixable proper lossℓ\\ell\. Ifβℓ⋆=\+∞\\beta\_\{\\ell\}^\{\\star\}=\+\\infty,[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)giveswℓ=0w\_\{\\ell\}=0almost everywhere, soℓ\\ellis independent of the prediction and its regret is zero\. Otherwise, we takeβ=βℓ⋆\\beta=\\beta\_\{\\ell\}^\{\\star\}\. The mixability characterization in[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)then givesℓ∈ℒmix​\(β\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(\\beta\\right\)\. The learning rate0<η≤1/20<\\eta\\leq 1/2and predictorf∈ℱf\\in\{\\mathcal\{F\}\}are arbitrary and fixed\. Averaging[Lemma3\.7](https://arxiv.org/html/2610.41023#S3.Thmtheorem7)and[Lemma3\.6](https://arxiv.org/html/2610.41023#S3.Thmtheorem6)overDtD\_\{t\}and applying[Lemma3\.5](https://arxiv.org/html/2610.41023#S3.Thmtheorem5)gives

∑t=1T𝔼P∼Dt​\[ℓ⁡\(P,yt\)−ℓ⁡\(f⁡\(xt\),yt\)\]≤𝒪⁡\(1β​\{Mδ​log⁡\(e​\|ℱ\|\)η\+T​η2\+T​δ\}\)\.\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[\\ell\(P,y\_\{t\}\)\-\\ell\(f\(x\_\{t\}\),y\_\{t\}\)\]\\leq\\mathcal\{O\}\\left\(\\frac\{1\}\{\\beta\}\\left\\\{\\frac\{M\_\{\\delta\}\\log\(e\|\{\\mathcal\{F\}\}\|\)\}\{\\eta\}\+T\\eta^\{2\}\+T\\delta\\right\\\}\\right\)\.Balancing the complexity and variance terms gives the choice

η=min⁡\{12,\(Mδ​log⁡\(e​\|ℱ\|\)T\)1/3\}\.\\eta=\\min\\left\\\{\\frac\{1\}\{2\},\\left\(\\frac\{M\_\{\\delta\}\\log\(e\|\{\\mathcal\{F\}\}\|\)\}\{T\}\\right\)^\{1/3\}\\right\\\}\.\(32\)SinceMδ=2​log⁡\(2​T\+1\)M\_\{\\delta\}=2\\log\(2T\+1\)andT​δ≤1/2T\\delta\\leq 1/2, comparison with the bestf∈ℱf\\in\{\\mathcal\{F\}\}yields

Regℓ,ℱ​\(T\)≤𝒪⁡\(T1/3​\(log⁡\(T\)​log⁡\(\|ℱ\|\)\)2/3\+log⁡\(T\)​log⁡\(\|ℱ\|\)β\)\.\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)\\leq\\mathcal\{O\}\\left\(\\frac\{T^\{1/3\}\(\\log\(T\)\\log\(\|\{\\mathcal\{F\}\}\|\)\)^\{2/3\}\+\\log\(T\)\\log\(\|\{\\mathcal\{F\}\}\|\)\}\{\\beta\}\\right\)\.Sinceβ=βℓ⋆\\beta=\\beta\_\{\\ell\}^\{\\star\}, this gives the claimed dependence on the mixability constant\. ∎

### A\.4The best\-of\-all\-worlds guarantee

See[1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4)

###### Proof of[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4)\.

The learning rates satisfy0<ηbd≤1/40<\\eta^\{\\mathrm\{bd\}\}\\leq 1/4and0<ηMIX≤1/20<\\eta^\{\\mathrm\{MIX\}\}\\leq 1/2\. If a loss hasβℓ⋆=\+∞\\beta\_\{\\ell\}^\{\\star\}=\+\\infty,[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)makes it independent of the prediction, so its regret is zero\. For these losses, the factorsJℓyJ\_\{\\ell\}^\{y\}are taken to be one,b¯ℓ=0\\overline\{b\}\_\{\\ell\}=0, and the weightsWt,f,ℓmixW^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}remain one\.

We first bound the total weightΦt\\Phi\_\{t\}using the component weights and test functions defined in[Section3\.3](https://arxiv.org/html/2610.41023#S3.SS3)\. For the bounded component, everyℓ∈ℒbd\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{bd\}\}takes values in\[0,1\]\[0,1\], so\|Δℓ​\(q\)\|≤1\|\\Delta\_\{\\ell\}\(q\)\|\\leq 1and hence\|bℓ​\(p,q\)\|≤2\|b\_\{\\ell\}\(p,q\)\|\\leq 2\. Since\|yt−P\|≤1\|y\_\{t\}\-P\|\\leq 1, this gives\|zt,f,ℓbd\|≤2\|z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\|\\leq 2\. Thus\|ηbd​zt,f,ℓbd\|≤1/2\|\\eta^\{\\mathrm\{bd\}\}z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\|\\leq 1/2, andex−x2≤1\+xe^\{x\-x^\{2\}\}\\leq 1\+xfor\|x\|≤1/2\|x\|\\leq 1/2gives

Wt\+1,f,ℓbdWt,f,ℓbd\\displaystyle\\frac\{W^\{\\mathrm\{bd\}\}\_\{t\+1,f,\\ell\}\}\{W^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\}≤1\+ηbd​zt,f,ℓbd=1\+ηbd​𝔼P∼Dt​\[bℓ​\(P,f⁡\(xt\)\)​\(yt−P\)\]\.\\displaystyle\\leq 1\+\\eta^\{\\mathrm\{bd\}\}z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}=1\+\\eta^\{\\mathrm\{bd\}\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[b\_\{\\ell\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\]\.Multiplying byWt,f,ℓbdW^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}and averaging overπbd\\pi^\{\\mathrm\{bd\}\}gives the bound below forc=bdc=\\mathrm\{bd\}\. Forc=MIXc=\\mathrm\{MIX\}, the same bound follows from Jensen’s inequality and[Lemma3\.4](https://arxiv.org/html/2610.41023#S3.Thmtheorem4), as in the proof of[Lemma3\.5](https://arxiv.org/html/2610.41023#S3.Thmtheorem5)\. Forc=mixc=\\mathrm\{mix\}, it follows from Jensen’s inequality and[Lemma3\.8](https://arxiv.org/html/2610.41023#S3.Thmtheorem8)\. In each case, the definition ofGtcG\_\{t\}^\{c\}therefore gives that for eachc∈𝒞c\\in\\mathcal\{C\}

Φt\+1c≤Φtc​\(1\+𝔼P∼Dt​\[Gtc​\(P\)​\(yt−P\)\]\)\.\\Phi\_\{t\+1\}^\{c\}\\leq\\Phi\_\{t\}^\{c\}\\left\(1\+\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[G\_\{t\}^\{c\}\(P\)\(y\_\{t\}\-P\)\]\\right\)\.The same factor bounds give1\+Gtc​\(p\)​\(y−p\)≥01\+G\_\{t\}^\{c\}\(p\)\(y\-p\)\\geq 0forc=mix,MIXc=\\mathrm\{mix\},\\mathrm\{MIX\}\. Forc=bdc=\\mathrm\{bd\}, this follows from\|Gtbd​\(p\)\|≤2​ηbd≤1/2\|G\_\{t\}^\{\\mathrm\{bd\}\}\(p\)\|\\leq 2\\eta^\{\\mathrm\{bd\}\}\\leq 1/2\. Their weighted averageGtG\_\{t\}in[Eq\. \(29\)](https://arxiv.org/html/2610.41023#S3.E29)consequently satisfies1\+Gt​\(p\)​\(y−p\)≥01\+G\_\{t\}\(p\)\(y\-p\)\\geq 0\. Sinceδ/\(1−δ\)=1/\(2​T\+1\)<ϵt=1/T\\delta/\(1\-\\delta\)=1/\(2T\+1\)<\\epsilon\_\{t\}=1/T,[Lemma2\.2](https://arxiv.org/html/2610.41023#S2.Thmtheorem2)supplies the required forecast distribution\.

Summing the component inequalities and applying[Eq\. \(29\)](https://arxiv.org/html/2610.41023#S3.E29)and the EVI condition yields

Φt\+1≤Φt​\(1\+𝔼P∼Dt​\[Gt​\(P\)​\(yt−P\)\]\)≤\(1\+ϵt\)​Φt\.\\Phi\_\{t\+1\}\\leq\\Phi\_\{t\}\\left\(1\+\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[G\_\{t\}\(P\)\(y\_\{t\}\-P\)\]\\right\)\\leq\(1\+\\epsilon\_\{t\}\)\\Phi\_\{t\}\.Each component starts with weights equal to one and has a probability prior, soΦ1c=1\\Phi\_\{1\}^\{c\}=1andΦ1=3\\Phi\_\{1\}=3\. Therefore

ΦT\+1c≤ΦT\+1≤3​∏t=1T\(1\+ϵt\)≤3​e\(c∈𝒞\)\.\\Phi\_\{T\+1\}^\{c\}\\leq\\Phi\_\{T\+1\}\\leq 3\\prod\_\{t=1\}^\{T\}\(1\+\\epsilon\_\{t\}\)\\leq 3e\\qquad\(c\\in\\mathcal\{C\}\)\.
The three guarantees follow in the order stated in[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4)\. For the finite mixable component,\(f,ℓ\)∈ℱ×ℒmix\(f,\\ell\)\\in\{\\mathcal\{F\}\}\\times\\mathcal\{L\}\_\{\\mathrm\{mix\}\}is arbitrary and fixed, with0<βℓ⋆<\+∞0<\\beta\_\{\\ell\}^\{\\star\}<\+\\infty\. The priorπmix\\pi^\{\\mathrm\{mix\}\}assigns mass1/\(\|ℱ\|​\|ℒmix\|\)1/\(\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\|\)to this expert, so

WT\+1,f,ℓmix\|ℱ\|​\|ℒmix\|≤ΦT\+1mix≤3​e\.\\frac\{W^\{\\mathrm\{mix\}\}\_\{T\+1,f,\\ell\}\}\{\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\|\}\\leq\\Phi\_\{T\+1\}^\{\\mathrm\{mix\}\}\\leq 3e\.Taking logarithms and expanding the update in[Algorithm3](https://arxiv.org/html/2610.41023#alg3)gives

∑t=1Tzt,f,ℓmix≤1\+log⁡\(3​\|ℱ\|​\|ℒmix\|\)βℓ⋆\.\\sum\_\{t=1\}^\{T\}z^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\\leq\\frac\{1\+\\log\(3\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\|\)\}\{\\beta\_\{\\ell\}^\{\\star\}\}\.By the definition ofzt,f,ℓmixz^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}and[Lemma3\.7](https://arxiv.org/html/2610.41023#S3.Thmtheorem7),

∑t𝔼P∼Dt​\[ℓ⁡\(P,yt\)−ℓ⁡\(f⁡\(xt\),yt\)\]≤∑tzt,f,ℓmix\+2​T​δβℓ⋆≤1\+log⁡\(3​\|ℒmix\|​\|ℱ\|\)\+2​T​δβℓ⋆\.\\displaystyle\\sum\_\{t\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\ell\(P,y\_\{t\}\)\-\\ell\(f\(x\_\{t\}\),y\_\{t\}\)\\right\]\\leq\\sum\_\{t\}z^\{\\mathrm\{mix\}\}\_\{t,f,\\ell\}\+\\frac\{2T\\delta\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\leq\\frac\{1\+\\log\(3\|\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\|\\,\|\{\\mathcal\{F\}\}\|\)\+2T\\delta\}\{\\beta\_\{\\ell\}^\{\\star\}\}\.The use ofβℓ⋆\\beta\_\{\\ell\}^\{\\star\}in the update and clipping bound is justified by[Lemma3\.8](https://arxiv.org/html/2610.41023#S3.Thmtheorem8), which includes the finite optimal constant\. SinceT​δ≤1/2T\\delta\\leq 1/2, taking the best benchmark proves[Eq\. \(2\)](https://arxiv.org/html/2610.41023#S1.E2)\.

For the universal mixable component,ℓ\\ellis an arbitrary mixable proper loss andf∈ℱf\\in\{\\mathcal\{F\}\}is arbitrary and fixed\. The caseβℓ⋆=\+∞\\beta\_\{\\ell\}^\{\\star\}=\+\\inftyagain gives zero regret\. Otherwise, we takeβ=βℓ⋆\\beta=\\beta\_\{\\ell\}^\{\\star\}, which is allowed by the mixability characterization in[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)\. The curvature mass ismδ=∫Iδwℓ​\(v\)​𝑑v≤Mδ/βm\_\{\\delta\}=\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\,\\mathrm\{d\}v\\leq M\_\{\\delta\}/\\beta\. Whenmδ\>0m\_\{\\delta\}\>0, we use the distributionQ=δf⊗νδQ=\\delta\_\{f\}\\otimes\\nu\_\{\\delta\}onℱ×Iδ\{\\mathcal\{F\}\}\\times I\_\{\\delta\}, where

νδ​\(d​v\)=wℓ​\(v\)mδ​d​v\(v∈Iδ\)\.\\nu\_\{\\delta\}\(\\,\\mathrm\{d\}v\)=\\frac\{w\_\{\\ell\}\(v\)\}\{m\_\{\\delta\}\}\\,\\mathrm\{d\}v\\quad\(v\\in I\_\{\\delta\}\)\.As in the proof of[Lemma3\.5](https://arxiv.org/html/2610.41023#S3.Thmtheorem5), the curvature bound andπMIX=πδ\\pi^\{\\mathrm\{MIX\}\}=\\pi\_\{\\delta\}yield

KL\(Q∥πMIX\)=log\|ℱ\|\+KL\(νδ∥μδ\)≤log\|ℱ\|\+logMδβ​mδ\.\{\\text\{\\rm KL\}\}\(Q\\\|\\pi^\{\\mathrm\{MIX\}\}\)=\\log\|\{\\mathcal\{F\}\}\|\+\{\\text\{\\rm KL\}\}\(\\nu\_\{\\delta\}\\\|\\mu\_\{\\delta\}\)\\leq\\log\|\{\\mathcal\{F\}\}\|\+\\log\\frac\{M\_\{\\delta\}\}\{\\beta m\_\{\\delta\}\}\.The argument of[Eq\. \(19\)](https://arxiv.org/html/2610.41023#S3.E19), applied toWT\+1,fMIX​\(v\)W^\{\\mathrm\{MIX\}\}\_\{T\+1,f\}\(v\)and its component prior, gives

∫Iδwℓ​\(v\)​log⁡WT\+1,fMIX​\(v\)​𝑑v\\displaystyle\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\log W^\{\\mathrm\{MIX\}\}\_\{T\+1,f\}\(v\)\\,\\mathrm\{d\}v≤mδ\[KL\(Q∥πMIX\)\+logΦT\+1MIX\]\\displaystyle\\leq m\_\{\\delta\}\\left\[\{\\text\{\\rm KL\}\}\(Q\\\|\\pi^\{\\mathrm\{MIX\}\}\)\+\\log\\Phi\_\{T\+1\}^\{\\mathrm\{MIX\}\}\\right\]≤mδ​\[1\+log⁡\(3​\|ℱ\|\)\+log⁡Mδβ​mδ\]\\displaystyle\\leq m\_\{\\delta\}\\left\[1\+\\log\(3\|\{\\mathcal\{F\}\}\|\)\+\\log\\frac\{M\_\{\\delta\}\}\{\\beta m\_\{\\delta\}\}\\right\]≤Mδβ​\(1\+log⁡\(3​\|ℱ\|\)\+1/e\)\.\\displaystyle\\leq\\frac\{M\_\{\\delta\}\}\{\\beta\}\(1\+\\log\(3\|\{\\mathcal\{F\}\}\|\)\+1/e\)\.The second inequality usesΦT\+1MIX≤3​e\\Phi\_\{T\+1\}^\{\\mathrm\{MIX\}\}\\leq 3e\. The last usesmδ≤Mδ/βm\_\{\\delta\}\\leq M\_\{\\delta\}/\\betaandu​log⁡\(1/u\)≤1/eu\\log\(1/u\)\\leq 1/eforu=β​mδ/Mδ∈\(0,1\]u=\\beta m\_\{\\delta\}/M\_\{\\delta\}\\in\(0,1\]\. Expanding the log update in[Algorithm3](https://arxiv.org/html/2610.41023#alg3)gives

∑t∫Iδwℓ​\(v\)​zt,f,vMIX​𝑑v≤\\displaystyle\\sum\_\{t\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)z^\{\\mathrm\{MIX\}\}\_\{t,f,v\}\\,\\mathrm\{d\}v\\leq\{\}Mδ​\(1\+log⁡\(3​\|ℱ\|\)\+1/e\)β​ηMIX\\displaystyle\\frac\{M\_\{\\delta\}\(1\+\\log\(3\|\{\\mathcal\{F\}\}\|\)\+1/e\)\}\{\\beta\\eta^\{\\mathrm\{MIX\}\}\}\+ηMIX∑t𝔼P∼Dt∫Iδwℓ\(v\)P\(1−P\)bv\(P,f\(xt\)\)2dv\.\\displaystyle\+\\eta^\{\\mathrm\{MIX\}\}\\sum\_\{t\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\\,\\mathrm\{d\}v\.Whenmδ=0m\_\{\\delta\}=0, the threshold integrals vanish, so this bound still holds\. By the definition ofzt,f,vMIXz^\{\\mathrm\{MIX\}\}\_\{t,f,v\},[Eq\. \(26\)](https://arxiv.org/html/2610.41023#S3.E26)and[Lemma3\.6](https://arxiv.org/html/2610.41023#S3.Thmtheorem6)then yield

∑t𝔼P∼Dt​\[ℓ⁡\(P,yt\)−ℓ⁡\(f⁡\(xt\),yt\)\]\\displaystyle\\sum\_\{t\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\ell\(P,y\_\{t\}\)\-\\ell\(f\(x\_\{t\}\),y\_\{t\}\)\\right\]≤∑t∫Iδwℓ​\(v\)​zt,f,vMIX​𝑑v−∑t𝔼P∼Dt​∫Iδwℓ​\(v\)​dv​\(P,f⁡\(xt\)\)​𝑑v\+2​T​δβ\\displaystyle\\leq\\sum\_\{t\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)z^\{\\mathrm\{MIX\}\}\_\{t,f,v\}\\,\\mathrm\{d\}v\-\\sum\_\{t\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)d\_\{v\}\(P,f\(x\_\{t\}\)\)\\,\\mathrm\{d\}v\+\\frac\{2T\\delta\}\{\\beta\}≤Mδ​\(1\+log⁡\(3​\|ℱ\|\)\+1/e\)β​ηMIX\\displaystyle\\leq\\frac\{M\_\{\\delta\}\(1\+\\log\(3\|\{\\mathcal\{F\}\}\|\)\+1/e\)\}\{\\beta\\eta^\{\\mathrm\{MIX\}\}\}\+∑t𝔼P∼Dt∫Iδwℓ\(v\)\[ηMIXP\(1−P\)bv\(P,f\(xt\)\)2−dv\(P,f\(xt\)\)\]dv\+2​T​δβ\\displaystyle\\quad\+\\sum\_\{t\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\left\[\\eta^\{\\mathrm\{MIX\}\}P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\-d\_\{v\}\(P,f\(x\_\{t\}\)\)\\right\]\\,\\mathrm\{d\}v\+\\frac\{2T\\delta\}\{\\beta\}≤Mδ​\(1\+log⁡\(3​\|ℱ\|\)\+1/e\)β​ηMIX\+T​\(ηMIX\)2\+2​T​δβ\.\\displaystyle\\leq\\frac\{M\_\{\\delta\}\(1\+\\log\(3\|\{\\mathcal\{F\}\}\|\)\+1/e\)\}\{\\beta\\eta^\{\\mathrm\{MIX\}\}\}\+\\frac\{T\(\\eta^\{\\mathrm\{MIX\}\}\)^\{2\}\+2T\\delta\}\{\\beta\}\.WithηMIX\\eta^\{\\mathrm\{MIX\}\}chosen as in[Eq\. \(32\)](https://arxiv.org/html/2610.41023#A1.E32), comparison with the bestf∈ℱf\\in\{\\mathcal\{F\}\}gives

Regℓ,ℱ​\(T\)≤𝒪⁡\(T1/3​\(Mδ​log⁡\(\|ℱ\|\)\)2/3\+Mδ​log⁡\(\|ℱ\|\)βℓ⋆\),\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)\\leq\\mathcal\{O\}\\left\(\\frac\{T^\{1/3\}\(M\_\{\\delta\}\\log\(\|\{\\mathcal\{F\}\}\|\)\)^\{2/3\}\+M\_\{\\delta\}\\log\(\|\{\\mathcal\{F\}\}\|\)\}\{\\beta\_\{\\ell\}^\{\\star\}\}\\right\),which proves[Eq\. \(3\)](https://arxiv.org/html/2610.41023#S1.E3)\. For the bounded component,\(f,ℓ\)∈ℱ×ℒbd\(f,\\ell\)\\in\{\\mathcal\{F\}\}\\times\\mathcal\{L\}\_\{\\mathrm\{bd\}\}is arbitrary and fixed\. Its prior mass underπbd\\pi^\{\\mathrm\{bd\}\}is1/\(\|ℱ\|​\|ℒbd\|\)1/\(\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\), so

WT\+1,f,ℓbd\|ℱ\|​\|ℒbd\|≤ΦT\+1bd≤3​e\.\\frac\{W^\{\\mathrm\{bd\}\}\_\{T\+1,f,\\ell\}\}\{\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\}\\leq\\Phi\_\{T\+1\}^\{\\mathrm\{bd\}\}\\leq 3e\.Taking logarithms and expanding the update gives

∑tzt,f,ℓbd≤1\+log⁡\(3​\|ℱ\|​\|ℒbd\|\)ηbd\+ηbd​∑t\(zt,f,ℓbd\)2\.\\sum\_\{t\}z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\\leq\\frac\{1\+\\log\(3\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\)\}\{\\eta^\{\\mathrm\{bd\}\}\}\+\\eta^\{\\mathrm\{bd\}\}\\sum\_\{t\}\(z^\{\\mathrm\{bd\}\}\_\{t,f,\\ell\}\)^\{2\}\.The learning rate is chosen as

ηbd=min⁡\{14,log⁡\(3​e​\|ℒbd\|​\|ℱ\|\)4​T\}\.\\eta^\{\\mathrm\{bd\}\}=\\min\\left\\\{\\frac\{1\}\{4\},\\sqrt\{\\frac\{\\log\(3e\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\\,\|\{\\mathcal\{F\}\}\|\)\}\{4T\}\}\\right\\\}\.\(33\)Together with\|zt,f,ℓbd\|≤2\|z\_\{t,f,\\ell\}^\{\\mathrm\{bd\}\}\|\\leq 2, the preceding comparison yields

∑tzt,f,ℓbd≤min⁡\{2​T,1\+log⁡\(3​\|ℒbd\|​\|ℱ\|\)ηbd\+4​T​ηbd\}≤C​T​log⁡\(e​\|ℒbd\|​\|ℱ\|\)\.\\sum\_\{t\}z\_\{t,f,\\ell\}^\{\\mathrm\{bd\}\}\\leq\\min\\left\\\{2T,\\frac\{1\+\\log\(3\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\\,\|\{\\mathcal\{F\}\}\|\)\}\{\\eta^\{\\mathrm\{bd\}\}\}\+4T\\eta^\{\\mathrm\{bd\}\}\\right\\\}\\leq C\\sqrt\{T\\log\(e\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\\,\|\{\\mathcal\{F\}\}\|\)\}\.\(34\)The last inequality uses[Eq\. \(33\)](https://arxiv.org/html/2610.41023#A1.E33), with the trivial bound2​T2Twhen the logarithm is comparable toTTor larger\. By[Eq\. \(6\)](https://arxiv.org/html/2610.41023#S1.E6), this also bounds the cumulative loss comparison with everyf∈ℱf\\in\{\\mathcal\{F\}\}, proving[Eq\. \(4\)](https://arxiv.org/html/2610.41023#S1.E4)\. All three guarantees hold simultaneously for the common sequence of forecast distributions\. ∎

### A\.5Consequences for the five stated loss families

See[1\.5](https://arxiv.org/html/2610.41023#S1.Thmtheorem5)

###### Proof of[Corollary1\.5](https://arxiv.org/html/2610.41023#S1.Thmtheorem5)\.

A single choice of the finite input families suffices\. The finite mixable family contains squared loss and log loss, whose optimal mixability constants are22and11, respectively\. The bounded family contains the00\-11loss in the statement, all the auxiliary losses in[Eq\. \(35\)](https://arxiv.org/html/2610.41023#A4.E35), and their action\-reversed versions

ℓk,j−\(q,0\)=θj𝕀\{q<k/T\},ℓk,j−\(q,1\)=\(1−θj\)𝕀\{q≥k/T\}\.\\ell^\{\-\}\_\{k,j\}\(q,0\)=\\theta\_\{j\}\\mathbb\{I\}\{\\left\\\{\{q<k/T\}\\right\\\}\},\\qquad\\ell^\{\-\}\_\{k,j\}\(q,1\)=\(1\-\\theta\_\{j\}\)\\mathbb\{I\}\{\\left\\\{\{q\\geq k/T\}\\right\\\}\}\.
There are at most4​T​\(T\+1\)\+14T\(T\+1\)\+1bounded losses\. We restrict the EVI solver in[Algorithm3](https://arxiv.org/html/2610.41023#alg3)to forecast distributions supported onΓT=\{i​δ:1≤i≤2​T\+1\}\\Gamma\_\{T\}=\\\{i\\delta:1\\leq i\\leq 2T\+1\\\}, whereδ=1/\[2​\(T\+1\)\]\\delta=1/\[2\(T\+1\)\]\. This grid contains both endpoints ofIδI\_\{\\delta\}and has adjacent gaps equal toδ\\delta, so[Lemma2\.2](https://arxiv.org/html/2610.41023#S2.Thmtheorem2)supplies the EVI condition with the same tolerance\. The first, second, and fifth claims now follow from[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4): the two mixable losses haveO⁡\(log⁡\(e​\|ℱ\|\)\)O\(\\log\(e\|\{\\mathcal\{F\}\}\|\)\)regret, the universal mixable bound is retained, and the00\-11loss hasO⁡\(T​log⁡\(e​\|ℱ\|​T\)\)O\(\\sqrt\{T\\log\(e\|\{\\mathcal\{F\}\}\|T\)\}\)regret\.

##### The GLM consequence\.

Fix a continuous convex functionAAandf∈ℱf\\in\{\\mathcal\{F\}\}\. By[PropositionD\.1](https://arxiv.org/html/2610.41023#A4.Thmtheorem1), there are coefficientsck,j≥0c\_\{k,j\}\\geq 0with∑k,jck,j=1\\sum\_\{k,j\}c\_\{k,j\}=1such that[Eq\. \(36\)](https://arxiv.org/html/2610.41023#A4.E36)holds for every forecast inΓT\\Gamma\_\{T\}\. Substitutingq=f⁡\(xt\)q=f\(x\_\{t\}\), taking expectations overP∼DtP\\sim D\_\{t\}, and summing over the rounds gives

∑t=1T𝔼P∼Dt​\[ℓA​\(πA​\(P\),yt\)−ℓA​\(f⁡\(xt\),yt\)\]\\displaystyle\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[\\ell\_\{A\}\(\\pi\_\{A\}\(P\),y\_\{t\}\)\-\\ell\_\{A\}\(f\(x\_\{t\}\),y\_\{t\}\)\]≤∑k=1T∑j=12​T\+2ck,j​Regℓk,j,ℱ​\(T\)\+T​δ2\+1\\displaystyle\\leq\\sum\_\{k=1\}^\{T\}\\sum\_\{j=1\}^\{2T\+2\}c\_\{k,j\}\{\\text\{\\rm Reg\}\}\_\{\\ell\_\{k,j\},\{\\mathcal\{F\}\}\}\(T\)\+\\frac\{T\\delta\}\{2\}\+1≤C​T​log⁡\(e​\|ℱ\|​\|ℒbd\|\)\+T​δ2\+1\.\\displaystyle\\leq C\\sqrt\{T\\log\(e\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\)\}\+\\frac\{T\\delta\}\{2\}\+1\.The coefficients are the same in every round, and each cumulative auxiliary loss difference againstffis at most its regret againstℱ\{\\mathcal\{F\}\}\. The last inequality therefore follows from[Theorem1\.4](https://arxiv.org/html/2610.41023#S1.Thmtheorem4)and∑k,jck,j=1\\sum\_\{k,j\}c\_\{k,j\}=1\. SinceT​δ/2\+1<2T\\delta/2\+1<2and\|ℒbd\|≤4​T​\(T\+1\)\+1\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\\leq 4T\(T\+1\)\+1, taking the bestf∈ℱf\\in\{\\mathcal\{F\}\}proves the fourth claim, with regret𝒪⁡\(T​log⁡\(e​\|ℱ\|​T\)\)\\mathcal\{O\}\(\\sqrt\{T\\log\(e\|\{\\mathcal\{F\}\}\|T\)\}\)for every GLM loss\.

It remains to establish the third claim for arbitraryLL\-Lipschitz losses\.

##### The Lipschitz consequence\.

We use the threshold\-basis construction of\([Okoroafor et al\., 2025](https://arxiv.org/html/2610.41023#bib.bib3), Lemma 5\.4\)for the outcome differenceg⁡\(q\)=ℓ⁡\(q,0\)−ℓ⁡\(q,1\)=−Δℓ​\(q\)g\(q\)=\\ell\(q,0\)\-\\ell\(q,1\)=\-\\Delta\_\{\\ell\}\(q\), and express the resulting approximation using the auxiliary losses above\. Both partial losses areLL\-Lipschitz, soggis absolutely continuous andV=∫01\|g′​\(s\)\|​𝑑s≤2​LV=\\int\_\{0\}^\{1\}\|g^\{\\prime\}\(s\)\|\\,\\mathrm\{d\}s\\leq 2L\. IfV=0V=0, all reports have the same outcome difference, and a Bayes action minimizes both partial losses, giving nonpositive regret\. ForV\>0V\>0, we write

g¯=g⁡\(0\)−∫01\[−g′​\(s\)\]\+​𝑑s,ψ⁡\(p\)=g​\(πℓ​\(p\)\)−g¯V\.\\underline\{g\}=g\(0\)\-\\int\_\{0\}^\{1\}\[\-g^\{\\prime\}\(s\)\]\_\{\+\}\\,\\mathrm\{d\}s,\\qquad\\psi\(p\)=\\frac\{g\(\\pi\_\{\\ell\}\(p\)\)\-\\underline\{g\}\}\{V\}\.These definitions giveg¯≤g⁡\(q\)≤g¯\+V\\underline\{g\}\\leq g\(q\)\\leq\\underline\{g\}\+Vfor allq∈\[0,1\]q\\in\[0,1\], so0≤ψ≤10\\leq\\psi\\leq 1\. Bayes optimality implies thatg​\(πℓ​\(p\)\)g\(\\pi\_\{\\ell\}\(p\)\)is nondecreasing inpp\. Indeed, adding the optimality inequalities atp<p′p<p^\{\\prime\}gives

\(p′−p\)​\[g⁡\(πℓ​\(p′\)\)−g⁡\(πℓ​\(p\)\)\]≥0\.\(p^\{\\prime\}\-p\)\\left\[g\(\\pi\_\{\\ell\}\(p^\{\\prime\}\)\)\-g\(\\pi\_\{\\ell\}\(p\)\)\\right\]\\geq 0\.Thusψ\\psiis also nondecreasing\. LetJ=2​T\+1J=2T\+1,pi=i​δp\_\{i\}=i\\delta, and letai=ψ⁡\(pi\)a\_\{i\}=\\psi\(p\_\{i\}\)for1≤i≤J1\\leq i\\leq J, witha0=0a\_\{0\}=0andaJ\+1=1a\_\{J\+1\}=1\. The nonnegative numbersωj=aj−aj−1\\omega\_\{j\}=a\_\{j\}\-a\_\{j\-1\}sum to one and satisfyψ\(p\)=∑j=1J\+1ωj𝕀\{p\>θj\}\\psi\(p\)=\\sum\_\{j=1\}^\{J\+1\}\\omega\_\{j\}\\mathbb\{I\}\{\\left\\\{\{p\>\\theta\_\{j\}\}\\right\\\}\}forp∈ΓTp\\in\\Gamma\_\{T\}\. For1≤k≤T1\\leq k\\leq T, let

dk\+=∫\(k−1\)/Tk/T\[g′​\(s\)\]\+​𝑑s,dk−=∫\(k−1\)/Tk/T\[−g′​\(s\)\]\+​𝑑s\.d\_\{k\}^\{\+\}=\\int\_\{\(k\-1\)/T\}^\{k/T\}\[g^\{\\prime\}\(s\)\]\_\{\+\}\\,\\mathrm\{d\}s,\\qquad d\_\{k\}^\{\-\}=\\int\_\{\(k\-1\)/T\}^\{k/T\}\[\-g^\{\\prime\}\(s\)\]\_\{\+\}\\,\\mathrm\{d\}s\.Their total mass isVV\. In these two indicator orientations, the threshold approximation on the mesh1/T1/Tis

\|g\(q\)−g¯−∑k=1T\[dk\+𝕀\{q≥k/T\}\+dk−𝕀\{q<k/T\}\]\|≤2​LT\(q∈\[0,1\]\)\.\\left\|g\(q\)\-\\underline\{g\}\-\\sum\_\{k=1\}^\{T\}\\left\[d\_\{k\}^\{\+\}\\mathbb\{I\}\{\\left\\\{\{q\\geq k/T\}\\right\\\}\}\+d\_\{k\}^\{\-\}\\mathbb\{I\}\{\\left\\\{\{q<k/T\}\\right\\\}\}\\right\]\\right\|\\leq\\frac\{2L\}\{T\}\\qquad\(q\\in\[0,1\]\)\.Withℓk,j\+=ℓk,j\\ell^\{\+\}\_\{k,j\}=\\ell\_\{k,j\}for the original auxiliary loss, the two orientations have coefficients

bℓk,j\+\(p,q\)=𝕀\{q≥k/T\}−𝕀\{p\>θj\},bℓk,j−\(p,q\)=𝕀\{q<k/T\}−𝕀\{p\>θj\}b\_\{\\ell^\{\+\}\_\{k,j\}\}\(p,q\)=\\mathbb\{I\}\{\\left\\\{\{q\\geq k/T\}\\right\\\}\}\-\\mathbb\{I\}\{\\left\\\{\{p\>\\theta\_\{j\}\}\\right\\\}\},\\qquad b\_\{\\ell^\{\-\}\_\{k,j\}\}\(p,q\)=\\mathbb\{I\}\{\\left\\\{\{q<k/T\}\\right\\\}\}\-\\mathbb\{I\}\{\\left\\\{\{p\>\\theta\_\{j\}\}\\right\\\}\}on the forecast grid\. Sincebℓ​\(p,q\)=g⁡\(q\)−g⁡\(πℓ​\(p\)\)b\_\{\\ell\}\(p,q\)=g\(q\)\-g\(\\pi\_\{\\ell\}\(p\)\), we obtain

\|bℓ​\(p,q\)−∑k=1T∑j=1J\+1ωj​\[dk\+​bℓk,j\+​\(p,q\)\+dk−​bℓk,j−​\(p,q\)\]\|≤2​LT\.\\left\|b\_\{\\ell\}\(p,q\)\-\\sum\_\{k=1\}^\{T\}\\sum\_\{j=1\}^\{J\+1\}\\omega\_\{j\}\\left\[d\_\{k\}^\{\+\}b\_\{\\ell^\{\+\}\_\{k,j\}\}\(p,q\)\+d\_\{k\}^\{\-\}b\_\{\\ell^\{\-\}\_\{k,j\}\}\(p,q\)\\right\]\\right\|\\leq\\frac\{2L\}\{T\}\.All coefficients are nonnegative, independent of the round, and have total massV≤2​LV\\leq 2L\. The bounded\-component estimate[Eq\. \(34\)](https://arxiv.org/html/2610.41023#A1.E34)controls each auxiliary correlation byC​T​log⁡\(e​\|ℱ\|​\|ℒbd\|\)C\\sqrt\{T\\log\(e\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\)\}\. After multiplication byyt−Py\_\{t\}\-P, summation, and forecast expectation, the approximation costs at most2​L2L\. The loss comparison in[Eq\. \(6\)](https://arxiv.org/html/2610.41023#S1.E6)therefore yields

Regℓ,ℱ​\(T\)≤2​C​L​T​log⁡\(e​\|ℱ\|​\|ℒbd\|\)\+2​L=O⁡\(L​T​log⁡\(e​\|ℱ\|​T\)\)\.\{\\text\{\\rm Reg\}\}\_\{\\ell,\{\\mathcal\{F\}\}\}\(T\)\\leq 2CL\\sqrt\{T\\log\(e\|\{\\mathcal\{F\}\}\|\|\\mathcal\{L\}\_\{\\mathrm\{bd\}\}\|\)\}\+2L=O\\left\(L\\sqrt\{T\\log\(e\|\{\\mathcal\{F\}\}\|T\)\}\\right\)\.This proves the remaining claim\. Since the finite families and the forecasts do not depend on the targetLL,ℓ\\ell, or GLM potential, the five guarantees hold simultaneously\. ∎

## Appendix BAuxiliary Analysis

###### Proof of[Lemma3\.5](https://arxiv.org/html/2610.41023#S3.Thmtheorem5)\.

The parameters0<η≤1/20<\\eta\\leq 1/2,β\>0\\beta\>0,ℓ∈ℒmix​\(β\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(\\beta\\right\), andf∈ℱf\\in\{\\mathcal\{F\}\}are arbitrary and fixed\. By the definition ofzt,f,vz\_\{t,f,v\}, the weight update in[Algorithm2](https://arxiv.org/html/2610.41023#alg2)and Jensen’s inequality give

Wt\+1,f​\(v\)Wt,f​\(v\)\\displaystyle\\frac\{W\_\{t\+1,f\}\(v\)\}\{W\_\{t,f\}\(v\)\}=exp⁡\(𝔼P∼Dt​\[η​bv​\(P,f⁡\(xt\)\)​\(yt−P\)−η2​P​\(1−P\)​bv​\(P,f⁡\(xt\)\)2\]\)\\displaystyle=\\exp\\left\(\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[\\eta b\_\{v\}\(P,f\(x\_\{t\}\)\)\(y\_\{t\}\-P\)\-\\eta^\{2\}P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\\right\]\\right\)≤𝔼P∼Dt​\[FP,ytη​\(bv​\(P,f⁡\(xt\)\)\)\]\.\\displaystyle\\leq\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\left\[F^\{\\eta\}\_\{P,y\_\{t\}\}\(b\_\{v\}\(P,f\(x\_\{t\}\)\)\)\\right\]\.By[Lemma3\.4](https://arxiv.org/html/2610.41023#S3.Thmtheorem4)and[Eq\. \(24\)](https://arxiv.org/html/2610.41023#S3.E24),

0<Fp,yη​\(bv​\(p,q\)\)≤1\+b~vη​\(p,q\)​\(y−p\)\.0<F^\{\\eta\}\_\{p,y\}\(b\_\{v\}\(p,q\)\)\\leq 1\+\\widetilde\{b\}^\{\\eta\}\_\{v\}\(p,q\)\(y\-p\)\.Averaging with the normalized weights in[Eq\. \(25\)](https://arxiv.org/html/2610.41023#S3.E25)shows that1\+Gt​\(p\)​\(y−p\)≥01\+G\_\{t\}\(p\)\(y\-p\)\\geq 0\. Sinceδ/\(1−δ\)=1/\(2​T\+1\)<ϵt=1/T\\delta/\(1\-\\delta\)=1/\(2T\+1\)<\\epsilon\_\{t\}=1/T,[Lemma2\.2](https://arxiv.org/html/2610.41023#S2.Thmtheorem2)supplies the required forecast distribution\. Applying the same factor bound to the update inequality above, multiplying byWt,f​\(v\)W\_\{t,f\}\(v\), and averaging overπδ\\pi\_\{\\delta\}therefore gives, by the definition ofGtG\_\{t\}and the EVI condition,

Φt\+1≤Φt​\(1\+𝔼P∼Dt​\[Gt​\(P\)​\(yt−P\)\]\)≤\(1\+ϵt\)​Φt\.\\Phi\_\{t\+1\}\\leq\\Phi\_\{t\}\\left\(1\+\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\[G\_\{t\}\(P\)\(y\_\{t\}\-P\)\]\\right\)\\leq\(1\+\\epsilon\_\{t\}\)\\Phi\_\{t\}\.SinceΦ1=1\\Phi\_\{1\}=1and∑t=1Tϵt=1\\sum\_\{t=1\}^\{T\}\\epsilon\_\{t\}=1, iteration givesΦT\+1≤∏t=1T\(1\+ϵt\)≤e\\Phi\_\{T\+1\}\\leq\\prod\_\{t=1\}^\{T\}\(1\+\\epsilon\_\{t\}\)\\leq e\.

The curvature bound givesm=∫Iδwℓ​\(v\)​𝑑v≤Mδ/βm=\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\,\\mathrm\{d\}v\\leq M\_\{\\delta\}/\\beta\. Ifm=0m=0, both threshold integrals in the claimed inequality vanish\. Form\>0m\>0, we useQ=δf⊗νℓQ=\\delta\_\{f\}\\otimes\\nu\_\{\\ell\}, whereνℓ​\(d​v\)=wℓ​\(v\)​d​v/m\\nu\_\{\\ell\}\(\\,\\mathrm\{d\}v\)=w\_\{\\ell\}\(v\)\\,\\mathrm\{d\}v/m, so that averaging overQQgives the desired weighting bywℓw\_\{\\ell\}\. The prior comparison in[Eq\. \(19\)](https://arxiv.org/html/2610.41023#S3.E19), applied withπδ\\pi\_\{\\delta\}, yields

∫Iδwℓ\(v\)logWT\+1,f\(v\)dv=m𝔼Q\[logWT\+1\]≤m\(KL\(Q∥πδ\)\+1\)\.\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\log W\_\{T\+1,f\}\(v\)\\,\\mathrm\{d\}v=m\\mathbb\{E\}\_\{Q\}\[\\log W\_\{T\+1\}\]\\leq m\\left\(\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{\\delta\}\)\+1\\right\)\.Because the weights start at one, their logarithms are the sums of the update exponents\. Substitution into this inequality gives

η∑t=1T∫Iδwℓ\(v\)zt,f,vdv−η2∑t=1T𝔼P∼Dt∫Iδwℓ\(v\)P\(1−P\)bv\(P,f\(xt\)\)2dv≤m\(KL\(Q∥πδ\)\+1\)\.\\displaystyle\\eta\\sum\_\{t=1\}^\{T\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)z\_\{t,f,v\}\\,\\mathrm\{d\}v\-\\eta^\{2\}\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{P\\sim D\_\{t\}\}\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)P\(1\-P\)b\_\{v\}\(P,f\(x\_\{t\}\)\)^\{2\}\\,\\mathrm\{d\}v\\leq m\\left\(\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{\\delta\}\)\+1\\right\)\.After division byη\\eta, the second term gives the variance correction in the statement\. It remains to bound the right\-hand side\.

The priorπδ=Unif⁡\(ℱ\)⊗μδ\\pi\_\{\\delta\}=\\operatorname\{Unif\}\(\{\\mathcal\{F\}\}\)\\otimes\\mu\_\{\\delta\}assigns mass1/\|ℱ\|1/\|\{\\mathcal\{F\}\}\|to the fixed predictorff, andμδ​\(d​v\)=d​v/\[Mδ​v​\(1−v\)\]\\mu\_\{\\delta\}\(\\,\\mathrm\{d\}v\)=\\,\\mathrm\{d\}v/\[M\_\{\\delta\}v\(1\-v\)\]\. Thus

KL\(Q∥πδ\)\\displaystyle\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{\\delta\}\)=log\|ℱ\|\+KL\(νℓ∥μδ\)\\displaystyle=\\log\|\{\\mathcal\{F\}\}\|\+\{\\text\{\\rm KL\}\}\(\\nu\_\{\\ell\}\\\|\\mu\_\{\\delta\}\)=log⁡\|ℱ\|\+∫Iδlog⁡Mδ​v​\(1−v\)​wℓ​\(v\)m​νℓ​\(dv\)\\displaystyle=\\log\|\{\\mathcal\{F\}\}\|\+\\int\_\{I\_\{\\delta\}\}\\log\\frac\{M\_\{\\delta\}v\(1\-v\)w\_\{\\ell\}\(v\)\}\{m\}\\,\\nu\_\{\\ell\}\(\\,\\mathrm\{d\}v\)≤log⁡\|ℱ\|\+log⁡Mδβ​m,\\displaystyle\\leq\\log\|\{\\mathcal\{F\}\}\|\+\\log\\frac\{M\_\{\\delta\}\}\{\\beta m\},where the last inequality useswℓ​\(v\)≤1/\[β​v​\(1−v\)\]w\_\{\\ell\}\(v\)\\leq 1/\[\\beta v\(1\-v\)\]\. Withs=β​m/Mδ∈\(0,1\]s=\\beta m/M\_\{\\delta\}\\in\(0,1\], we obtain

m\(KL\(Q∥πδ\)\+1\)\\displaystyle m\\left\(\{\\text\{\\rm KL\}\}\(Q\\\|\\pi\_\{\\delta\}\)\+1\\right\)≤m​log⁡\(e​\|ℱ\|\)\+m​log⁡Mδβ​m\\displaystyle\\leq m\\log\(e\|\{\\mathcal\{F\}\}\|\)\+m\\log\\frac\{M\_\{\\delta\}\}\{\\beta m\}=Mδβ​\[s​log⁡\(e​\|ℱ\|\)\+s​log⁡\(1/s\)\]\\displaystyle=\\frac\{M\_\{\\delta\}\}\{\\beta\}\\left\[s\\log\(e\|\{\\mathcal\{F\}\}\|\)\+s\\log\(1/s\)\\right\]≤Mδβ​\[log⁡\(e​\|ℱ\|\)\+1/e\],\\displaystyle\\leq\\frac\{M\_\{\\delta\}\}\{\\beta\}\\left\[\\log\(e\|\{\\mathcal\{F\}\}\|\)\+1/e\\right\],usings≤1s\\leq 1ands​log⁡\(1/s\)≤1/es\\log\(1/s\)\\leq 1/e\. Substitution and division byη\\etagive the comparison bound\. Its complexity term satisfies

Mδ​\[log⁡\(e​\|ℱ\|\)\+1/e\]β​η=𝒪⁡\(log⁡\(T\)​log⁡\(\|ℱ\|\)β​η\),\\frac\{M\_\{\\delta\}\[\\log\(e\|\{\\mathcal\{F\}\}\|\)\+1/e\]\}\{\\beta\\eta\}=\\mathcal\{O\}\\left\(\\frac\{\\log\(T\)\\log\(\|\{\\mathcal\{F\}\}\|\)\}\{\\beta\\eta\}\\right\),sinceMδ=2​log⁡\(2​T\+1\)M\_\{\\delta\}=2\\log\(2T\+1\)\. ForT≥2T\\geq 2and\|ℱ\|≥2\|\{\\mathcal\{F\}\}\|\\geq 2, this is thelog⁡T​log​\|ℱ\|/\(β​η\)\\log T\\log\|\{\\mathcal\{F\}\}\|/\(\\beta\\eta\)dependence stated in the lemma\. ∎

###### Proof of[Lemma3\.6](https://arxiv.org/html/2610.41023#S3.Thmtheorem6)\.

The parametersβ,ℓ,η,p,q\\beta,\\ell,\\eta,p,qare arbitrary and fixed as in the statement\. Withh=η​p​\(1−p\)h=\\eta p\(1\-p\), the positive part is confined to\|v−p\|<h\|v\-p\|<h, wherev≥\(1−η\)​pv\\geq\(1\-\\eta\)pand1−v≥\(1−η\)​\(1−p\)1\-v\\geq\(1\-\\eta\)\(1\-p\)\. Thusv⁡\(1−v\)≥p⁡\(1−p\)/4v\(1\-v\)\\geq p\(1\-p\)/4, and the curvature bound gives

∫Iδwℓ​\(v\)​\[η​p​\(1−p\)​bv​\(p,q\)2−dv​\(p,q\)\]​𝑑v\\displaystyle\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\left\[\\eta p\(1\-p\)b\_\{v\}\(p,q\)^\{2\}\-d\_\{v\}\(p,q\)\\right\]\\,\\mathrm\{d\}v≤4β​p​\(1−p\)​∫−hh\[h−\|s\|\]\+​𝑑s\\displaystyle\\leq\\frac\{4\}\{\\beta p\(1\-p\)\}\\int\_\{\-h\}^\{h\}\[h\-\|s\|\]\_\{\+\}\\,\\mathrm\{d\}s=4​η2​p​\(1−p\)β≤η2β\.∎\\displaystyle=\\frac\{4\\eta^\{2\}p\(1\-p\)\}\{\\beta\}\\leq\\frac\{\\eta^\{2\}\}\{\\beta\}\.\\qed

###### Proof of[Lemma3\.7](https://arxiv.org/html/2610.41023#S3.Thmtheorem7)\.

The parametersβ,ℓ,q,y\\beta,\\ell,q,yare arbitrary and fixed as in the statement, andq¯=clip⁡\(q,Iδ\)\\bar\{q\}=\\operatorname\{clip\}\(q,I\_\{\\delta\}\)\. Ifq∈Iδq\\in I\_\{\\delta\}, clipping leaves the report unchanged, so its cost is zero\. For reports outsideIδI\_\{\\delta\}, the representation in[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)shows thatℓ⁡\(⋅,0\)\\ell\(\\cdot,0\)is nondecreasing andℓ⁡\(⋅,1\)\\ell\(\\cdot,1\)is nonincreasing\.

Whenq<δq<\\delta, we haveq¯=δ\\bar\{q\}=\\delta, and monotonicity givesℓ⁡\(δ,1\)≤ℓ⁡\(q,1\)\\ell\(\\delta,1\)\\leq\\ell\(q,1\)\. To bound the increase for outcome00, applying[Eq\. \(28\)](https://arxiv.org/html/2610.41023#S3.E28)at reportp=δp=\\deltagives

\(1−δ\)​exp⁡\(β⁡\[ℓ⁡\(δ,0\)−ℓ⁡\(q,0\)\]\)\+δ​exp⁡\(β⁡\[ℓ⁡\(δ,1\)−ℓ⁡\(q,1\)\]\)≤1\.\\displaystyle\(1\-\\delta\)\\exp\\left\(\\beta\[\\ell\(\\delta,0\)\-\\ell\(q,0\)\]\\right\)\+\\delta\\exp\\left\(\\beta\[\\ell\(\\delta,1\)\-\\ell\(q,1\)\]\\right\)\\leq 1\.Both terms are nonnegative, so the first is at most one\. Taking logarithms yields

ℓ⁡\(δ,0\)−ℓ⁡\(q,0\)≤−log⁡\(1−δ\)β≤δβ⁡\(1−δ\)≤2​δβ,\\ell\(\\delta,0\)\-\\ell\(q,0\)\\leq\-\\frac\{\\log\(1\-\\delta\)\}\{\\beta\}\\leq\\frac\{\\delta\}\{\\beta\(1\-\\delta\)\}\\leq\\frac\{2\\delta\}\{\\beta\},where the last inequality usesδ<1/2\\delta<1/2\.

Whenq\>1−δq\>1\-\\delta, we haveq¯=1−δ\\bar\{q\}=1\-\\deltaandℓ⁡\(1−δ,0\)≤ℓ⁡\(q,0\)\\ell\(1\-\\delta,0\)\\leq\\ell\(q,0\)\. Applying[Eq\. \(28\)](https://arxiv.org/html/2610.41023#S3.E28)atp=1−δp=1\-\\deltanow bounds its outcome\-11term by one, giving\(1−δ\)​exp⁡\(β⁡\[ℓ⁡\(1−δ,1\)−ℓ⁡\(q,1\)\]\)≤1\(1\-\\delta\)\\exp\\left\(\\beta\[\\ell\(1\-\\delta,1\)\-\\ell\(q,1\)\]\\right\)\\leq 1\. The same logarithmic bound therefore controls the increase for outcome11\. Ifℓ⁡\(q,y\)=\+∞\\ell\(q,y\)=\+\\infty, the corresponding loss difference is−∞\-\\infty, so the claimed bound also holds\.

For the second claim,p∈Iδp\\in I\_\{\\delta\}is arbitrary and fixed\. For everyv∈\(δ,1−δ\)v\\in\(\\delta,1\-\\delta\), clipping does not change whetherqqlies belowvv\. Hencebv​\(p,q\)=bv​\(p,q¯\)b\_\{v\}\(p,q\)=b\_\{v\}\(p,\\bar\{q\}\), anddv​\(p,q\)=dv​\(p,q¯\)d\_\{v\}\(p,q\)=d\_\{v\}\(p,\\bar\{q\}\)follows fromdv​\(p,q\)=\|v−p\|​bv​\(p,q\)2d\_\{v\}\(p,q\)=\|v\-p\|b\_\{v\}\(p,q\)^\{2\}\. The possible differences at the two endpoint thresholds do not affect the integrals\. Sincep,q¯∈Iδp,\\bar\{q\}\\in I\_\{\\delta\}, only thresholds inIδI\_\{\\delta\}contribute to their loss difference in[Eq\. \(15\)](https://arxiv.org/html/2610.41023#S2.E15)\. Combining this representation with the clipping bound gives

ℓ⁡\(p,y\)−ℓ⁡\(q,y\)\\displaystyle\\ell\(p,y\)\-\\ell\(q,y\)=ℓ⁡\(p,y\)−ℓ⁡\(q¯,y\)\+ℓ⁡\(q¯,y\)−ℓ⁡\(q,y\)\\displaystyle=\\ell\(p,y\)\-\\ell\(\\bar\{q\},y\)\+\\ell\(\\bar\{q\},y\)\-\\ell\(q,y\)≤∫Iδwℓ​\(v\)​\[bv​\(p,q\)​\(y−p\)−dv​\(p,q\)\]​dv\+2​δβ,\\displaystyle\\leq\\int\_\{I\_\{\\delta\}\}w\_\{\\ell\}\(v\)\\left\[b\_\{v\}\(p,q\)\(y\-p\)\-d\_\{v\}\(p,q\)\\right\]\\,\\mathrm\{d\}v\+\\frac\{2\\delta\}\{\\beta\},which proves[Eq\. \(26\)](https://arxiv.org/html/2610.41023#S3.E26)\.

Finally, the corrected test in[Eq\. \(24\)](https://arxiv.org/html/2610.41023#S3.E24)depends onqqonly throughbv​\(p,q\)b\_\{v\}\(p,q\), and the weight update in[Algorithm2](https://arxiv.org/html/2610.41023#alg2)uses onlybv​\(p,q\)b\_\{v\}\(p,q\)and its square\. Thus clipping the benchmark reports leaves the weight densities unchanged almost everywhere and the EVI tests unchanged\. ∎

## Appendix CPreliminary Facts

###### Proof of[Lemma2\.1](https://arxiv.org/html/2610.41023#S2.Thmtheorem1)\.

The representation in[Eq\. \(13\)](https://arxiv.org/html/2610.41023#S2.E13)\.Letℓ\\ellbe a Lipschitz or mixable proper loss, and recall thatΔℓ​\(p\)=ℓ⁡\(p,1\)−ℓ⁡\(p,0\)\\Delta\_\{\\ell\}\(p\)=\\ell\(p,1\)\-\\ell\(p,0\)\. We first show thatΔℓ\\Delta\_\{\\ell\}is Lipschitz on every closed subinterval of\(0,1\)\(0,1\), which will allow us to construct the density and recover the partial losses by integration\. Properness atppandqqgives, for all0<p<q<10<p<q<1,

p⁡\[Δℓ​\(p\)−Δℓ​\(q\)\]≤ℓ⁡\(q,0\)−ℓ⁡\(p,0\)≤q⁡\[Δℓ​\(p\)−Δℓ​\(q\)\]\.p\\left\[\\Delta\_\{\\ell\}\(p\)\-\\Delta\_\{\\ell\}\(q\)\\right\]\\leq\\ell\(q,0\)\-\\ell\(p,0\)\\leq q\\left\[\\Delta\_\{\\ell\}\(p\)\-\\Delta\_\{\\ell\}\(q\)\\right\]\.HenceΔℓ\\Delta\_\{\\ell\}is nonincreasing\. Ifℓ\\ellisLL\-Lipschitz, its two partial losses areLL\-Lipschitz, so their differenceΔℓ\\Delta\_\{\\ell\}is2​L2L\-Lipschitz\.

Ifℓ∈ℒmix​\(β\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(\\beta\\right\), concavity of the logarithm applied to[Eq\. \(28\)](https://arxiv.org/html/2610.41023#S3.E28), with comparatorrrand weights1−r1\-randrr, gives, for allr,p∈\(0,1\)r,p\\in\(0,1\),

βdℓ\(r,p\)≤KL\(Ber\(r\)∥Ber\(p\)\)\.\\beta d\_\{\\ell\}\(r,p\)\\leq\{\\text\{\\rm KL\}\}\\left\(\\texttt\{Ber\}\(r\)\\\|\\texttt\{Ber\}\(p\)\\right\)\.For all0<p<q<10<p<q<1, the definition ofdℓd\_\{\\ell\}gives

dℓ​\(q,p\)\+dℓ​\(p,q\)=\(q−p\)​\[Δℓ​\(p\)−Δℓ​\(q\)\],d\_\{\\ell\}\(q,p\)\+d\_\{\\ell\}\(p,q\)=\(q\-p\)\\left\[\\Delta\_\{\\ell\}\(p\)\-\\Delta\_\{\\ell\}\(q\)\\right\],since the termsℓ⁡\(p,0\)\\ell\(p,0\)andℓ⁡\(q,0\)\\ell\(q,0\)cancel\. The definition of Bernoulli KL divergence also gives

KL\(Ber\(q\)∥Ber\(p\)\)\+KL\(Ber\(p\)∥Ber\(q\)\)\\displaystyle\{\\text\{\\rm KL\}\}\(\\texttt\{Ber\}\(q\)\\\|\\texttt\{Ber\}\(p\)\)\+\{\\text\{\\rm KL\}\}\(\\texttt\{Ber\}\(p\)\\\|\\texttt\{Ber\}\(q\)\)=\(q−p\)​log⁡q⁡\(1−p\)p⁡\(1−q\)\\displaystyle=\(q\-p\)\\log\\frac\{q\(1\-p\)\}\{p\(1\-q\)\}=\(q−p\)​∫pqd​ss⁡\(1−s\)\.\\displaystyle=\(q\-p\)\\int\_\{p\}^\{q\}\\frac\{\\,\\mathrm\{d\}s\}\{s\(1\-s\)\}\.Adding the preceding KL bounds in the two directions and dividing byβ⁡\(q−p\)\>0\\beta\(q\-p\)\>0therefore yields

0≤Δℓ​\(p\)−Δℓ​\(q\)≤1β​∫pqd​ss⁡\(1−s\)\.0\\leq\\Delta\_\{\\ell\}\(p\)\-\\Delta\_\{\\ell\}\(q\)\\leq\\frac\{1\}\{\\beta\}\\int\_\{p\}^\{q\}\\frac\{\\,\\mathrm\{d\}s\}\{s\(1\-s\)\}\.Since1/\[s⁡\(1−s\)\]1/\[s\(1\-s\)\]is bounded on every closed subinterval of\(0,1\)\(0,1\), this inequality makesΔℓ\\Delta\_\{\\ell\}Lipschitz on each such interval\.

Thus, in either case,Δℓ\\Delta\_\{\\ell\}is differentiable almost everywhere\. We now definewℓ​\(p\):=−Δℓ′​\(p\)w\_\{\\ell\}\(p\):=\-\\Delta\_\{\\ell\}^\{\\prime\}\(p\)wherever the derivative exists, andwℓ​\(p\)=0w\_\{\\ell\}\(p\)=0elsewhere\. This gives a nonnegative measurable function becauseΔℓ\\Delta\_\{\\ell\}is nonincreasing\. On every closed subinterval of\(0,1\)\(0,1\), the Lipschitz bound forΔℓ\\Delta\_\{\\ell\}and the properness inequalities above also show thatℓ⁡\(⋅,0\)\\ell\(\\cdot,0\)is Lipschitz\. The identityℓ⁡\(⋅,1\)=ℓ⁡\(⋅,0\)\+Δℓ\\ell\(\\cdot,1\)=\\ell\(\\cdot,0\)\+\\Delta\_\{\\ell\}gives the same property forℓ⁡\(⋅,1\)\\ell\(\\cdot,1\)\. Dividing the properness inequalities byq−pq\-pand taking the limit gives the first identity below, while differentiating the relation between the partial losses gives the second:

∂pℓ\(p,0\)=pwℓ\(p\),∂pℓ\(p,1\)=−\(1−p\)wℓ\(p\)almost everywhere\.\\partial\_\{p\}\\ell\(p,0\)=pw\_\{\\ell\}\(p\),\\qquad\\partial\_\{p\}\\ell\(p,1\)=\-\(1\-p\)w\_\{\\ell\}\(p\)\\quad\\text\{almost everywhere\}\.On each such interval, Lipschitz continuity ensures that integrating these derivatives recovers the differences of the partial losses\. The derivative signs and the endpoint convention in[Section2](https://arxiv.org/html/2610.41023#S2)imply thatℓ⁡\(0,0\)\\ell\(0,0\)andℓ⁡\(1,1\)\\ell\(1,1\)are finite: each is at most an interior loss value, and the endpoint convention excludes−∞\-\\infty\. Passing to these endpoint limits gives

ℓ⁡\(p,0\)=ℓ⁡\(0,0\)\+∫0pv​wℓ​\(v\)​𝑑v,ℓ⁡\(p,1\)=ℓ⁡\(1,1\)\+∫p1\(1−v\)​wℓ​\(v\)​𝑑v\.\\ell\(p,0\)=\\ell\(0,0\)\+\\int\_\{0\}^\{p\}vw\_\{\\ell\}\(v\)\\,\\mathrm\{d\}v,\\qquad\\ell\(p,1\)=\\ell\(1,1\)\+\\int\_\{p\}^\{1\}\(1\-v\)w\_\{\\ell\}\(v\)\\,\\mathrm\{d\}v\.The equalities extend top=0,1p=0,1by the same endpoint convention, allowing an integral to equal\+∞\+\\infty\. By the definition ofλv\\lambda\_\{v\}, the integral for outcome00is∫01wℓ​\(v\)​λv​\(p,0\)​𝑑v\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\lambda\_\{v\}\(p,0\)\\,\\mathrm\{d\}v, and the integral for outcome11is∫01wℓ​\(v\)​λv​\(p,1\)​𝑑v\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\lambda\_\{v\}\(p,1\)\\,\\mathrm\{d\}v\. Thus, withCℓ​\(y\)=\(1−y\)​ℓ​\(0,0\)\+y​ℓ​\(1,1\)C\_\{\\ell\}\(y\)=\(1\-y\)\\ell\(0,0\)\+y\\ell\(1,1\),

ℓ⁡\(p,y\)=Cℓ​\(y\)\+∫01wℓ​\(v\)​λv​\(p,y\)​𝑑v\.\\ell\(p,y\)=C\_\{\\ell\}\(y\)\+\\int\_\{0\}^\{1\}w\_\{\\ell\}\(v\)\\lambda\_\{v\}\(p,y\)\\,\\mathrm\{d\}v\.This proves[Eq\. \(13\)](https://arxiv.org/html/2610.41023#S2.E13)\.

The Lipschitz equivalence\.For the remaining claims, letℓ\\ellbe any proper loss admitting[Eq\. \(13\)](https://arxiv.org/html/2610.41023#S2.E13)\. Since its partial losses are finite on\(0,1\)\(0,1\), the densitywℓw\_\{\\ell\}is integrable on every closed subinterval of\(0,1\)\(0,1\)\. Differentiating the representation gives∂pℓ⁡\(p,0\)=p​wℓ​\(p\)\\partial\_\{p\}\\ell\(p,0\)=pw\_\{\\ell\}\(p\)and∂pℓ⁡\(p,1\)=−\(1−p\)​wℓ​\(p\)\\partial\_\{p\}\\ell\(p,1\)=\-\(1\-p\)w\_\{\\ell\}\(p\)almost everywhere\. Thuswℓ=−Δℓ′w\_\{\\ell\}=\-\\Delta\_\{\\ell\}^\{\\prime\}almost everywhere\.

Ifℓ∈ℒLip​\(L\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{Lip\}\}\\left\(L\\right\), both partial\-loss derivatives have absolute value at mostLL\. Thusp​wℓ​\(p\)≤Lpw\_\{\\ell\}\(p\)\\leq Land\(1−p\)​wℓ​\(p\)≤L\(1\-p\)w\_\{\\ell\}\(p\)\\leq L, which givewℓ​\(p\)≤L/max⁡\{p,1−p\}w\_\{\\ell\}\(p\)\\leq L/\\max\\\{p,1\-p\\\}almost everywhere\.

Conversely, ifwℓ​\(v\)≤L/max⁡\{v,1−v\}w\_\{\\ell\}\(v\)\\leq L/\\max\\\{v,1\-v\\\}almost everywhere, thenv​wℓ​\(v\)≤Lvw\_\{\\ell\}\(v\)\\leq Land\(1−v\)​wℓ​\(v\)≤L\(1\-v\)w\_\{\\ell\}\(v\)\\leq L\. For all0≤p<q≤10\\leq p<q\\leq 1, the representation gives

0≤ℓ⁡\(q,0\)−ℓ⁡\(p,0\)\\displaystyle 0\\leq\\ell\(q,0\)\-\\ell\(p,0\)=∫pqv​wℓ​\(v\)​dv≤L⁡\(q−p\),\\displaystyle=\\int\_\{p\}^\{q\}vw\_\{\\ell\}\(v\)\\,\\mathrm\{d\}v\\leq L\(q\-p\),0≤ℓ⁡\(p,1\)−ℓ⁡\(q,1\)\\displaystyle 0\\leq\\ell\(p,1\)\-\\ell\(q,1\)=∫pq\(1−v\)​wℓ​\(v\)​dv≤L⁡\(q−p\)\.\\displaystyle=\\int\_\{p\}^\{q\}\(1\-v\)w\_\{\\ell\}\(v\)\\,\\mathrm\{d\}v\\leq L\(q\-p\)\.Thus both partial losses areLL\-Lipschitz\.

The mixability equivalence\.Fixβ\>0\\beta\>0\. Ifℓ∈ℒmix​\(β\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(\\beta\\right\), the bound onΔℓ​\(p\)−Δℓ​\(q\)\\Delta\_\{\\ell\}\(p\)\-\\Delta\_\{\\ell\}\(q\)established in the first part gives

0≤Δℓ​\(p\)−Δℓ​\(q\)q−p≤1β⁡\(q−p\)​∫pqd​ss⁡\(1−s\)\.0\\leq\\frac\{\\Delta\_\{\\ell\}\(p\)\-\\Delta\_\{\\ell\}\(q\)\}\{q\-p\}\\leq\\frac\{1\}\{\\beta\(q\-p\)\}\\int\_\{p\}^\{q\}\\frac\{\\,\\mathrm\{d\}s\}\{s\(1\-s\)\}\.Takingq↓pq\\downarrow p\(i\.e\.,qqapproachesppfrom above\) and usingwℓ=−Δℓ′w\_\{\\ell\}=\-\\Delta\_\{\\ell\}^\{\\prime\}giveswℓ​\(p\)≤1/\[β​p​\(1−p\)\]w\_\{\\ell\}\(p\)\\leq 1/\[\\beta p\(1\-p\)\]almost everywhere\.

Conversely, supposewℓ​\(s\)≤1/\[β​s​\(1−s\)\]w\_\{\\ell\}\(s\)\\leq 1/\[\\beta s\(1\-s\)\]almost everywhere\. For allr,p∈\(0,1\)r,p\\in\(0,1\), the definition ofdℓd\_\{\\ell\}givesdℓ​\(r,p\)=\(1−r\)​\[ℓ⁡\(p,0\)−ℓ⁡\(r,0\)\]\+r⁡\[ℓ⁡\(p,1\)−ℓ⁡\(r,1\)\]d\_\{\\ell\}\(r,p\)=\(1\-r\)\[\\ell\(p,0\)\-\\ell\(r,0\)\]\+r\[\\ell\(p,1\)\-\\ell\(r,1\)\]\. The derivative identities above imply\(1−r\)​∂sℓ⁡\(s,0\)\+r​∂sℓ⁡\(s,1\)=\(s−r\)​wℓ​\(s\)\(1\-r\)\\partial\_\{s\}\\ell\(s,0\)\+r\\partial\_\{s\}\\ell\(s,1\)=\(s\-r\)w\_\{\\ell\}\(s\)almost everywhere\. Integrating fromrrtoppand using the bound onwℓw\_\{\\ell\}therefore yields

βdℓ\(r,p\)=β∫rp\(s−r\)wℓ\(s\)ds≤∫rps−rs⁡\(1−s\)ds=KL\(Ber\(r\)∥Ber\(p\)\)\.\\beta d\_\{\\ell\}\(r,p\)=\\beta\\int\_\{r\}^\{p\}\(s\-r\)w\_\{\\ell\}\(s\)\\,\\mathrm\{d\}s\\leq\\int\_\{r\}^\{p\}\\frac\{s\-r\}\{s\(1\-s\)\}\\,\\mathrm\{d\}s=\{\\text\{\\rm KL\}\}\\left\(\\texttt\{Ber\}\(r\)\\\|\\texttt\{Ber\}\(p\)\\right\)\.Consequently, for allp,q∈\(0,1\)p,q\\in\(0,1\),

log𝔼Y∼Ber​\(p\)exp\(β\[ℓ\(p,Y\)−ℓ\(q,Y\)\]\)=supr∈\(0,1\)\{β\[dℓ\(r,p\)−dℓ\(r,q\)\]−KL\(Ber\(r\)∥Ber\(p\)\)\}≤0,\\log\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(p\)\}\\exp\\left\(\\beta\[\\ell\(p,Y\)\-\\ell\(q,Y\)\]\\right\)=\\sup\_\{r\\in\(0,1\)\}\\left\\\{\\beta\[d\_\{\\ell\}\(r,p\)\-d\_\{\\ell\}\(r,q\)\]\-\{\\text\{\\rm KL\}\}\(\\texttt\{Ber\}\(r\)\\\|\\texttt\{Ber\}\(p\)\)\\right\\\}\\leq 0,where the equality follows by maximizing overrr, and the inequality uses the preceding bound anddℓ​\(r,q\)≥0d\_\{\\ell\}\(r,q\)\\geq 0\.

For any0<ε<1/20<\\varepsilon<1/2and any distributionDDsupported on\[ε,1−ε\]\[\\varepsilon,1\-\\varepsilon\], continuity ofq↦exp⁡\(−β​ℓ​\(q,0\)\)q\\mapsto\\exp\\left\(\-\\beta\\ell\(q,0\)\\right\)gives a reportp∈\[ε,1−ε\]p\\in\[\\varepsilon,1\-\\varepsilon\]satisfying

exp⁡\(−β​ℓ​\(p,0\)\)=𝔼Q∼D​exp⁡\(−β​ℓ​\(Q,0\)\)\.\\exp\\left\(\-\\beta\\ell\(p,0\)\\right\)=\\mathbb\{E\}\_\{Q\\sim D\}\\exp\\left\(\-\\beta\\ell\(Q,0\)\\right\)\.Exponentiating the logarithmic bound above and averaging overq∼Dq\\sim Dthen gives

exp⁡\(−β​ℓ​\(p,1\)\)≥𝔼Q∼D​exp⁡\(−β​ℓ​\(Q,1\)\),\\exp\\left\(\-\\beta\\ell\(p,1\)\\right\)\\geq\\mathbb\{E\}\_\{Q\\sim D\}\\exp\\left\(\-\\beta\\ell\(Q,1\)\\right\),since the outcome\-00term in the averaged inequality equals1−p1\-p\. For an arbitrary distributionDDon\[0,1\]\[0,1\], the same argument applies after clipping its reports to\[ε,1−ε\]\[\\varepsilon,1\-\\varepsilon\]\. Asε↓0\\varepsilon\\downarrow 0, compactness of\[0,1\]\[0,1\]gives a convergent subsequence of the resulting reports\. The integral representation makesq↦exp⁡\(−β​ℓ​\(q,y\)\)q\\mapsto\\exp\\left\(\-\\beta\\ell\(q,y\)\\right\)bounded and continuous on\[0,1\]\[0,1\]for both outcomes, withexp⁡\(−∞\)=0\\exp\\left\(\-\\infty\\right\)=0\. Bounded convergence for the clipped reports therefore implies that the limiting report satisfies both inequalities for the originalDD\. This isβ\\beta\-mixability\.

The consequence for Lipschitz losses\.Finally, ifℓ∈ℒLip​\(L\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{Lip\}\}\\left\(L\\right\), the Lipschitz curvature bound gives

v⁡\(1−v\)​wℓ​\(v\)≤L​min⁡\{v,1−v\}≤L/2almost everywhere\.v\(1\-v\)w\_\{\\ell\}\(v\)\\leq L\\min\\\{v,1\-v\\\}\\leq L/2\\quad\\text\{almost everywhere\}\.The sufficient condition for mixability withβ=2/L\\beta=2/Ltherefore givesℓ∈ℒmix​\(2/L\)\\ell\\in\\mathcal\{L\}\_\{\\mathrm\{mix\}\}\\left\(2/L\\right\)\. ∎

###### Proof of[Lemma2\.2](https://arxiv.org/html/2610.41023#S2.Thmtheorem2)\.

The parametersδ,G,ϵ\\delta,G,\\epsilonare arbitrary and fixed as in the statement\. The assumption1\+G⁡\(p\)​\(y−p\)≥01\+G\(p\)\(y\-p\)\\geq 0aty=0y=0andy=1y=1gives1−p​G​\(p\)≥01\-pG\(p\)\\geq 0and1\+\(1−p\)​G​\(p\)≥01\+\(1\-p\)G\(p\)\\geq 0, respectively\. Hence, for everyp∈Iδ∩\(0,1\)p\\in I\_\{\\delta\}\\cap\(0,1\),

−11−p≤G⁡\(p\)≤1p\.\-\\frac\{1\}\{1\-p\}\\leq G\(p\)\\leq\\frac\{1\}\{p\}\.
We use a finite gridΓ⊆Iδ\\Gamma\\subseteq I\_\{\\delta\}, meaning a finite set of possible forecast values, containing both endpoints ofIδI\_\{\\delta\}\. The gaps between consecutive values will be specified below\. We first show that, for everyr∈\[0,1\]r\\in\[0,1\], somep∈Γp\\in\\GammasatisfiesG​\(p\)​\(r−p\)≤ϵG\(p\)\(r\-p\)\\leq\\epsilon\. Hererris the probability of outcome11, and

𝔼Y∼Ber​\(r\)​\[G⁡\(p\)​\(Y−p\)\]=G⁡\(p\)​\(r−p\)\.\\mathbb\{E\}\_\{Y\\sim\\texttt\{Ber\}\(r\)\}\[G\(p\)\(Y\-p\)\]=G\(p\)\(r\-p\)\.The minimax theorem will then give a single distribution onΓ\\Gammathat satisfies the desired bound for both outcomes\.

The valuer∈\[0,1\]r\\in\[0,1\]is arbitrary and fixed\. Ifr∈Γr\\in\\Gamma, thenrritself is an allowed forecast, andp=rp=rgivesG​\(p\)​\(r−p\)=0G\(p\)\(r\-p\)=0\. Two grid points are consecutive if no point ofΓ\\Gammalies strictly between them\. Whenrrlies between consecutive grid pointsp−,p\+∈Γp\_\{\-\},p\_\{\+\}\\in\\Gammasatisfying0<p−<r<p\+<10<p\_\{\-\}<r<p\_\{\+\}<1, the bounds onGGgive, withd=p\+−p−d=p\_\{\+\}\-p\_\{\-\},

min⁡\{G⁡\(p−\)​\(r−p−\),G⁡\(p\+\)​\(r−p\+\)\}\\displaystyle\\min\\left\\\{G\(p\_\{\-\}\)\(r\-p\_\{\-\}\),G\(p\_\{\+\}\)\(r\-p\_\{\+\}\)\\right\\\}≤min⁡\{r−p−p−,p\+−r1−p\+\}\\displaystyle\\leq\\min\\left\\\{\\frac\{r\-p\_\{\-\}\}\{p\_\{\-\}\},\\frac\{p\_\{\+\}\-r\}\{1\-p\_\{\+\}\}\\right\\\}≤\(r−p−\)\+\(p\+−r\)p−\+\(1−p\+\)=d1−d\.\\displaystyle\\leq\\frac\{\(r\-p\_\{\-\}\)\+\(p\_\{\+\}\-r\)\}\{p\_\{\-\}\+\(1\-p\_\{\+\}\)\}=\\frac\{d\}\{1\-d\}\.The first inequality uses the upper bound onG⁡\(p−\)G\(p\_\{\-\}\)and the lower bound onG⁡\(p\+\)G\(p\_\{\+\}\), sincer−p−\>0r\-p\_\{\-\}\>0andr−p\+<0r\-p\_\{\+\}<0\. For the second inequality, the fraction is a weighted average of the two preceding fractions, with positive weightsp−p\_\{\-\}and1−p\+1\-p\_\{\+\}, and is therefore at least their minimum\. Thus one of the two forecasts gives a value at mostd/\(1−d\)d/\(1\-d\)\.

Whenδ\>0\\delta\>0,Γ\\Gammais any grid allowed by the statement\. All its points lie in\(0,1\)\(0,1\), and every consecutive gap satisfiesd≤δd\\leq\\delta\. The preceding argument therefore gives a forecast with value at mostδ/\(1−δ\)<ϵ\\delta/\(1\-\\delta\)<\\epsilonfor everyr∈Iδr\\in I\_\{\\delta\}\. Ifrris outsideIδI\_\{\\delta\}, the nearest endpoint gives the same bound\. More explicitly,

G​\(δ\)​\(r−δ\)\\displaystyle G\(\\delta\)\(r\-\\delta\)≤δ−r1−δ≤δ1−δ\\displaystyle\\leq\\frac\{\\delta\-r\}\{1\-\\delta\}\\leq\\frac\{\\delta\}\{1\-\\delta\}if​r<δ,\\displaystyle\\text\{if \}r<\\delta,G​\(1−δ\)​\(r−1\+δ\)\\displaystyle G\(1\-\\delta\)\(r\-1\+\\delta\)≤r−1\+δ1−δ≤δ1−δ\\displaystyle\\leq\\frac\{r\-1\+\\delta\}\{1\-\\delta\}\\leq\\frac\{\\delta\}\{1\-\\delta\}if​r\>1−δ\.\\displaystyle\\text\{if \}r\>1\-\\delta\.
Whenδ=0\\delta=0, we choose a finite gridΓ\\Gammacontaining00and11whose consecutive gaps are at mostϵ/\(1\+ϵ\+\|G⁡\(0\)\|\+\|G⁡\(1\)\|\)\\epsilon/\(1\+\\epsilon\+\|G\(0\)\|\+\|G\(1\)\|\)\. Such a grid exists becauseG⁡\(0\)G\(0\)andG⁡\(1\)G\(1\)are finite\. This gap bound is at mostϵ/\(1\+ϵ\)\\epsilon/\(1\+\\epsilon\), sod/\(1−d\)≤ϵd/\(1\-d\)\\leq\\epsilonwheneverrrlies between consecutive points strictly inside\(0,1\)\(0,1\)\. Ifrrlies between00and the next grid point, the forecastp=0p=0gives

G⁡\(0\)​r≤\|G⁡\(0\)\|​r≤ϵ​\|G⁡\(0\)\|1\+ϵ\+\|G⁡\(0\)\|\+\|G⁡\(1\)\|≤ϵ\.G\(0\)r\\leq\|G\(0\)\|r\\leq\\frac\{\\epsilon\|G\(0\)\|\}\{1\+\\epsilon\+\|G\(0\)\|\+\|G\(1\)\|\}\\leq\\epsilon\.Similarly, ifrrlies between the last grid point below11and11, the forecastp=1p=1gives

G⁡\(1\)​\(r−1\)≤\|G⁡\(1\)\|​\(1−r\)≤ϵ​\|G⁡\(1\)\|1\+ϵ\+\|G⁡\(0\)\|\+\|G⁡\(1\)\|≤ϵ\.G\(1\)\(r\-1\)\\leq\|G\(1\)\|\(1\-r\)\\leq\\frac\{\\epsilon\|G\(1\)\|\}\{1\+\\epsilon\+\|G\(0\)\|\+\|G\(1\)\|\}\\leq\\epsilon\.Together with the caser∈Γr\\in\\Gamma, this covers everyr∈\[0,1\]r\\in\[0,1\]\.

In both cases, we have shown thatminp∈Γ⁡G⁡\(p\)​\(r−p\)≤ϵ\\min\_\{p\\in\\Gamma\}G\(p\)\(r\-p\)\\leq\\epsilonfor everyr∈\[0,1\]r\\in\[0,1\]\. Applying the finite minimax theorem to the forecast choicesp∈Γp\\in\\Gammaand outcome choicesy∈\{0,1\}y\\in\\\{0,1\\\}gives

minD​on​Γ⁡maxy∈\{0,1\}​𝔼P∼D​\[G⁡\(P\)​\(y−P\)\]=maxr∈\[0,1\]⁡minp∈Γ⁡G⁡\(p\)​\(r−p\)≤ϵ,\\displaystyle\\min\_\{D\\text\{ on \}\\Gamma\}\\max\_\{y\\in\\\{0,1\\\}\}\\mathbb\{E\}\_\{P\\sim D\}\[G\(P\)\(y\-P\)\]=\\max\_\{r\\in\[0,1\]\}\\min\_\{p\\in\\Gamma\}G\(p\)\(r\-p\)\\leq\\epsilon,where the minimum is over probability distributionsDDonΓ\\Gamma\. Thus the same distributionDDsatisfies the bound for bothy=0y=0andy=1y=1\.

It remains to limitDDto at most three forecast values\. The pair of expectations fory=0y=0andy=1y=1is an average, with weights given byDD, of the vectors\(−p​G​\(p\),\(1−p\)​G​\(p\)\)\(\-pG\(p\),\(1\-p\)G\(p\)\)forp∈Γp\\in\\Gamma\. Carathéodory’s theorem inℝ2\\mathbb\{R\}^\{2\}states that this average can be expressed using at most three of these vectors with nonnegative weights summing to one\. These weights give a distribution on at most three points ofΓ\\Gamma, again denoted byDD, with the same two expectations\. Hence the required bound holds for bothy=0y=0andy=1y=1\. ∎

## Appendix DApproximate Bases for GLM Losses

For an integerT≥1T\\geq 1, letδ=1/\[2​\(T\+1\)\]\\delta=1/\[2\(T\+1\)\],ΓT=\{i​δ:1≤i≤2​T\+1\}\\Gamma\_\{T\}=\\\{i\\delta:1\\leq i\\leq 2T\+1\\\}, andθj=\(j−12\)​δ\\theta\_\{j\}=\(j\-\\tfrac\{1\}\{2\}\)\\deltafor1≤j≤2​T\+21\\leq j\\leq 2T\+2\. For1≤k≤T1\\leq k\\leq Tand1≤j≤2​T\+21\\leq j\\leq 2T\+2, define the auxiliary losses

ℓk,j\(q,0\)=θj𝕀\{q≥k/T\},ℓk,j\(q,1\)=\(1−θj\)𝕀\{q<k/T\}\.\\ell\_\{k,j\}\(q,0\)=\\theta\_\{j\}\\mathbb\{I\}\{\\left\\\{\{q\\geq k/T\}\\right\\\}\},\\qquad\\ell\_\{k,j\}\(q,1\)=\(1\-\\theta\_\{j\}\)\\mathbb\{I\}\{\\left\\\{\{q<k/T\}\\right\\\}\}\.\(35\)These2​T​\(T\+1\)2T\(T\+1\)losses take values in\[0,1\]\[0,1\]and admit the Bayes actionπℓk,j\(p\)=𝕀\{p\>θj\}\\pi\_\{\\ell\_\{k,j\}\}\(p\)=\\mathbb\{I\}\{\\left\\\{\{p\>\\theta\_\{j\}\}\\right\\\}\}onΓT\\Gamma\_\{T\}\. Both the grid and the auxiliary losses depend only onTT\.

###### Proposition D\.1\(Finite threshold approximation for GLM loss comparisons\)\.

For every continuous convex functionA:\[0,1\]→ℝA:\[0,1\]\\to\\mathbb\{R\}, letℓA​\(q,y\)=A⁡\(q\)−y​q\\ell\_\{A\}\(q,y\)=A\(q\)\-yqand fix a Bayes post\-processingπA\\pi\_\{A\}\. There exist coefficientsck,j≥0c\_\{k,j\}\\geq 0with∑k=1T∑j=12​T\+2ck,j=1\\sum\_\{k=1\}^\{T\}\\sum\_\{j=1\}^\{2T\+2\}c\_\{k,j\}=1such that, for everyp∈ΓTp\\in\\Gamma\_\{T\},q∈\[0,1\]q\\in\[0,1\], andy∈\{0,1\}y\\in\\\{0,1\\\},

ℓA​\(πA​\(p\),y\)−ℓA​\(q,y\)≤∑k=1T∑j=12​T\+2ck,j​\[ℓk,j​\(πℓk,j​\(p\),y\)−ℓk,j​\(q,y\)\]\+δ2\+1T\.\\displaystyle\\ell\_\{A\}\(\\pi\_\{A\}\(p\),y\)\-\\ell\_\{A\}\(q,y\)\\leq\\sum\_\{k=1\}^\{T\}\\sum\_\{j=1\}^\{2T\+2\}c\_\{k,j\}\\left\[\\ell\_\{k,j\}\(\\pi\_\{\\ell\_\{k,j\}\}\(p\),y\)\-\\ell\_\{k,j\}\(q,y\)\\right\]\+\\frac\{\\delta\}\{2\}\+\\frac\{1\}\{T\}\.\(36\)

###### Proof of[PropositionD\.1](https://arxiv.org/html/2610.41023#A4.Thmtheorem1)\.

Fix a continuous convex functionAAand its Bayes post\-processingπA\\pi\_\{A\}\. WriteJ=2​T\+1J=2T\+1,pi=i​δp\_\{i\}=i\\delta, andqi=πA​\(pi\)q\_\{i\}=\\pi\_\{A\}\(p\_\{i\}\)for1≤i≤J1\\leq i\\leq J, whereδ=1/\[2​\(T\+1\)\]\\delta=1/\[2\(T\+1\)\]\. We also setp0=0p\_\{0\}=0,pJ\+1=1p\_\{J\+1\}=1,q0=0q\_\{0\}=0, andqJ\+1=1q\_\{J\+1\}=1to include the endpoints in the partitions below\.

Approximating the loss differences\.Fori<i′i<i^\{\\prime\}, Bayes optimality gives

A⁡\(qi\)−pi​qi\\displaystyle A\(q\_\{i\}\)\-p\_\{i\}q\_\{i\}≤A⁡\(qi′\)−pi​qi′,\\displaystyle\\leq A\(q\_\{i^\{\\prime\}\}\)\-p\_\{i\}q\_\{i^\{\\prime\}\},A⁡\(qi′\)−pi′​qi′\\displaystyle A\(q\_\{i^\{\\prime\}\}\)\-p\_\{i^\{\\prime\}\}q\_\{i^\{\\prime\}\}≤A⁡\(qi\)−pi′​qi\.\\displaystyle\\leq A\(q\_\{i\}\)\-p\_\{i^\{\\prime\}\}q\_\{i\}\.Adding these inequalities yields\(pi′−pi\)​\(qi′−qi\)≥0\(p\_\{i^\{\\prime\}\}\-p\_\{i\}\)\(q\_\{i^\{\\prime\}\}\-q\_\{i\}\)\\geq 0\. Sincepi′\>pip\_\{i^\{\\prime\}\}\>p\_\{i\}, we obtainqi≤qi′q\_\{i\}\\leq q\_\{i^\{\\prime\}\}\. Thusq0≤⋯≤qJ\+1q\_\{0\}\\leq\\cdots\\leq q\_\{J\+1\}, and the intervals\(qj−1,qj\]\(q\_\{j\-1\},q\_\{j\}\]partition\(0,1\]\(0,1\]\.

Bayes optimality also givesA⁡\(q\)≥A⁡\(qi\)\+pi​\(q−qi\)A\(q\)\\geq A\(q\_\{i\}\)\+p\_\{i\}\(q\-q\_\{i\}\)for everyq∈\[0,1\]q\\in\[0,1\]\. We therefore define

A¯​\(q\)=max1≤i≤J⁡\{A⁡\(qi\)\+pi​\(q−qi\)\}\.\\bar\{A\}\(q\)=\\max\_\{1\\leq i\\leq J\}\\\{A\(q\_\{i\}\)\+p\_\{i\}\(q\-q\_\{i\}\)\\\}\.Each line in this maximum lies belowAA, soA¯≤A\\bar\{A\}\\leq A\. Atqiq\_\{i\}, theiith line equalsA⁡\(qi\)A\(q\_\{i\}\), givingA¯​\(qi\)=A⁡\(qi\)\\bar\{A\}\(q\_\{i\}\)=A\(q\_\{i\}\)\. Consequently, for any comparator reportq∈\[0,1\]q\\in\[0,1\]and outcomey∈\{0,1\}y\\in\\\{0,1\\\},

ℓA​\(qi,y\)−ℓA​\(q,y\)≤A¯​\(qi\)−A¯​\(q\)−y⁡\(qi−q\)\.\\ell\_\{A\}\(q\_\{i\},y\)\-\\ell\_\{A\}\(q,y\)\\leq\\bar\{A\}\(q\_\{i\}\)\-\\bar\{A\}\(q\)\-y\(q\_\{i\}\-q\)\.
We next bound the slopes ofA¯\\bar\{A\}between consecutiveqiq\_\{i\}\. For2≤j≤J2\\leq j\\leq J, consider a nonempty interval\(qj−1,qj\)\(q\_\{j\-1\},q\_\{j\}\)\. Atqj−1q\_\{j\-1\}, every line in the maximum is at mostA⁡\(qj−1\)A\(q\_\{j\-1\}\)\. The lines indexed byi≤j−1i\\leq j\-1have slopes at mostpj−1p\_\{j\-1\}, so to the right ofqj−1q\_\{j\-1\}they are no larger than the line indexed byj−1j\-1\. Similarly, to the left ofqjq\_\{j\}, the lines indexed byi≥ji\\geq jare no larger than the line indexed byjj\. HenceA¯\\bar\{A\}is the maximum of these two lines on\(qj−1,qj\)\(q\_\{j\-1\},q\_\{j\}\), and its derivative lies betweenpj−1p\_\{j\-1\}andpjp\_\{j\}wherever it exists\. On\(0,q1\)\(0,q\_\{1\}\)the first line attains the maximum, and on\(qJ,1\)\(q\_\{J\},1\)the last line attains the maximum\. Thus, for every1≤j≤J\+11\\leq j\\leq J\+1,

pj−1≤A¯′​\(s\)≤pjfor almost every​s∈\(qj−1,qj\)\.p\_\{j\-1\}\\leq\\bar\{A\}^\{\\prime\}\(s\)\\leq p\_\{j\}\\quad\\text\{for almost every \}s\\in\(q\_\{j\-1\},q\_\{j\}\)\.Sinceθj=\(pj−1\+pj\)/2\\theta\_\{j\}=\(p\_\{j\-1\}\+p\_\{j\}\)/2andpj−pj−1=δp\_\{j\}\-p\_\{j\-1\}=\\delta, this gives\|A¯′​\(s\)−θj\|≤δ/2\|\\bar\{A\}^\{\\prime\}\(s\)\-\\theta\_\{j\}\|\\leq\\delta/2on that interval\.

The functionA¯\\bar\{A\}is continuous and piecewise linear, so integrating its derivative gives

A¯​\(qi\)−A¯​\(q\)−y⁡\(qi−q\)\\displaystyle\\bar\{A\}\(q\_\{i\}\)\-\\bar\{A\}\(q\)\-y\(q\_\{i\}\-q\)=∑j=1J\+1∫qj−1qj\(A¯′\(s\)−y\)\[𝕀\{s≤qi\}−𝕀\{s≤q\}\]ds\\displaystyle=\\sum\_\{j=1\}^\{J\+1\}\\int\_\{q\_\{j\-1\}\}^\{q\_\{j\}\}\(\\bar\{A\}^\{\\prime\}\(s\)\-y\)\[\\mathbb\{I\}\{\\left\\\{\{s\\leq q\_\{i\}\}\\right\\\}\}\-\\mathbb\{I\}\{\\left\\\{\{s\\leq q\}\\right\\\}\}\]\\,\\mathrm\{d\}s≤∑j=1J\+1\(θj−y\)∫qj−1qj\[𝕀\{s≤qi\}−𝕀\{s≤q\}\]ds\+δ2\.\\displaystyle\\leq\\sum\_\{j=1\}^\{J\+1\}\(\\theta\_\{j\}\-y\)\\int\_\{q\_\{j\-1\}\}^\{q\_\{j\}\}\[\\mathbb\{I\}\{\\left\\\{\{s\\leq q\_\{i\}\}\\right\\\}\}\-\\mathbb\{I\}\{\\left\\\{\{s\\leq q\}\\right\\\}\}\]\\,\\mathrm\{d\}s\+\\frac\{\\delta\}\{2\}\.Indeed, the indicator difference is nonzero only betweenqiq\_\{i\}andqq, where its absolute value is one\. ReplacingA¯′​\(s\)\\bar\{A\}^\{\\prime\}\(s\)byθj\\theta\_\{j\}therefore introduces an error at most\(δ/2\)​\|qi−q\|≤δ/2\(\\delta/2\)\|q\_\{i\}\-q\|\\leq\\delta/2\.

Letq−=⌊T​q⌋/Tq^\{\-\}=\\lfloor Tq\\rfloor/T\. Replacing𝕀\{s≤q\}\\mathbb\{I\}\{\\left\\\{\{s\\leq q\}\\right\\\}\}by𝕀\{s≤q−\}\\mathbb\{I\}\{\\left\\\{\{s\\leq q^\{\-\}\}\\right\\\}\}changes the integrand only on\(q−,q\]\(q^\{\-\},q\], whose length is at most1/T1/T\. Since\|θj−y\|≤1\|\\theta\_\{j\}\-y\|\\leq 1, this introduces an additional error at most1/T1/T\.

Expressing the approximation using the auxiliary losses\.For1≤k≤T1\\leq k\\leq Tand1≤j≤J\+11\\leq j\\leq J\+1, let

ck,j=Leb⁡\(\(\(k−1\)/T,k/T\]∩\(qj−1,qj\]\)\.c\_\{k,j\}=\\operatorname\{Leb\}\\bigl\(\(\(k\-1\)/T,k/T\]\\cap\(q\_\{j\-1\},q\_\{j\}\]\\bigr\)\.Thusck,jc\_\{k,j\}is the length of the intersection of the two indicated intervals\. Both collections of intervals partition\(0,1\]\(0,1\], sock,j≥0c\_\{k,j\}\\geq 0and∑k,jck,j=1\\sum\_\{k,j\}c\_\{k,j\}=1\. These coefficients depend only onAA, its fixed post\-processingπA\\pi\_\{A\}, and the grids\.

On each nonempty intersection, the two indicators in the preceding approximation are constant\. Sinceq−q^\{\-\}is a multiple of1/T1/T, we have𝕀\{s≤q−\}=𝕀\{q≥k/T\}\\mathbb\{I\}\{\\left\\\{\{s\\leq q^\{\-\}\}\\right\\\}\}=\\mathbb\{I\}\{\\left\\\{\{q\\geq k/T\}\\right\\\}\}there\. Likewise, the ordering of theqiq\_\{i\}gives𝕀\{s≤qi\}=𝕀\{j≤i\}\\mathbb\{I\}\{\\left\\\{\{s\\leq q\_\{i\}\}\\right\\\}\}=\\mathbb\{I\}\{\\left\\\{\{j\\leq i\}\\right\\\}\}\. Becausepi=i​δp\_\{i\}=i\\deltaandθj=\(j−12\)​δ\\theta\_\{j\}=\(j\-\\tfrac\{1\}\{2\}\)\\delta, this last indicator equals𝕀\{pi\>θj\}\\mathbb\{I\}\{\\left\\\{\{p\_\{i\}\>\\theta\_\{j\}\}\\right\\\}\}\. Splitting the integrals over these intersections therefore gives

∑j=1J\+1\(θj−y\)∫qj−1qj\[𝕀\{s≤qi\}−𝕀\{s≤q−\}\]ds=∑k=1T∑j=1J\+1ck,j\(θj−y\)\[𝕀\{pi\>θj\}−𝕀\{q≥k/T\}\]\.\\displaystyle\\sum\_\{j=1\}^\{J\+1\}\(\\theta\_\{j\}\-y\)\\int\_\{q\_\{j\-1\}\}^\{q\_\{j\}\}\[\\mathbb\{I\}\{\\left\\\{\{s\\leq q\_\{i\}\}\\right\\\}\}\-\\mathbb\{I\}\{\\left\\\{\{s\\leq q^\{\-\}\}\\right\\\}\}\]\\,\\mathrm\{d\}s=\\sum\_\{k=1\}^\{T\}\\sum\_\{j=1\}^\{J\+1\}c\_\{k,j\}\(\\theta\_\{j\}\-y\)\[\\mathbb\{I\}\{\\left\\\{\{p\_\{i\}\>\\theta\_\{j\}\}\\right\\\}\}\-\\mathbb\{I\}\{\\left\\\{\{q\\geq k/T\}\\right\\\}\}\]\.
To identify these terms as auxiliary loss differences, the definition in[Eq\. \(35\)](https://arxiv.org/html/2610.41023#A4.E35)gives

ℓk,j\(q,y\)=\(1−θj\)y\+\(θj−y\)𝕀\{q≥k/T\}\.\\ell\_\{k,j\}\(q,y\)=\(1\-\\theta\_\{j\}\)y\+\(\\theta\_\{j\}\-y\)\\mathbb\{I\}\{\\left\\\{\{q\\geq k/T\}\\right\\\}\}\.Its Bayes action on the forecast grid isπℓk,j\(pi\)=𝕀\{pi\>θj\}\\pi\_\{\\ell\_\{k,j\}\}\(p\_\{i\}\)=\\mathbb\{I\}\{\\left\\\{\{p\_\{i\}\>\\theta\_\{j\}\}\\right\\\}\}\. This action is either zero or one, and0<k/T≤10<k/T\\leq 1, so

ℓk,j\(πℓk,j\(pi\),y\)−ℓk,j\(q,y\)=\(θj−y\)\[𝕀\{pi\>θj\}−𝕀\{q≥k/T\}\]\.\\displaystyle\\ell\_\{k,j\}\(\\pi\_\{\\ell\_\{k,j\}\}\(p\_\{i\}\),y\)\-\\ell\_\{k,j\}\(q,y\)=\(\\theta\_\{j\}\-y\)\[\\mathbb\{I\}\{\\left\\\{\{p\_\{i\}\>\\theta\_\{j\}\}\\right\\\}\}\-\\mathbb\{I\}\{\\left\\\{\{q\\geq k/T\}\\right\\\}\}\]\.Combining the preceding identities with the two approximation errors proves[Eq\. \(36\)](https://arxiv.org/html/2610.41023#A4.E36)\. ∎

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