基于上一轮效用预测的 IR-PRM+：常数遗憾与最佳迭代点收敛性

# IR-PRM+ with Previous-Utility Prediction: Constant Regret and Best-Iterate Convergence
Source: [https://arxiv.org/html/2610.19016](https://arxiv.org/html/2610.19016)
Michael SullinsandIan A\. KashAffiliation:University of Illinois Chicago,Chicago,Illinois,USAemail:[iankash@uic\.edu](mailto:[email protected])

###### Abstract\.

Regret matching \(and its variants\) and counterfactual regret minimization are central algorithms in practice for normal\-form and extensive\-form game solving\. Yet these have historically lacked the stronger theoretical convergence guarantees available to optimistic and extra\-gradient methods\. Recent work introduced IREG\-PRM\+, a regret matching variant which does have strong theoretical convergence guarantees\. However, it relies on a simultaneous, extra\-gradient update\. IREG\-PRM\+ is an instance of the more general IR\-PRM framework, where empirically optimistic and alternating updates often have better performance\. However, the prior extra\-gradient analysis does not extend to these cases\. We introduce a new approach that covers both settings\. For simultaneous and alternating IR\-PRM\+ in finite two\-player zero\-sum matrix games, previous\-utility prediction recovers the constant\-regret and convergence\-rate guarantees previously obtained with extra\-gradient prediction: external regret bounded independently ofTT, anO⁡\(T−1\)O\(T^\{\-1\}\)average duality gap, and anO\(T−1/2\)O\(T^\{\-1/2\}\)best\-current\-iterate gap\. Moreover, the current duality gap vanishes and the current profile converges to the Nash equilibrium set\. The key step bounds the component of consecutive strategy movement not controlled by the existing IR\-PRM\+ regret inequality using the relative growth of the nondecreasing regret\-vector normρt\\rho\_\{t\}\. Coupling this bound across the two players yields recurrences whose effective coefficient scales as1/\(ρtX​ρtY\)1/\(\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\), preventing unbounded norm growth\. We extend the analysis to finitely many simplex learners under a weighted Minty condition and, through the corresponding CFR variant IR\-PCFR\+, recover analogous guarantees for finite two\-player zero\-sum perfect\-recall extensive\-form games under the same full\-reach condition as prior work\.

###### Keywords:

regret minimization, zero\-sum games, counterfactual regret minimization, extensive\-form games

## 1\.Introduction

Regret minimization is a standard approach to equilibrium computation in games\. Regret matching \(RM\), counterfactual regret minimization \(CFR\), and their variants are among the most successful algorithms for normal\-form and extensive\-form game solving\. CFR\+ combines RM\+ with alternating updates and weighted averaging\([Tammelin et al\., 2015](https://arxiv.org/html/2610.19016#bib.bib2)\), while predictive RM\+ \(PRM\+\) uses optimism to obtain substantial empirical speedups within the same basic regret\-matching framework\([Farina et al\., 2021](https://arxiv.org/html/2610.19016#bib.bib3)\)\. This practical success has long coexisted with a theoretical gap: in two\-player zero\-sum games, first\-order methods such as Mirror\-Prox, the excessive\-gap technique, and optimistic mirror descent admitO⁡\(T−1\)O\(T^\{\-1\}\)ergodic convergence guarantees\([Nemirovski, 2004](https://arxiv.org/html/2610.19016#bib.bib16);[Nesterov, 2005](https://arxiv.org/html/2610.19016#bib.bib18);[Rakhlin and Sridharan, 2013](https://arxiv.org/html/2610.19016#bib.bib19);[Chiang et al\., 2012](https://arxiv.org/html/2610.19016#bib.bib20)\), whereas standard RM\+\-based methods do not generally attain this rate; in particular, PRM\+ can exhibitΩ\(T−1/2\)\\Omega\(T^\{\-1/2\}\)convergence even in self\-play\([Farina et al\., 2023](https://arxiv.org/html/2610.19016#bib.bib4);[Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)\.

Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)recently introduced the IR\-PRM\+ framework to address this limitation\. Its extra\-gradient instantiation, IREG\-PRM\+, was designed to retain the parameter\-free, scale\-invariant structure of regret matching while maintaining a nondecreasing regret\-vector norm and satisfying an RVU\-like\([Syrgkanis et al\., 2015](https://arxiv.org/html/2610.19016#bib.bib1)\)bound\. IREG\-PRM\+ obtains constant regret in two\-player zero\-sum self\-play, the optimalO⁡\(T−1\)O\(T^\{\-1\}\)average\-iterate rate, and anO\(T−1/2\)O\(T^\{\-1/2\}\)best\-iterate guarantee, while remaining competitive with PRM\+ empirically\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)\. Their analysis further connects IREG\-PRM\+ to optimistic gradient descent with an adaptive learning rate, providing a direct bridge between practical regret matching and the stronger convergence theory of first\-order methods\.

The existing IR\-PRM\+ guarantees, however, do not cover two practically important settings\. First, the constant\-regret analysis of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)relies on extra\-gradient prediction\. Both players first form an intermediate profile\(x~t,y~t\)\(\\widetilde\{x\}\_\{t\},\\widetilde\{y\}\_\{t\}\)by normalizing their regret vectors\. The game is evaluated there to construct the current predictions, and the resulting played profile is then evaluated again, giving two evaluations per update\. By contrast, previous\-utility optimism forms the prediction directly from previously observed feedback and therefore requires only a single evaluation per update\. This is the form of optimism used by PRM\+\([Farina et al\., 2021](https://arxiv.org/html/2610.19016#bib.bib3)\)\. Second, their fast\-convergence theory does not cover alternating IR\-PRM\+ updates\. Alternating updates are already a standard ingredient of CFR\+, where they are associated with its strong practical performance\([Tammelin et al\., 2015](https://arxiv.org/html/2610.19016#bib.bib2)\)\. Subsequent work has likewise reported substantial empirical benefits from alternation, including strong performance of alternating PRM\+ across several benchmark games\([Cai et al\., 2025a](https://arxiv.org/html/2610.19016#bib.bib5)\)\. Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)explicitly leave both previous\-utility prediction and alternating variants as open theoretical questions\. Their empirical results suggest that these previously unanalyzed settings are practically important: across all six reported benchmarks, when performance is measured by gradient evaluations, the best\-performing update is either alternating or simultaneous rather than extra\-gradient\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)\.

The two settings pose different technical obstacles: previous\-utility prediction requires control of a portion of strategy movement not controlled by the RVU bound used by Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\), while alternation breaks the structure that lets the two players’ RVU inequalities be combined directly\. In greater detail, under previous\-utility prediction,

utX−mtX=A⁡\(yt−yt−1\),utY−mtY=−A⊤​\(xt−xt−1\),u\_\{t\}^\{X\}\-m\_\{t\}^\{X\}=A\(y\_\{t\}\-y\_\{t\-1\}\),\\qquad u\_\{t\}^\{Y\}\-m\_\{t\}^\{Y\}=\-A^\{\\top\}\(x\_\{t\}\-x\_\{t\-1\}\),so the prediction error depends on the full movement between consecutive played strategies\. The IR\-PRM\+ RVU bound, however, contains a negative term only for the prediction displacementxt−x~tx\_\{t\}\-\\widetilde\{x\}\_\{t\}, which captures only one part of that movement\. Alternating updates introduce a separate difficulty: the second player responds to the first player’s newly updated strategy, so the two players’ updates are evaluated at different profiles\. Previous\-utility optimism therefore requires control of the missing part of consecutive strategy movement, while alternation additionally requires handling the staggered timing of the two updates\.

Our main observation is that growth of the regret\-vector norm itself controls the missing post\-update displacementx~t−xt−1\\widetilde\{x\}\_\{t\}\-x\_\{t\-1\}, a property not used by the extra\-gradient argument\. IR\-PRM\+ obtains its strategy by normalizing a nonnegative regret vector\. Ifρt\\rho\_\{t\}denotes its Euclidean norm, then a fixed\-size change in the regret vector has less effect on the normalized strategy whenρt\\rho\_\{t\}is large\. Quantitatively, the post\-update displacement is controlled by the relative growthρt−ρt−1\\rho\_\{t\}\-\\rho\_\{t\-1\}of the regret\-vector norm\. In self\-play, the resulting feedback is doubly self\-damping: a large value of a player’s ownρt\\rho\_\{t\}limits how much a given prediction error can increase its norm, while a large opponent norm limits the opponent’s post\-update displacement and therefore reduces that prediction error\. At the level of the coupled recurrence, the key inequality for the row playerXXis, schematically,

\(1\)ρt\+1X−ρtX≤CρtX​ρtY​\[ρtY​‖yt−y~t‖22\+\(ρtY−ρt−1Y\)\],\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}\\leq\\frac\{C\}\{\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\}\\left\[\\rho\_\{t\}^\{Y\}\\left\\lVert y\_\{t\}\-\\widetilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\(\\rho\_\{t\}^\{Y\}\-\\rho\_\{t\-1\}^\{Y\}\)\\right\],with a symmetric inequality for the column playerYYand a time\-independent, but problem\-specific, constantCC\. The factor1/ρtX1/\\rho\_\{t\}^\{X\}comes from converting the row player’s prediction error into growth of its own regret\-vector norm; the factor1/ρtY1/\\rho\_\{t\}^\{Y\}comes from controlling the column player’s post\-update displacement\. Hence the coefficient1/\(ρtX​ρtY\)1/\(\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\)becomes smaller whenever either regret\-vector norm grows\. Once their product is sufficiently large, the positive prediction\-error contribution is dominated by the negative prediction displacement term in the RVU inequality, preventing sustained further growth\. Summing the two players’ bounds and using the fact that their cumulative regrets equal a nonnegative multiple of the duality gap of the averaged profile then bounds the accumulated weighted squared prediction displacements by the regret\-vector norms themselves\. Substituting this bound back into the norm\-growth inequality shows that bothρtX\\rho\_\{t\}^\{X\}andρtY\\rho\_\{t\}^\{Y\}remain bounded for alltt\.

Our results show that, for simultaneous IR\-PRM\+, previous\-utility prediction recovers the constant\-regret and convergence\-rate guarantees of IREG\-PRM\+ without the extra\-gradient evaluation\. In finite two\-player zero\-sum matrix games we obtain external regret bounded by constants independent ofTT, anO⁡\(T−1\)O\(T^\{\-1\}\)average duality gap, convergence of the current duality gap to zero, convergence of the current profile to the Nash equilibrium set, and anO\(T−1/2\)O\(T^\{\-1/2\}\)best\-current\-iterate guarantee\. We then extend the analysis to alternating updates\. In this setting, the two players’ prediction errors arise at different points within each full update because one player updates before the other\. The proof therefore works with the intermediate profile after the first player’s update and shows that the corresponding bounds combine into regret and current\-profile guarantees once both players have updated\. Thus the same regret\-vector norm\-growth argument extends to both previous\-utility prediction and alternating updates\. It also yields a two\-displacement RVU bound for IR\-PRM\+ that controls both the prediction displacementxt−x~tx\_\{t\}\-\\widetilde\{x\}\_\{t\}and the post\-update displacementx~t\+1−xt\\widetilde\{x\}\_\{t\+1\}\-x\_\{t\}, extending the previous RVU form with the additional control needed for previous\-utility prediction\. Although our convergence analysis does not rely on this refined RVU bound, it shows that the same theory recovers the missing form directly\.

Finally, we extend the argument from the two players of a matrix game to the many local learners used by CFR\. A counterfactual utility at one information set can depend on strategies at many other information sets, so the two\-player recurrence must be replaced by a product\-of\-simplices coupling\. We show that, under the weighted Minty condition of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\), previous\-utility prediction keeps all local regret\-vector norms bounded\. In their CFR setting, the full\-reach assumption ensures that the weights in this condition are strictly positive\. Our boundedness result can therefore be combined with their IR\-PCFR\+ reduction to obtain analogous guarantees for finite two\-player zero\-sum perfect\-recall extensive\-form games\. The result applies to both simultaneous and alternating IR\-PCFR\+: bounded sequence\-form external regret, anO⁡\(T−1\)O\(T^\{\-1\}\)average duality gap, convergence of the current sequence\-form gap to zero and of the current realization\-plan profile to the sequence\-form Nash set, and anO\(T−1/2\)O\(T^\{\-1/2\}\)best\-current\-iterate guarantee\.

## 2\.Related Work

CFR decomposes global external regret into local counterfactual regrets and thereby makes regret matching practical in large extensive\-form games\([Zinkevich et al\., 2007](https://arxiv.org/html/2610.19016#bib.bib12)\)\. CFR\+ replaces regret matching with RM\+ and combines it with alternating updates and weighted averaging, producing substantial empirical improvements while retaining the usualO\(T−1/2\)O\(T^\{\-1/2\}\)worst\-case convergence rate\([Tammelin et al\., 2015](https://arxiv.org/html/2610.19016#bib.bib2)\)\. Farina et al\.\([Farina et al\., 2021](https://arxiv.org/html/2610.19016#bib.bib3)\)subsequently developed predictive RM and RM\+ through predictive Blackwell approachability and showed that PRM\+ can substantially accelerate CFR in practice\. These works established predictive regret matching as a practical route to faster CFR convergence, while retaining the classical regret\-matching framework\.

A central framework for fast convergence through stability and predictability is the Regret bounded by Variation in Utilities \(RVU\) property, introduced by Syrgkanis et al\.\([Syrgkanis et al\., 2015](https://arxiv.org/html/2610.19016#bib.bib1)\)\. An RVU bound controls regret through variation in the utility sequence together with a negative penalty on the learner’s own movement; Syrgkanis et al\. showed that recency\-biased regularized learners satisfy such bounds and that, when these learners interact, the variation and movement terms can be combined to obtain faster\-than\-worst\-case convergence in games\. Farina et al\.\([Farina et al\., 2023](https://arxiv.org/html/2610.19016#bib.bib4)\)showed that RM\+ and predictive RM\+ can be unstable and obtained faster convergence through stabilized variants, including Smooth PRM\+, which require parameter tuning\. Cai et al\.\([Cai et al\., 2025a](https://arxiv.org/html/2610.19016#bib.bib5)\)studied last\-iterate convergence directly: they exhibit nonconvergent behavior for several standard RM\+\-based dynamics and prove last\-iterate guarantees for extra\-gradient RM\+ and smooth predictive RM\+\. Meng et al\.\([Meng et al\., 2025b](https://arxiv.org/html/2610.19016#bib.bib17)\)established last\-iterate and finite\-time best\-iterate convergence for a smooth RM\+ variant under the weak Minty condition\. More broadly, Cai et al\.\([Cai et al\., 2025b](https://arxiv.org/html/2610.19016#bib.bib6)\)gave a black\-box reduction from average\-iterate to last\-iterate convergence for bilinear games, while Meng et al\.\([Meng et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib7)\)introduced a different parameter\-free smooth predictive RM\+ variant with anO⁡\(T−1\)O\(T^\{\-1\}\)average\-iterate convergence guarantee\. In extensive\-form games, Lee et al\.\([Lee et al\., 2021](https://arxiv.org/html/2610.19016#bib.bib8)\)established last\-iterate convergence for optimistic regret\-minimization methods operating directly on treeplexes\. Chakrabarti et al\.\([Chakrabarti et al\., 2024](https://arxiv.org/html/2610.19016#bib.bib9)\)developed Blackwell\-approachability methods on treeplexes, including a stabilized variant with anO⁡\(T−1\)O\(T^\{\-1\}\)convergence rate, while Meng et al\.\([Meng et al\., 2025a](https://arxiv.org/html/2610.19016#bib.bib10)\)obtained last\-iterate convergence through an RM\-based CFR method for a sequence of perturbed regularized extensive\-form games\. Taken together, these works obtain stronger convergence guarantees through several modifications of regret\-matching dynamics and related first\-order methods\. Our results show that, within IR\-PRM\+, previous\-utility prediction and alternating updates retain constant regret, the𝒪T​\(T−1\)\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)average\-gap rate, the𝒪T\(T−1/2\)\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)best\-current\-iterate rate, and current\-profile convergence without extra\-gradient prediction\.

## 3\.Preliminaries and IR\-PRM\+

LetΔk:=\{p∈ℝ\+k:⟨p,𝟏⟩=1\}\\Delta\_\{k\}:=\\\{p\\in\\mathbb\{R\}\_\{\+\}^\{k\}:\\left\\langle p,\\mathbf\{1\}\\right\\rangle=1\\\}denote thekk\-action simplex, and letA∈ℝkX×kYA\\in\\mathbb\{R\}^\{k^\{X\}\\times k^\{Y\}\}be the payoff matrix of a two\-player zero\-sum game\. At roundtt, the row player choosesxt∈ΔkXx\_\{t\}\\in\\Delta\_\{k^\{X\}\}and the column player choosesyt∈ΔkYy\_\{t\}\\in\\Delta\_\{k^\{Y\}\}; the row player receives expected payoffxt⊤​A​ytx\_\{t\}^\{\\top\}Ay\_\{t\}and the column player its negative\. The utilities of each player’s pure actions against the opponent’s current mixed strategy are

utX=A​yt,utY=−A⊤​xt\.u\_\{t\}^\{X\}=Ay\_\{t\},\\qquad u\_\{t\}^\{Y\}=\-A^\{\\top\}x\_\{t\}\.We distinguish two update schedules\. Under*simultaneous*updates, both players choose their round\-ttstrategies from information available before the joint play and then update after observing the round\-ttutilities\. Under*alternating*updates, the players update sequentially, so the second update can use the first player’s newly computed strategy\. The analysis in this paper is simultaneous unless stated otherwise\. For later use, letx¯T=T−1​∑t=1Txt\\bar\{x\}\_\{T\}=T^\{\-1\}\\sum\_\{t=1\}^\{T\}x\_\{t\}andy¯T=T−1​∑t=1Tyt\\bar\{y\}\_\{T\}=T^\{\-1\}\\sum\_\{t=1\}^\{T\}y\_\{t\}denote the average strategies\. The duality gap is

Gap⁡\(x,y\):=maxx′∈ΔkX⁡\(x′\)⊤​A​y−miny′∈ΔkY⁡x⊤​A​y′\.\\operatorname\{Gap\}\(x,y\):=\\max\_\{x^\{\\prime\}\\in\\Delta\_\{k^\{X\}\}\}\(x^\{\\prime\}\)^\{\\top\}Ay\-\\min\_\{y^\{\\prime\}\\in\\Delta\_\{k^\{Y\}\}\}x^\{\\top\}Ay^\{\\prime\}\.
Suppressing player superscripts, consider a generickk\-action learner\. It choosesxt∈Δkx\_\{t\}\\in\\Delta\_\{k\}, observes a utility vectorut∈ℝku\_\{t\}\\in\\mathbb\{R\}^\{k\}, and incurs external regret

Reg⁡\(T\)=max⁡∑t=1Tp∈Δk⁡⟨ut,p−xt⟩\.\\operatorname\{Reg\}\(T\)=\\max\_\{p\\in\\Delta\_\{k\}\}\\sum\_\{t=1\}^\{T\}\\left\\langle u\_\{t\},p\-x\_\{t\}\\right\\rangle\.We now describe one IR\-PRM\+ learner\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)\. Each round the learner begins with a nonnegative cumulative regret vectorr~t∈ℝ\+k\\tilde\{r\}\_\{t\}\\in\\mathbb\{R\}\_\{\+\}^\{k\}produced at the end of the preceding round, and we letρt:=‖r~t‖2\\rho\_\{t\}:=\\left\\lVert\\tilde\{r\}\_\{t\}\\right\\rVert\_\{2\}denote its Euclidean norm\. For akk\-action learner, we use the symmetric unit\-norm initialization, using𝟏\\mathbf\{1\}to denote the all\-ones vector

\(2\)r~1=1k​𝟏,ρ1:=‖r~1‖2=1\.\\tilde\{r\}\_\{1\}=\\frac\{1\}\{\\sqrt\{k\}\}\\mathbf\{1\},\\qquad\\rho\_\{1\}:=\\left\\lVert\\tilde\{r\}\_\{1\}\\right\\rVert\_\{2\}=1\.The initialization in \([2](https://arxiv.org/html/2610.19016#S3.E2)\) is used only to avoid division by zero in the main normalization formulas\. IR\-PRM\+ permits any nonnegative initial regret vector, and Appendix F extends all results to the standard zero initializationr~1=0\\tilde\{r\}\_\{1\}=0; only rounds withr~t=0\\tilde\{r\}\_\{t\}=0and the first transition toρt\>0\\rho\_\{t\}\>0require separate treatment\.

The cumulative regret vector induces

x~t=r~t‖r~t‖1\.\\tilde\{x\}\_\{t\}=\\frac\{\\tilde\{r\}\_\{t\}\}\{\\left\\lVert\\tilde\{r\}\_\{t\}\\right\\rVert\_\{1\}\}\.Before choosing its strategy, the learner receives a predictionmt∈ℝkm\_\{t\}\\in\\mathbb\{R\}^\{k\}of the utility vectorutu\_\{t\}\. The prediction step addsmtm\_\{t\}to the regret vector and applies a common scalar shiftγt\\gamma\_\{t\}, chosen uniquely so that the positive part preserves its Euclidean norm:

‖\[r~t\+mt−γt​𝟏\]\+‖2=ρt\.\\left\\lVert\\left\[\\tilde\{r\}\_\{t\}\+m\_\{t\}\-\\gamma\_\{t\}\\mathbf\{1\}\\right\]\_\{\+\}\\right\\rVert\_\{2\}=\\rho\_\{t\}\.Here\[z\]\+\\left\[z\\right\]\_\{\+\}denotes the coordinatewise positive part\. Set

rt=r~t\+mt−γt​𝟏,xt=\[rt\]\+‖\[rt\]\+‖1\.r\_\{t\}=\\tilde\{r\}\_\{t\}\+m\_\{t\}\-\\gamma\_\{t\}\\mathbf\{1\},\\qquad x\_\{t\}=\\frac\{\\left\[r\_\{t\}\\right\]\_\{\+\}\}\{\\left\\lVert\\left\[r\_\{t\}\\right\]\_\{\+\}\\right\\rVert\_\{1\}\}\.The distinction betweenx~t\\tilde\{x\}\_\{t\}andxtx\_\{t\}will matter later:x~t\\tilde\{x\}\_\{t\}is the strategy induced byr~t\\tilde\{r\}\_\{t\}before prediction, whereasxtx\_\{t\}is the strategy actually played after incorporatingmtm\_\{t\}\. After observingutu\_\{t\}, define the prediction error

\(3\)gt=\(ut−mt\)−⟨ut−mt,xt⟩​𝟏,g\_\{t\}=\(u\_\{t\}\-m\_\{t\}\)\-\\left\\langle u\_\{t\}\-m\_\{t\},x\_\{t\}\\right\\rangle\\mathbf\{1\},and update

\(4\)r~t\+1=\[rt\+gt\]\+\.\\tilde\{r\}\_\{t\+1\}=\\left\[r\_\{t\}\+g\_\{t\}\\right\]\_\{\+\}\.Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)establish that the regret\-vector norm is nondecreasing, so \([2](https://arxiv.org/html/2610.19016#S3.E2)\) implies

\(5\)ρt≥1for every​t\.\\rho\_\{t\}\\geq 1\\qquad\\text\{for every \}t\.
For their constant\-regret analysis, Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)require an extra\-gradient evaluation before the round\-ttplay\. Starting from the current regret vectors, IR\-PRM\+ first gives the strategiesx~t\\tilde\{x\}\_\{t\}andy~t\\tilde\{y\}\_\{t\}\. Zhang et al\. then evaluate the game at this fresh pair and use the resulting utility vectors as the predictions

mtX=A​y~t,mtY=−A⊤​x~t\.m\_\{t\}^\{X\}=A\\tilde\{y\}\_\{t\},\\qquad m\_\{t\}^\{Y\}=\-A^\{\\top\}\\tilde\{x\}\_\{t\}\.These predictions are fed into the predictive step above to produce the strategiesxtx\_\{t\}andyty\_\{t\}that are actually played\. The game is then evaluated again at\(xt,yt\)\(x\_\{t\},y\_\{t\}\)to obtain the round\-ttutilities used in the update\. Thus extra\-gradient prediction requires a preliminary game evaluation at\(x~t,y~t\)\(\\tilde\{x\}\_\{t\},\\tilde\{y\}\_\{t\}\)before the usual evaluation at the played strategies\.

With previous\-utility prediction, there is no preliminary game evaluation\. We setm1X=m1Y=0m\_\{1\}^\{X\}=m\_\{1\}^\{Y\}=0, and from roundt=2t=2onward simply reuse the utility observed on the preceding round:

mtX=A​yt−1,mtY=−A⊤​xt−1\.m\_\{t\}^\{X\}=Ay\_\{t\-1\},\\qquad m\_\{t\}^\{Y\}=\-A^\{\\top\}x\_\{t\-1\}\.IR\-PRM\+ then uses this prediction to producextx\_\{t\}andyty\_\{t\}and evaluates the game only at the played pair\.

#### Asymptotic notation\.

Throughout,𝒪T​\(⋅\)\\mathcal\{O\}\_\{T\}\(\\cdot\)suppresses constants independent ofTT, including dependence on action dimensions, weights in the weighted Minty condition, initialization, and the game\.

## 4\.Starting from the IR\-PRM\+ RVU Bound

The constant\-regret analysis of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)is based on an RVU\-type inequality for IR\-PRM\+\. We use the following intermediate form from their proof, which keeps the terminal regret\-vector norm explicit\.

###### Fact 1 \(Intermediate inequality from Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)\)\.

For IR\-PRM\+ with any nonnegative initial regret vector,

\(6\)Reg⁡\(T\)≤ρT\+1−∑t=1Tρt2​k​‖xt−x~t‖22\.\\operatorname\{Reg\}\(T\)\\leq\\rho\_\{T\+1\}\-\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\}\}\{2k\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\.

The negative term in \([6](https://arxiv.org/html/2610.19016#S4.E6)\) controls the prediction displacement from the intermediate strategyx~t\\tilde\{x\}\_\{t\}to the played strategyxtx\_\{t\}\. Under extra\-gradient prediction, this is directly aligned with the prediction error: for the two players, the corresponding errors areA⁡\(yt−y~t\)A\(y\_\{t\}\-\\tilde\{y\}\_\{t\}\)and−A⊤​\(xt−x~t\)\-A^\{\\top\}\(x\_\{t\}\-\\tilde\{x\}\_\{t\}\)\. Previous\-utility prediction changes this relationship\. Fort≥2t\\geq 2,

\(7\)utX−mtX\\displaystyle u\_\{t\}^\{X\}\-m\_\{t\}^\{X\}=A⁡\(yt−yt−1\),\\displaystyle=A\(y\_\{t\}\-y\_\{t\-1\}\),\(8\)utY−mtY\\displaystyle u\_\{t\}^\{Y\}\-m\_\{t\}^\{Y\}=−A⊤​\(xt−xt−1\)\.\\displaystyle=\-A^\{\\top\}\(x\_\{t\}\-x\_\{t\-1\}\)\.Thus the prediction error depends on the full movement between consecutive played strategies, whereas \([6](https://arxiv.org/html/2610.19016#S4.E6)\) directly controls only the prediction displacement fromx~t\\tilde\{x\}\_\{t\}toxtx\_\{t\}\. The remaining part is the post\-update displacement fromxt−1x\_\{t\-1\}tox~t\\tilde\{x\}\_\{t\}\. The next section relates this post\-update displacement to growth of the same regret\-vector normρt\\rho\_\{t\}that appears in \([6](https://arxiv.org/html/2610.19016#S4.E6)\)\.

Proofs for the results in Sections[4](https://arxiv.org/html/2610.19016#S4)–[6](https://arxiv.org/html/2610.19016#S6)that do not appear in the main text are collected in Appendix A\.

## 5\.Single\-Learner Displacement and Movement Bounds

Consider a single generic IR\-PRM\+ learner\. The negative term in Eq\. \([6](https://arxiv.org/html/2610.19016#S4.E6)\) controls the prediction displacement fromx~t\\tilde\{x\}\_\{t\}to the played strategyxtx\_\{t\}, but previous\-utility prediction depends on the full movement between consecutive played strategies\. The missing piece is therefore the post\-update displacement fromxtx\_\{t\}tox~t\+1\\tilde\{x\}\_\{t\+1\}\. The key observation is that this post\-update displacement is controlled by the increase in the regret\-vector norm\. Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)makes this relation quantitative\.

###### Lemma 0 \(Post\-update displacement bound\)\.

For akk\-action IR\-PRM\+ learner and everyt≥1t\\geq 1,

\(9\)‖x~t\+1−xt‖22≤2​\(1\+k\)2​\(ρt\+1−ρt\)ρt\+1\.\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\frac\{2\(1\+\\sqrt\{k\}\)^\{2\}\(\\rho\_\{t\+1\}\-\\rho\_\{t\}\)\}\{\\rho\_\{t\+1\}\}\.

Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)establish that the IR\-PRM\+ regret\-vector norm is nondecreasing,ρt\+1≥ρt\\rho\_\{t\+1\}\\geq\\rho\_\{t\}\. Their update also gives

‖\[rt\]\+‖2=ρt,‖r~t\+1‖2=ρt\+1,\\left\\lVert\\left\[r\_\{t\}\\right\]\_\{\+\}\\right\\rVert\_\{2\}=\\rho\_\{t\},\\qquad\\left\\lVert\\tilde\{r\}\_\{t\+1\}\\right\\rVert\_\{2\}=\\rho\_\{t\+1\},and their monotonicity argument yields

\(10\)⟨r~t\+1,\[rt\]\+⟩≥ρt2\.\\left\\langle\\tilde\{r\}\_\{t\+1\},\\left\[r\_\{t\}\\right\]\_\{\+\}\\right\\rangle\\geq\\rho\_\{t\}^\{2\}\.We use these properties to relate the increase in the regret\-vector norm to the corresponding post\-update displacement\.

###### Proof\.

Sincextx\_\{t\}andx~t\+1\\tilde\{x\}\_\{t\+1\}are theℓ1\\ell\_\{1\}normalizations of\[rt\]\+/ρt\\left\[r\_\{t\}\\right\]\_\{\+\}/\\rho\_\{t\}andr~t\+1/ρt\+1\\tilde\{r\}\_\{t\+1\}/\\rho\_\{t\+1\}, respectively, an elementary normalization inequality for nonnegative Euclidean unit vectors gives

\(11\)‖x~t\+1−xt‖22≤\(1\+k\)2​‖r~t\+1ρt\+1−\[rt\]\+ρt‖22\.\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq\(1\+\\sqrt\{k\}\)^\{2\}\\left\\lVert\\frac\{\\tilde\{r\}\_\{t\+1\}\}\{\\rho\_\{t\+1\}\}\-\\frac\{\\left\[r\_\{t\}\\right\]\_\{\+\}\}\{\\rho\_\{t\}\}\\right\\rVert\_\{2\}^\{2\}\.See the normalization bound in Appendix G\.

As we have Euclidean unit vectors,

‖r~t\+1ρt\+1−\[rt\]\+ρt‖22\\displaystyle\\left\\lVert\\frac\{\\tilde\{r\}\_\{t\+1\}\}\{\\rho\_\{t\+1\}\}\-\\frac\{\\left\[r\_\{t\}\\right\]\_\{\+\}\}\{\\rho\_\{t\}\}\\right\\rVert\_\{2\}^\{2\}=‖r~t\+1ρt\+1‖22\+‖\[rt\]\+ρt‖22−2​⟨r~t\+1ρt\+1,\[rt\]\+ρt⟩\\displaystyle=\\left\\lVert\\frac\{\\tilde\{r\}\_\{t\+1\}\}\{\\rho\_\{t\+1\}\}\\right\\rVert\_\{2\}^\{2\}\+\\left\\lVert\\frac\{\\left\[r\_\{t\}\\right\]\_\{\+\}\}\{\\rho\_\{t\}\}\\right\\rVert\_\{2\}^\{2\}\-2\\left\\langle\\frac\{\\tilde\{r\}\_\{t\+1\}\}\{\\rho\_\{t\+1\}\},\\frac\{\\left\[r\_\{t\}\\right\]\_\{\+\}\}\{\\rho\_\{t\}\}\\right\\rangle=1\+1−2​⟨r~t\+1,\[rt\]\+⟩ρt​ρt\+1\\displaystyle=1\+1\-\\frac\{2\\left\\langle\\tilde\{r\}\_\{t\+1\},\\left\[r\_\{t\}\\right\]\_\{\+\}\\right\\rangle\}\{\\rho\_\{t\}\\rho\_\{t\+1\}\}≤2−2​ρt2ρt​ρt\+1\\displaystyle\\leq 2\-\\frac\{2\\rho\_\{t\}^\{2\}\}\{\\rho\_\{t\}\\rho\_\{t\+1\}\}\(12\)=2​\(ρt\+1−ρt\)ρt\+1\.\\displaystyle=\\frac\{2\(\\rho\_\{t\+1\}\-\\rho\_\{t\}\)\}\{\\rho\_\{t\+1\}\}\.The inequality uses \([10](https://arxiv.org/html/2610.19016#S5.E10)\)\. Substituting \([12](https://arxiv.org/html/2610.19016#S5.E12)\) into \([11](https://arxiv.org/html/2610.19016#S5.E11)\) gives \([9](https://arxiv.org/html/2610.19016#S5.E9)\)\. ∎

Together with the prediction displacement already controlled by \([6](https://arxiv.org/html/2610.19016#S4.E6)\), Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)gives the two pieces needed to bound movement between consecutive played strategies\. This also leads to a two\-displacement RVU bound with explicit terms forxt−x~tx\_\{t\}\-\\tilde\{x\}\_\{t\}andx~t\+1−xt\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}; see the two\-displacement RVU bound in Appendix H\. This strengthened RVU bound is not used in the main convergence analysis, which works directly with regret\-vector norm growth\.

###### Corollary 0 \(Consecutive\-iterate movement\)\.

For akk\-action learner and everyt≥1t\\geq 1,

\(13\)‖xt\+1−xt‖22≤2​‖xt\+1−x~t\+1‖22\+4​\(1\+k\)2​\(ρt\+1−ρt\)ρt\+1\.\\left\\lVert x\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq 2\\left\\lVert x\_\{t\+1\}\-\\tilde\{x\}\_\{t\+1\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{4\(1\+\\sqrt\{k\}\)^\{2\}\(\\rho\_\{t\+1\}\-\\rho\_\{t\}\)\}\{\\rho\_\{t\+1\}\}\.

###### Proof\.

Insertx~t\+1\\tilde\{x\}\_\{t\+1\}betweenxtx\_\{t\}andxt\+1x\_\{t\+1\}and use‖a\+b‖22≤2​‖a‖22\+2​‖b‖22\\left\\lVert a\+b\\right\\rVert\_\{2\}^\{2\}\\leq 2\\left\\lVert a\\right\\rVert\_\{2\}^\{2\}\+2\\left\\lVert b\\right\\rVert\_\{2\}^\{2\}:

‖xt\+1−xt‖22≤2​‖xt\+1−x~t\+1‖22\+2​‖x~t\+1−xt‖22\.\\left\\lVert x\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq 2\\left\\lVert x\_\{t\+1\}\-\\tilde\{x\}\_\{t\+1\}\\right\\rVert\_\{2\}^\{2\}\+2\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\.Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)bounds the last term and gives \([13](https://arxiv.org/html/2610.19016#S5.E13)\)\. ∎

## 6\.Two\-Player Analysis

We now return to the two\-player game\. Our goal is to control the growth of the two regret\-vector norms\. Under previous\-utility prediction, each player’s prediction error is driven by the opponent’s consecutive strategy movement; Corollary[2](https://arxiv.org/html/2610.19016#S5.Thmtheorem2)expresses that movement in terms of prediction displacement and the opponent’s norm growth, which controls the post\-update displacement\. The proof below combines this movement bound with a one\-step IR\-PRM\+ norm\-growth estimate to obtain the coupled recurrences\. Here and below,‖A‖2\\\|A\\\|\_\{2\}denotes the spectral norm of the game matrix\.

###### Lemma 0 \(Regret\-vector norm growth in self\-play\)\.

LetκA:=2​\(1\+kX\)2​\(1\+kY\)2​‖A‖22\\kappa\_\{A\}:=2\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\\|A\\\|\_\{2\}^\{2\}\. For everyt≥2t\\geq 2,

\(14\)ρt\+1X−ρtX\\displaystyle\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}≤κAρtX​ρtY​\[ρtY2​kY​‖yt−y~t‖22\+ρtY−ρt−1Y\],\\displaystyle\\leq\\frac\{\\kappa\_\{A\}\}\{\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\}\\left\[\\frac\{\\rho\_\{t\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\\rho\_\{t\}^\{Y\}\-\\rho\_\{t\-1\}^\{Y\}\\right\],\(15\)ρt\+1Y−ρtY\\displaystyle\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}≤κAρtX​ρtY​\[ρtX2​kX​‖xt−x~t‖22\+ρtX−ρt−1X\]\.\\displaystyle\\leq\\frac\{\\kappa\_\{A\}\}\{\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\}\\left\[\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\\rho\_\{t\}^\{X\}\-\\rho\_\{t\-1\}^\{X\}\\right\]\.

###### Proof\.

We first record a norm\-growth result used below\. Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)give

ρt\+12−ρt2≤‖gt‖22,\\rho\_\{t\+1\}^\{2\}\-\\rho\_\{t\}^\{2\}\\leq\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\},
and their monotonicity result givesρt\+1≥ρt\\rho\_\{t\+1\}\\geq\\rho\_\{t\}\. Therefore,

\(16\)0≤ρt\+1−ρt≤‖gt‖22ρt\+1\+ρt≤‖gt‖222​ρt\.0\\leq\\rho\_\{t\+1\}\-\\rho\_\{t\}\\leq\\frac\{\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\}\}\{\\rho\_\{t\+1\}\+\\rho\_\{t\}\}\\leq\\frac\{\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\}\}\{2\\rho\_\{t\}\}\.
Forv∈ℝhv\\in\\mathbb\{R\}^\{h\}andx∈Δhx\\in\\Delta\_\{h\}, the Cauchy–Schwarz inequality, together with‖x‖2≤1\\left\\lVert x\\right\\rVert\_\{2\}\\leq 1, gives

\(17\)‖v−⟨v,x⟩​𝟏‖2≤\(1\+h\)​‖v‖2\.\\left\\lVert v\-\\left\\langle v,x\\right\\rangle\\mathbf\{1\}\\right\\rVert\_\{2\}\\leq\(1\+\\sqrt\{h\}\)\\left\\lVert v\\right\\rVert\_\{2\}\.For the row player, previous\-utility prediction yieldsutX−mtX=A⁡\(yt−yt−1\)u\_\{t\}^\{X\}\-m\_\{t\}^\{X\}=A\(y\_\{t\}\-y\_\{t\-1\}\)\. Applying \([17](https://arxiv.org/html/2610.19016#S6.E17)\) and‖A​z‖2≤‖A‖2​‖z‖2\\left\\lVert Az\\right\\rVert\_\{2\}\\leq\\left\\lVert A\\right\\rVert\_\{2\}\\left\\lVert z\\right\\rVert\_\{2\}to \([16](https://arxiv.org/html/2610.19016#S6.E16)\), we obtain

\(18\)ρt\+1X−ρtX≤\(1\+kX\)2​‖A‖222​ρtX​‖yt−yt−1‖22\.\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}\\leq\\frac\{\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{2\\rho\_\{t\}^\{X\}\}\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\.Corollary[2](https://arxiv.org/html/2610.19016#S5.Thmtheorem2)for the column player at timet−1t\-1gives

‖yt−yt−1‖22≤2​‖yt−y~t‖22\+4​\(1\+kY\)2ρtY​\(ρtY−ρt−1Y\)\.\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\\leq 2\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{4\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\}\{\\rho\_\{t\}^\{Y\}\}\(\\rho\_\{t\}^\{Y\}\-\\rho\_\{t\-1\}^\{Y\}\)\.Substituting into \([18](https://arxiv.org/html/2610.19016#S6.E18)\) and using the specified value ofκA\\kappa\_\{A\}yields

ρt\+1X−ρtX\\displaystyle\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}≤κAρtX​ρtY\[ρtY2​\(1\+kY\)2‖yt−y~t‖22\\displaystyle\\leq\\frac\{\\kappa\_\{A\}\}\{\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\}\\biggl\[\\frac\{\\rho\_\{t\}^\{Y\}\}\{2\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+ρtY−ρt−1Y\]\.\\displaystyle\\hskip 49\.0pt\+\\rho\_\{t\}^\{Y\}\-\\rho\_\{t\-1\}^\{Y\}\\biggr\]\.Because\(1\+kY\)2≥kY\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\geq k^\{Y\}, this implies \([14](https://arxiv.org/html/2610.19016#S6.E14)\)\. Exchanging the players and using‖A⊤‖2=‖A‖2\\left\\lVert A^\{\\top\}\\right\\rVert\_\{2\}=\\left\\lVert A\\right\\rVert\_\{2\}proves \([15](https://arxiv.org/html/2610.19016#S6.E15)\)\. ∎

The coupled coefficient1/\(ρtX​ρtY\)1/\(\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\)is the self\-damping term used in the argument below\.

###### Theorem 2 \(Constant regret\)\.

In any finite two\-player zero\-sum matrix game, simultaneous IR\-PRM\+ with previous\-utility prediction has external regrets bounded by a finite game\-dependent constant independent ofTT\. Consequently,

Gap⁡\(x¯T,y¯T\)=𝒪T​\(T−1\)\.\\operatorname\{Gap\}\(\\bar\{x\}\_\{T\},\\bar\{y\}\_\{T\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)\.

###### Proof\.

By \([5](https://arxiv.org/html/2610.19016#S3.E5)\), both regret\-vector norms are at least one from the outset\. Since the norms are also nondecreasing, so is their product\. Consider the sufficient threshold

\(19\)ρtX​ρtY≥8​\(1\+kX\)2​\(1\+kY\)2​‖A‖22\.\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\\geq 8\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\\|A\\\|\_\{2\}^\{2\}\.If this threshold is never reached, then, sinceρtX,ρtY≥1\\rho\_\{t\}^\{X\},\\rho\_\{t\}^\{Y\}\\geq 1, each norm is bounded above by their product and hence by the right\-hand side of \([19](https://arxiv.org/html/2610.19016#S6.E19)\)\. If the threshold is reached, monotonicity ensures that it continues to hold\. Therefore, at all times after the threshold is reached the coefficient in \([14](https://arxiv.org/html/2610.19016#S6.E14)\) and \([15](https://arxiv.org/html/2610.19016#S6.E15)\) is at most

κAρtX​ρtY≤2​\(1\+kX\)2​\(1\+kY\)2​‖A‖228​\(1\+kX\)2​\(1\+kY\)2​‖A‖22=14\.\\frac\{\\kappa\_\{A\}\}\{\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\}\\leq\\frac\{2\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\\|A\\\|\_\{2\}^\{2\}\}\{8\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\\|A\\\|\_\{2\}^\{2\}\}=\\frac\{1\}\{4\}\.
To see why this is enough, defineSt:=ρtX\+ρtYS\_\{t\}:=\\rho\_\{t\}^\{X\}\+\\rho\_\{t\}^\{Y\}and

Ht:=ρtX2​kX​‖xt−x~t‖22\+ρtY2​kY​‖yt−y~t‖22\.H\_\{t\}:=\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{\\rho\_\{t\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\.
Adding \([14](https://arxiv.org/html/2610.19016#S6.E14)\) and \([15](https://arxiv.org/html/2610.19016#S6.E15)\) then gives, after the threshold is reached,

St\+1−St≤14​Ht\+14​\(St−St−1\)\.S\_\{t\+1\}\-S\_\{t\}\\leq\\frac\{1\}\{4\}H\_\{t\}\+\\frac\{1\}\{4\}\(S\_\{t\}\-S\_\{t\-1\}\)\.Letτ\\taudenote the first round from which the threshold holds\. Summing fromτ\\tautoTTgives

ST\+1−Sτ≤14​∑t=τTHt\+14​\(ST−Sτ−1\)\.S\_\{T\+1\}\-S\_\{\\tau\}\\leq\\frac\{1\}\{4\}\\sum\_\{t=\\tau\}^\{T\}H\_\{t\}\+\\frac\{1\}\{4\}\(S\_\{T\}\-S\_\{\\tau\-1\}\)\.SinceST≤ST\+1S\_\{T\}\\leq S\_\{T\+1\}, moving the resulting quarter ofST\+1−SτS\_\{T\+1\}\-S\_\{\\tau\}to the left and absorbing the finite threshold\-crossing terms into a constant gives

\(20\)ST\+1≤C0\+13​∑t=1THt,S\_\{T\+1\}\\leq C\_\{0\}\+\\frac\{1\}\{3\}\\sum\_\{t=1\}^\{T\}H\_\{t\},whereC0C\_\{0\}is finite and independent ofTT\.

The remaining step is to control the accumulated weighted squared prediction displacement terms\. Applying Fact[1](https://arxiv.org/html/2610.19016#Thmfact1)to each player and using the fact that the game is zero\-sum gives

\(21\)0≤T​Gap⁡\(x¯T,y¯T\)=RegX⁡\(T\)\+RegY⁡\(T\)≤ST\+1−∑t=1THt\.0\\leq T\\,\\operatorname\{Gap\}\(\\bar\{x\}\_\{T\},\\bar\{y\}\_\{T\}\)=\\operatorname\{Reg\}\_\{X\}\(T\)\+\\operatorname\{Reg\}\_\{Y\}\(T\)\\leq S\_\{T\+1\}\-\\sum\_\{t=1\}^\{T\}H\_\{t\}\.Hence∑t=1THt≤ST\+1\\sum\_\{t=1\}^\{T\}H\_\{t\}\\leq S\_\{T\+1\}\. Substituting this into \([20](https://arxiv.org/html/2610.19016#S6.E20)\) yieldsST\+1≤C0\+13​ST\+1S\_\{T\+1\}\\leq C\_\{0\}\+\\frac\{1\}\{3\}S\_\{T\+1\}, and thereforeST\+1≤32​C0S\_\{T\+1\}\\leq\\frac\{3\}\{2\}C\_\{0\}\. ∎

Theorem[2](https://arxiv.org/html/2610.19016#S6.Thmtheorem2)gives the desired𝒪T​\(T−1\)\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)average\-gap rate\. The same bounds also control the current simultaneous profiles\. The additional ingredient is a predicted\-utility residual bound \(Lemma 4 in Appendix A\.2\)\. For a utility vectorvvand strategyx∈Δkx\\in\\Delta\_\{k\}, the quantitymaxp∈Δk⁡⟨v,p−x⟩\\max\_\{p\\in\\Delta\_\{k\}\}\\left\\langle v,p\-x\\right\\rangleis the improvement available from switching to a best response\. When the regret\-vector norm is bounded, this residual for the predicted utility is controlled by the prediction displacement‖xt−x~t‖2\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\. Intuitively, if incorporating the prediction barely changes the normalized regret vector, then the prediction cannot cause a large profitable deviation\. Under previous\-utility prediction, Corollary[2](https://arxiv.org/html/2610.19016#S5.Thmtheorem2)also makes the predicted and current utilities asymptotically close by controlling consecutive strategy movement\. Combining these facts with the square\-summable strategy\-movement bounds underlying Theorem[2](https://arxiv.org/html/2610.19016#S6.Thmtheorem2)yields square summability of the current duality gaps\.

Let𝒵⋆\\mathcal\{Z\}^\{\\star\}denote the Nash equilibrium set\. Here and below,dist\\operatorname\{dist\}denotes Euclidean distance to a set\.

###### Theorem 3 \(Current\-profile convergence\)\.

In the same setting,

\(22\)∑t=1∞Gap⁡\(xt,yt\)2<∞\.\\sum\_\{t=1\}^\{\\infty\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)^\{2\}<\\infty\.Consequently,

Gap⁡\(xt,yt\)→0,dist⁡\(\(xt,yt\),𝒵⋆\)→0\.\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)\\to 0,\\qquad\\operatorname\{dist\}\\bigl\(\(x\_\{t\},y\_\{t\}\),\\mathcal\{Z\}^\{\\star\}\\bigr\)\\to 0\.Moreover,

\(23\)min1≤t≤TGap\(xt,yt\)=𝒪T\(T−1/2\)\.\\min\_\{1\\leq t\\leq T\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)\.

## 7\.Simultaneous IR\-PCFR\+ in Extensive\-Form Games

The CFR reduction follows Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\), who use the standard CFR decomposition to show that a full\-reach Nash equilibrium induces a weighted Minty condition for the counterfactual\-utility operator\. We retain this reduction but establish the convergence guarantees under previous\-utility prediction rather than extra\-gradient prediction\. This requires extending our two\-player norm\-growth analysis to finitely many coupled simplex learners\. Combined with the existing CFR reduction, the resulting bounds yield the simultaneous IR\-PCFR\+ guarantees below\. Proofs of the product\-of\-simplices and simultaneous CFR results established here are collected in Appendices B and C\.

### 7\.1\.Extensive\-form games and CFR

Consider a finite two\-player zero\-sum extensive\-form game with perfect recall\. LetℐX\\mathcal\{I\}\_\{X\}andℐY\\mathcal\{I\}\_\{Y\}denote the information sets of the two players, and letA⁡\(I\)A\(I\)be the actions available at information setII\. A behavioral strategy assigns a local distribution to every information set:

xI∈Δ\|A⁡\(I\)\|\(I∈ℐX\),yJ∈Δ\|A⁡\(J\)\|\(J∈ℐY\)\.x\_\{I\}\\in\\Delta\_\{\|A\(I\)\|\}\\quad\(I\\in\\mathcal\{I\}\_\{X\}\),\\qquad y\_\{J\}\\in\\Delta\_\{\|A\(J\)\|\}\\quad\(J\\in\\mathcal\{I\}\_\{Y\}\)\.We writex=\(xI\)I∈ℐXx=\(x\_\{I\}\)\_\{I\\in\\mathcal\{I\}\_\{X\}\}andy=\(yJ\)J∈ℐYy=\(y\_\{J\}\)\_\{J\\in\\mathcal\{I\}\_\{Y\}\}for the two behavioral strategies\.

We use the standard sequence\-form representation\([Koller et al\., 1994](https://arxiv.org/html/2610.19016#bib.bib14);[von Stengel, 1996](https://arxiv.org/html/2610.19016#bib.bib15)\)\. For a playerP∈\{X,Y\}P\\in\\\{X,Y\\\}, sequences record only that player’s actions\. ForI∈ℐPI\\in\\mathcal\{I\}\_\{P\}, letpIp\_\{I\}denote its parent sequence, writeI​aIafor the sequence obtained by extendingpIp\_\{I\}with actionaaatII, and letCσC\_\{\\sigma\}denote the information sets ofPPwhose parent sequence isσ\\sigma\. IfqXq^\{X\}is the realization plan induced byxx, then for everyXX\-sequenceσ\\sigma,

qX​\[σ\]=∏I​a⪯σI∈ℐXxI​\[a\],q^\{X\}\[\\sigma\]=\\prod\_\{\\begin\{subarray\}\{c\}Ia\\preceq\\sigma\\\\ I\\in\\mathcal\{I\}\_\{X\}\\end\{subarray\}\}x\_\{I\}\[a\],whereI​a⪯σIa\\preceq\\sigmameans that the sequenceI​aIais a prefix ofσ\\sigma, and analogously forqYq^\{Y\}\. Let𝒬X\\mathcal\{Q\}\_\{X\}and𝒬Y\\mathcal\{Q\}\_\{Y\}denote the corresponding sequence\-form polytopes\.

There is a matrixAseqA\_\{\\rm seq\}such that the expected payoff toXXis

U⁡\(qX,qY\)=\(qX\)⊤​Aseq​qY\.U\(q^\{X\},q^\{Y\}\)=\(q^\{X\}\)^\{\\top\}A\_\{\\rm seq\}q^\{Y\}\.The sequence\-form duality gap is

Gap𝒬⁡\(qX,qY\):=maxq^X∈𝒬X⁡\(q^X\)⊤​Aseq​qY−minq^Y∈𝒬Y⁡\(qX\)⊤​Aseq​q^Y\.\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q^\{X\},q^\{Y\}\):=\\max\_\{\\widehat\{q\}^\{X\}\\in\\mathcal\{Q\}\_\{X\}\}\(\\widehat\{q\}^\{X\}\)^\{\\top\}A\_\{\\rm seq\}q^\{Y\}\-\\min\_\{\\widehat\{q\}^\{Y\}\\in\\mathcal\{Q\}\_\{Y\}\}\(q^\{X\}\)^\{\\top\}A\_\{\\rm seq\}\\widehat\{q\}^\{Y\}\.
CFR runs one local learner at each information set\([Zinkevich et al\., 2007](https://arxiv.org/html/2610.19016#bib.bib12)\)\. The utility vector supplied to that learner is its counterfactual utility\. In the notation of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\), these vectors can be defined recursively\. ForI∈ℐXI\\in\\mathcal\{I\}\_\{X\},

\(24\)FIX​\(x,y\)​\[a\]=\(Aseq​qY\)​\[I​a\]\+∑K∈CI​a⟨FKX​\(x,y\),xK⟩,F\_\{I\}^\{X\}\(x,y\)\[a\]=\(A\_\{\\rm seq\}q^\{Y\}\)\[Ia\]\+\\sum\_\{K\\in C\_\{Ia\}\}\\left\\langle F\_\{K\}^\{X\}\(x,y\),x\_\{K\}\\right\\rangle,with theYY\-player counterfactual utilities defined analogously using−Aseq⊤​qX\-A\_\{\\rm seq\}^\{\\top\}q^\{X\}\. An empty child sum is zero\. Thus each information set induces a standard simplex regret\-minimization problem, coupled to the others through the current behavioral profile\.

The standard CFR decomposition relates these local counterfactual utilities to the sequence\-form game; we recall the underlying identity in Appendix C\.1\. A behavioral profile has full reach if each player’s realization plan assigns strictly positive probability to the parent sequence of every information set\. In particular, Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)show that if the game admits a full\-reach Nash equilibrium, then the counterfactual\-utility operator satisfies the weighted Minty condition used below, with weights given by the equilibrium parent\-sequence realization probabilities\. The full\-reach condition ensures that all of these weights are strictly positive\.

### 7\.2\.Simultaneous IR\-PRM\+ on a product of simplices

We now establish the product\-of\-simplices result needed for this reduction under previous\-utility prediction\. Following the product\-space framework of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\), specialized to the simplex domains on which IR\-PRM\+ operates, let

𝒳=∏i=1NΔki,F=\(F1,…,FN\)\.\\mathcal\{X\}=\\prod\_\{i=1\}^\{N\}\\Delta\_\{k\_\{i\}\},\\qquad F=\(F\_\{1\},\\ldots,F\_\{N\}\)\.In this subsection,x=\(x1,…,xN\)∈𝒳x=\(x\_\{1\},\\ldots,x\_\{N\}\)\\in\\mathcal\{X\}denotes the full product profile\.NNis the total number of local simplex learners andkik\_\{i\}is the number of actions of learnerii\. In the CFR application, each local learner corresponds to one information set, so

N=\|ℐX\|\+\|ℐY\|,ki=\|A⁡\(Ii\)\|,N=\|\\mathcal\{I\}\_\{X\}\|\+\|\\mathcal\{I\}\_\{Y\}\|,\\qquad k\_\{i\}=\|A\(I\_\{i\}\)\|,whereIiI\_\{i\}is the information set represented by learnerii\. Thus each componentxi∈Δkix\_\{i\}\\in\\Delta\_\{k\_\{i\}\}is one local information\-set strategy\.

The mapFi:𝒳→ℝkiF\_\{i\}:\\mathcal\{X\}\\to\\mathbb\{R\}^\{k\_\{i\}\}is bounded and, for some finiteLiL\_\{i\},

\(25\)‖Fi​\(x\)−Fi​\(x′\)‖2≤Li​‖x−x′‖2for all​x,x′∈𝒳\.\\left\\lVert F\_\{i\}\(x\)\-F\_\{i\}\(x^\{\\prime\}\)\\right\\rVert\_\{2\}\\leq L\_\{i\}\\left\\lVert x\-x^\{\\prime\}\\right\\rVert\_\{2\}\\qquad\\text\{for all \}x,x^\{\\prime\}\\in\\mathcal\{X\}\.Each learneriiuses the unit\-norm initializationr~i,1=ki−1/2𝟏\\tilde\{r\}\_\{i,1\}=k\_\{i\}^\{\-1/2\}\\mathbf\{1\}, soρi,1=1\\rho\_\{i,1\}=1\. We setmi,1=0m\_\{i,1\}=0and, fort≥2t\\geq 2, use

\(26\)ui,t=Fi​\(xt\),mi,t=Fi​\(xt−1\)\.u\_\{i,t\}=F\_\{i\}\(x\_\{t\}\),\\qquad m\_\{i,t\}=F\_\{i\}\(x\_\{t\-1\}\)\.Writeρi,t\\rho\_\{i,t\}for its regret\-vector norm andx~i,t\\tilde\{x\}\_\{i,t\}for the strategy induced before the round\-ttprediction\. Monotonicity givesρi,t≥1\\rho\_\{i,t\}\\geq 1for every learner and every round\.

The two\-player proof above used the fact that one learner’s prediction error is controlled by the other learner’s consecutive movement\. Here the feedback at learneriican depend on every component ofxx, so all local movements can contribute\. Lipschitzness first bounds learnerii’s prediction error by the movement of the full product profile; decomposing that movement across components and applying Corollary[2](https://arxiv.org/html/2610.19016#S5.Thmtheorem2)to each learner gives the following recurrence\.

###### Lemma 0 \(Product\-of\-simplices norm\-growth bound\)\.

For every learneriiand everyt≥2t\\geq 2,

ρi,t\+1−ρi,t≤∑j=1N\\displaystyle\\rho\_\{i,t\+1\}\-\\rho\_\{i,t\}\\leq\\sum\_\{j=1\}^\{N\}2​\(1\+kj\)2​\(1\+ki\)2​Li2ρi,t​ρj,t\[ρj,t2​kj‖xj,t−x~j,t‖22\\displaystyle\\frac\{2\(1\+\\sqrt\{k\_\{j\}\}\)^\{2\}\(1\+\\sqrt\{k\_\{i\}\}\)^\{2\}L\_\{i\}^\{2\}\}\{\\rho\_\{i,t\}\\rho\_\{j,t\}\}\\left\[\\frac\{\\rho\_\{j,t\}\}\{2k\_\{j\}\}\\left\\lVert x\_\{j,t\}\-\\tilde\{x\}\_\{j,t\}\\right\\rVert\_\{2\}^\{2\}\\right\.\(27\)\+\(ρj,t−ρj,t−1\)\]\.\\displaystyle\\left\.\\hskip 88\.00015pt\+\(\\rho\_\{j,t\}\-\\rho\_\{j,t\-1\}\)\\right\]\.

The proof extends the argument for Lemma[1](https://arxiv.org/html/2610.19016#S6.Thmtheorem1)to the coupled local learners on the product of simplices; the full proof is given in Appendix B\.1\.

To close these coupled recurrences, we use the weighted Minty condition of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\): there arex⋆∈𝒳x^\{\\star\}\\in\\mathcal\{X\}and fixed weightswi\>0w\_\{i\}\>0such that

\(28\)∑i=1Nwi​⟨Fi​\(x\),xi⋆−xi⟩≥0for every​x∈𝒳\.\\sum\_\{i=1\}^\{N\}w\_\{i\}\\left\\langle F\_\{i\}\(x\),x\_\{i\}^\{\\star\}\-x\_\{i\}\\right\\rangle\\geq 0\\qquad\\text\{for every \}x\\in\\mathcal\{X\}\.WithNNlocal learners, different regret\-vector norms may enter the large\-norm regime at different times\. The next theorem shows that this condition is nevertheless sufficient to bound all of them\.

###### Theorem 2 \(Constant regret under weighted Minty\)\.

Suppose eachFiF\_\{i\}is bounded and Lipschitz andFFsatisfies the weighted Minty condition\. Then, under simultaneous IR\-PRM\+ with previous\-utility prediction, every local regret\-vector norm remains bounded over time and every local learner’s external regret is bounded by a finite game\-dependent constant independent ofTT\.

###### Proof\.

Let

wmin:=mini⁡wi,wmax:=maxi⁡wi,w\_\{\\min\}:=\\min\_\{i\}w\_\{i\},\\qquad w\_\{\\max\}:=\\max\_\{i\}w\_\{i\},and write

ci​j:=2​\(1\+kj\)2​\(1\+ki\)2​Li2\.c\_\{ij\}:=2\(1\+\\sqrt\{k\_\{j\}\}\)^\{2\}\(1\+\\sqrt\{k\_\{i\}\}\)^\{2\}L\_\{i\}^\{2\}\.Sinceρj,t≥1\\rho\_\{j,t\}\\geq 1for every learnerjj, choose a finite thresholdΘi≥1\\Theta\_\{i\}\\geq 1such that

ci​jΘi≤wmin4​N​wmaxfor every​j\.\\frac\{c\_\{ij\}\}\{\\Theta\_\{i\}\}\\leq\\frac\{w\_\{\\min\}\}\{4Nw\_\{\\max\}\}\\qquad\\text\{for every \}j\.Hence, wheneverρi,t≥Θi\\rho\_\{i,t\}\\geq\\Theta\_\{i\}, every coefficient in \([27](https://arxiv.org/html/2610.19016#S7.E27)\) satisfies

ci​jρi,t​ρj,t≤wmin4​N​wmax\.\\frac\{c\_\{ij\}\}\{\\rho\_\{i,t\}\\rho\_\{j,t\}\}\\leq\\frac\{w\_\{\\min\}\}\{4Nw\_\{\\max\}\}\.If learneriinever reachesΘi\\Theta\_\{i\}, its norm is already bounded; otherwise monotonicity ensures that it remains aboveΘi\\Theta\_\{i\}after its first crossing\.

Summing the above\-threshold recurrences in \([27](https://arxiv.org/html/2610.19016#S7.E27)\) over theNNlearners andTTrounds removes the factor1/N1/Nand telescopes\. Thus, after absorbing the below\-threshold contributions into a finite constantCCindependent ofTT,

∑iρi,T\+1≤C\+wmin4​wmax​∑t=1T∑iρi,t2​ki​‖xi,t−x~i,t‖22\+wmin4​wmax​∑iρi,T\+1,\\sum\_\{i\}\\rho\_\{i,T\+1\}\\leq C\+\\frac\{w\_\{\\min\}\}\{4w\_\{\\max\}\}\\sum\_\{t=1\}^\{T\}\\sum\_\{i\}\\frac\{\\rho\_\{i,t\}\}\{2k\_\{i\}\}\\left\\lVert x\_\{i,t\}\-\\tilde\{x\}\_\{i,t\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{w\_\{\\min\}\}\{4w\_\{\\max\}\}\\sum\_\{i\}\\rho\_\{i,T\+1\},where the last term telescopes the norm\-growth terms in \([27](https://arxiv.org/html/2610.19016#S7.E27)\)\.

It remains to control the squared prediction displacements\. Sincexi⋆x\_\{i\}^\{\\star\}is a feasible comparator,

Regi⁡\(T\)≥∑t=1T⟨Fi​\(xt\),xi⋆−xi,t⟩\.\\operatorname\{Reg\}\_\{i\}\(T\)\\geq\\sum\_\{t=1\}^\{T\}\\left\\langle F\_\{i\}\(x\_\{t\}\),x\_\{i\}^\{\\star\}\-x\_\{i,t\}\\right\\rangle\.Multiplying bywiw\_\{i\}, summing overii, and using the weighted Minty condition gives

0≤∑iwi​Regi⁡\(T\)\.0\\leq\\sum\_\{i\}w\_\{i\}\\operatorname\{Reg\}\_\{i\}\(T\)\.Fact[1](https://arxiv.org/html/2610.19016#Thmfact1)therefore yields

∑t=1T∑iwi​ρi,t2​ki​‖xi,t−x~i,t‖22≤∑iwi​ρi,T\+1\.\\sum\_\{t=1\}^\{T\}\\sum\_\{i\}w\_\{i\}\\frac\{\\rho\_\{i,t\}\}\{2k\_\{i\}\}\\left\\lVert x\_\{i,t\}\-\\tilde\{x\}\_\{i,t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\sum\_\{i\}w\_\{i\}\\rho\_\{i,T\+1\}\.Usingwi≥wminw\_\{i\}\\geq w\_\{\\min\}on the left andwi≤wmaxw\_\{i\}\\leq w\_\{\\max\}on the right,

\(29\)∑t=1T∑iρi,t2​ki​‖xi,t−x~i,t‖22≤wmaxwmin​∑iρi,T\+1\.\\sum\_\{t=1\}^\{T\}\\sum\_\{i\}\\frac\{\\rho\_\{i,t\}\}\{2k\_\{i\}\}\\left\\lVert x\_\{i,t\}\-\\tilde\{x\}\_\{i,t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\frac\{w\_\{\\max\}\}\{w\_\{\\min\}\}\\sum\_\{i\}\\rho\_\{i,T\+1\}\.Substituting above gives

∑iρi,T\+1≤C\+14​∑iρi,T\+1\+14​∑iρi,T\+1,\\sum\_\{i\}\\rho\_\{i,T\+1\}\\leq C\+\\frac\{1\}\{4\}\\sum\_\{i\}\\rho\_\{i,T\+1\}\+\\frac\{1\}\{4\}\\sum\_\{i\}\\rho\_\{i,T\+1\},and hence

∑iρi,T\+1≤2​C\.\\sum\_\{i\}\\rho\_\{i,T\+1\}\\leq 2C\.Thus every local regret\-vector norm remains bounded\. Finally, Fact[1](https://arxiv.org/html/2610.19016#Thmfact1)gives

Regi⁡\(T\)≤ρi,T\+1=𝒪T​\(1\)\\operatorname\{Reg\}\_\{i\}\(T\)\\leq\\rho\_\{i,T\+1\}=\\mathcal\{O\}\_\{T\}\(1\)for every learnerii\. ∎

### 7\.3\.Simultaneous IR\-PCFR\+

We now apply Theorem[2](https://arxiv.org/html/2610.19016#S7.Thmtheorem2)to the CFR operator from Section[7\.1](https://arxiv.org/html/2610.19016#S7.SS1), which is bounded and Lipschitz on the behavioral strategy space\([Farina et al\., 2023](https://arxiv.org/html/2610.19016#bib.bib4)\)\. Run one IR\-PRM\+ learner at every information set, each with the unit\-norm initialization \([2](https://arxiv.org/html/2610.19016#S3.E2)\)\. Fort≥2t\\geq 2, the predictors at information setsI∈ℐXI\\in\\mathcal\{I\}\_\{X\}andJ∈ℐYJ\\in\\mathcal\{I\}\_\{Y\}are

mI,tX=FIX​\(xt−1,yt−1\),mJ,tY=FJY​\(xt−1,yt−1\)\.m\_\{I,t\}^\{X\}=F\_\{I\}^\{X\}\(x\_\{t\-1\},y\_\{t\-1\}\),\\qquad m\_\{J,t\}^\{Y\}=F\_\{J\}^\{Y\}\(x\_\{t\-1\},y\_\{t\-1\}\)\.The local strategies at all information sets are formed simultaneously\. The counterfactual utilitiesFIX​\(xt,yt\)F\_\{I\}^\{X\}\(x\_\{t\},y\_\{t\}\)andFJY​\(xt,yt\)F\_\{J\}^\{Y\}\(x\_\{t\},y\_\{t\}\)are then evaluated at that behavioral profile, and all local regret vectors are updated in parallel\. Setq¯TX=T−1​∑t=1TqtX\\bar\{q\}\_\{T\}^\{X\}=T^\{\-1\}\\sum\_\{t=1\}^\{T\}q\_\{t\}^\{X\}andq¯TY=T−1​∑t=1TqtY\\bar\{q\}\_\{T\}^\{Y\}=T^\{\-1\}\\sum\_\{t=1\}^\{T\}q\_\{t\}^\{Y\}\.

###### Theorem 3 \(Constant regret for simultaneous IR\-PCFR\+\)\.

In any finite two\-player zero\-sum perfect\-recall extensive\-form game admitting a full\-reach Nash equilibrium, simultaneous IR\-PCFR\+ with previous\-utility prediction at each information set has both players’ sequence\-form external regrets bounded by finite game\-dependent constants independent ofTT\. Consequently,

Gap𝒬⁡\(q¯TX,q¯TY\)=𝒪T​\(T−1\)\.\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(\\bar\{q\}\_\{T\}^\{X\},\\bar\{q\}\_\{T\}^\{Y\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)\.

Theorem[3](https://arxiv.org/html/2610.19016#S7.Thmtheorem3)uses the same local IR\-PRM\+ analysis as Theorem[2](https://arxiv.org/html/2610.19016#S7.Thmtheorem2); the standard CFR decomposition then transfers the local constant\-regret guarantees to the global sequence\-form game\. The full proof is given in Appendix C\.2\.

The product\-space movement argument also controls the current local counterfactual residuals; Appendix B\.2 makes this precise\. The pointwise CFR decomposition then transfers those local residual bounds to the current sequence\-form duality gap; the full proof of the result below is given in Appendix C\.3\.

Let𝒵𝒬⋆\\mathcal\{Z\}\_\{\\mathcal\{Q\}\}^\{\\star\}denote the sequence\-form Nash equilibrium set\.

###### Theorem 4 \(Current sequence\-form convergence\)\.

In the same setting,

\(30\)∑t=1∞Gap𝒬⁡\(qtX,qtY\)2<∞\.\\sum\_\{t=1\}^\{\\infty\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)^\{2\}<\\infty\.Consequently,

Gap𝒬⁡\(qtX,qtY\)→0,dist⁡\(\(qtX,qtY\),𝒵𝒬⋆\)→0\.\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\\to 0,\\qquad\\operatorname\{dist\}\\bigl\(\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\),\\mathcal\{Z\}\_\{\\mathcal\{Q\}\}^\{\\star\}\\bigr\)\\to 0\.Moreover,

\(31\)min1≤t≤TGap𝒬\(qtX,qtY\)=𝒪T\(T−1/2\)\.\\min\_\{1\\leq t\\leq T\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)\.

## 8\.Alternating Updates

We next consider alternating IR\-PRM\+\. Sety0=y~1y\_\{0\}=\\tilde\{y\}\_\{1\}and formx1x\_\{1\}using zero initial prediction\. For eacht≥1t\\geq 1, afterxtx\_\{t\}has been formed, the column player observes−A⊤​xt\-A^\{\\top\}x\_\{t\}againstyt−1y\_\{t\-1\}, updates its stored regret vector, and reuses this utility as the prediction producingyty\_\{t\}\. The row player then observesA​ytAy\_\{t\}againstxtx\_\{t\}, updates its stored regret vector, and reuses this utility as the prediction producingxt\+1x\_\{t\+1\}\. Thus each alternating round passes through the intermediate profile\(xt,yt−1\)\(x\_\{t\},y\_\{t\-1\}\)before producing the full profile\(xt,yt\)\(x\_\{t\},y\_\{t\}\)\.

This change in update order introduces two differences from the simultaneous analysis\. First, the two regret\-vector growth bounds are staggered across the row and column updates\. Second, the column player’s directly controlled regret is evaluated along the intermediate profiles\(xt,yt−1\)\(x\_\{t\},y\_\{t\-1\}\), whereas its external regret in Theorem[1](https://arxiv.org/html/2610.19016#S8.Thmtheorem1)is measured along the full profiles\(xt,yt\)\(x\_\{t\},y\_\{t\}\)\. The analysis below handles both effects while retaining the guarantees of the simultaneous case\.

###### Theorem 1 \(Constant regret with alternation\)\.

In any finite two\-player zero\-sum matrix game, alternating IR\-PRM\+ with previous\-utility prediction has both players’ external regrets bounded by game\-dependent constants independent ofTT\. Consequently,

Gap⁡\(x¯T,y¯T\)=𝒪T​\(T−1\)\.\\operatorname\{Gap\}\(\\bar\{x\}\_\{T\},\\bar\{y\}\_\{T\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)\.

#### Proof idea\.

The alternating updates lead to a staggered pair of norm\-growth recurrences\. The column recurrence contains the reciprocal factor1/\(ρtX​ρtY\)1/\(\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}\), while the row recurrence contains1/\(ρtX​ρt\+1Y\)1/\(\\rho\_\{t\}^\{X\}\\rho\_\{t\+1\}^\{Y\}\)\. SinceρtY\\rho\_\{t\}^\{Y\}is nondecreasing, the row coefficient is no larger than the column coefficient\. The same threshold used in the simultaneous analysis therefore controls both, and summing them in alternating order gives bounded regret\-vector norms\.

For the column player, this argument initially gives bounded regret on the intermediate profiles\(xt,yt−1\)\(x\_\{t\},y\_\{t\-1\}\)\. Replacingyt−1y\_\{t\-1\}byyty\_\{t\}changes the cumulative regret by an amount controlled byρt\+1Y−ρtY\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}at each indextt\. These increments telescope, so boundedness ofρtY\\rho\_\{t\}^\{Y\}yields bounded full\-profile regret as well\. The complete argument is given in Appendix D\.1\.

The same bounds imply square summability of the consecutive strategy movements needed to compare consecutive predictions, which gives convergence of the current profiles\. The full argument is given in Appendix D\.2\.

###### Theorem 2 \(Current\-profile convergence\)\.

In the same setting,

\(32\)∑t=1∞Gap⁡\(xt,yt\)2<∞\.\\sum\_\{t=1\}^\{\\infty\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)^\{2\}<\\infty\.Consequently,

Gap⁡\(xt,yt\)→0,dist⁡\(\(xt,yt\),𝒵⋆\)→0\.\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)\\to 0,\\qquad\\operatorname\{dist\}\\bigl\(\(x\_\{t\},y\_\{t\}\),\\mathcal\{Z\}^\{\\star\}\\bigr\)\\to 0\.Moreover,

\(33\)min1≤t≤TGap\(xt,yt\)=𝒪T\(T−1/2\)\.\\min\_\{1\\leq t\\leq T\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)\.

### 8\.1\.Alternating IR\-PCFR\+

The CFR decomposition itself is unchanged, but alternation staggers the local counterfactual utilities\. Define the intermediate and full behavioral profiles by

ht=\(xt,yt−1\),zt=\(xt,yt\)\.h\_\{t\}=\(x\_\{t\},y\_\{t\-1\}\),\\qquad z\_\{t\}=\(x\_\{t\},y\_\{t\}\)\.At each column\-player information setJJ, the learner observesFJY​\(ht\)F\_\{J\}^\{Y\}\(h\_\{t\}\)againstyJ,t−1y\_\{J,t\-1\}, updates its stored regret vector, and reuses this utility as the prediction producingyJ,ty\_\{J,t\}\. This forms the full profileztz\_\{t\}\. Each row\-player learner at information setIIthen observesFIX​\(zt\)F\_\{I\}^\{X\}\(z\_\{t\}\)againstxI,tx\_\{I,t\}, updates its stored regret vector, and reuses this utility as the prediction producingxI,t\+1x\_\{I,t\+1\}\.

Thus the row\-player local regret is controlled along the full profilesztz\_\{t\}, whereas the column\-player local regret is first controlled along the intermediate profileshth\_\{t\}\. The remaining step is to handle this staggered weighted comparison and transfer the column half\-step bounds to sequence\-form regret along the full profiles\.

###### Theorem 3 \(Constant regret for alternating IR\-PCFR\+\)\.

In any finite two\-player zero\-sum perfect\-recall extensive\-form game admitting a full\-reach Nash equilibrium, alternating IR\-PCFR\+ with previous\-utility prediction at each information set has both players’ sequence\-form external regrets bounded by finite game\-dependent constants independent ofTT\. Consequently,

Gap𝒬⁡\(q¯TX,q¯TY\)=𝒪T​\(T−1\)\.\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(\\bar\{q\}\_\{T\}^\{X\},\\bar\{q\}\_\{T\}^\{Y\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)\.

#### Proof idea\.

The staggered matrix\-game argument is applied to the local counterfactual\-utility learners\. The full\-reach equilibrium weights compare row\-player deviations atztz\_\{t\}with column\-player deviations athth\_\{t\}, so an additional term appears from changing the column strategy fromyt−1y\_\{t\-1\}toyty\_\{t\}\. For each column learner, this payoff change is bounded below by the negative increase in its regret\-vector norm\. Summing over time therefore telescopes, allowing the staggered weighted comparison to yield bounded local regret\. The standard CFR decomposition transfers the row bounds directly to full\-profile sequence\-form regret, while the same payoff\-shift bound converts the column half\-step regret to full\-profile regret\. Details are in Appendix E\.1\.

The same local bounds also yield control of the current full profiles; the proof is given in Appendix E\.2\.

###### Theorem 4 \(Current sequence\-form convergence\)\.

In the same setting,

\(34\)∑t=1∞Gap𝒬⁡\(qtX,qtY\)2<∞\.\\sum\_\{t=1\}^\{\\infty\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)^\{2\}<\\infty\.Consequently,

Gap𝒬⁡\(qtX,qtY\)→0,dist⁡\(\(qtX,qtY\),𝒵𝒬⋆\)→0\.\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\\to 0,\\qquad\\operatorname\{dist\}\\bigl\(\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\),\\mathcal\{Z\}\_\{\\mathcal\{Q\}\}^\{\\star\}\\bigr\)\\to 0\.Moreover,

\(35\)min1≤t≤TGap𝒬\(qtX,qtY\)=𝒪T\(T−1/2\)\.\\min\_\{1\\leq t\\leq T\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)\.

## 9\.Conclusion and Future Work

We showed that previous\-utility prediction lets IR\-PRM\+ recover the constant\-regret and convergence guarantees of extra\-gradient IREG\-PRM\+ under both simultaneous and alternating updates, with corresponding guarantees for IR\-PCFR\+\. The key step is a post\-update displacement bound linking strategy movement to regret\-vector growth\. We do not repeat an empirical evaluation, since the practical performance of these IR\-PRM\+ variants has already been evaluated by Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\); our contribution is to provide the missing theoretical guarantees\. Future work is to understand how far these guarantees extend beyond the zero\-sum and weighted\-Minty settings considered here, and whether the full\-reach assumption in the CFR analysis can be weakened\.

## Appendix AProofs for the Simultaneous Matrix\-Game Analysis

### A\.1\.Proof of Fact[1](https://arxiv.org/html/2610.19016#Thmfact1)

###### Proof\.

The IR\-PRM\+ RVU analysis yields the following bound before the negative prediction displacement term is discarded\. LetRTR\_\{T\}denote the cumulative regret vector\. The proof of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\), before eliminating the terminal regret\-vector norm, gives the coordinatewise inequality

RT≤r~T\+1−\(∑t=1Tρt2​k​‖xt−x~t‖22\)​𝟏\.R\_\{T\}\\leq\\tilde\{r\}\_\{T\+1\}\-\\left\(\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\}\}\{2k\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\right\)\\mathbf\{1\}\.External regret is the largest coordinate ofRTR\_\{T\}\. Since the quantity in parentheses is the same in every coordinate,

Reg⁡\(T\)\\displaystyle\\operatorname\{Reg\}\(T\)=maxa⁡RT,a\\displaystyle=\\max\_\{a\}R\_\{T,a\}≤‖r~T\+1‖∞−∑t=1Tρt2​k​‖xt−x~t‖22\\displaystyle\\leq\\left\\lVert\\tilde\{r\}\_\{T\+1\}\\right\\rVert\_\{\\infty\}\-\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\}\}\{2k\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}≤ρT\+1−∑t=1Tρt2​k​‖xt−x~t‖22,\\displaystyle\\leq\\rho\_\{T\+1\}\-\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\}\}\{2k\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\},where the last step uses‖r~T\+1‖∞≤‖r~T\+1‖2=ρT\+1\\left\\lVert\\tilde\{r\}\_\{T\+1\}\\right\\rVert\_\{\\infty\}\\leq\\left\\lVert\\tilde\{r\}\_\{T\+1\}\\right\\rVert\_\{2\}=\\rho\_\{T\+1\}\. This is \([6](https://arxiv.org/html/2610.19016#S4.E6)\)\. ∎

### A\.2\.Predicted\-utility residual bound

The following auxiliary lemma converts a small prediction displacement into approximate optimality for the predicted utility\. It is used only for the current\-profile results, not for the constant\-regret proof\.

###### Lemma 0 \(Predicted\-utility residual bound\)\.

For akk\-action IR\-PRM\+ learner,

\(A\.1\)maxp∈Δk⁡⟨mt,p−xt⟩≤2​2​k​ρt​‖xt−x~t‖2\.\\max\_\{p\\in\\Delta\_\{k\}\}\\left\\langle m\_\{t\},p\-x\_\{t\}\\right\\rangle\\leq 2\\sqrt\{2k\}\\,\\rho\_\{t\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\.

###### Proof\.

By \([5](https://arxiv.org/html/2610.19016#S3.E5)\),ρt\>0\\rho\_\{t\}\>0\. Fix an arbitrary comparatorp∈Δkp\\in\\Delta\_\{k\}\. We will first bound the residual for this fixedppand then maximize overpp\.

Fromrt=r~t\+mt−γt​𝟏r\_\{t\}=\\tilde\{r\}\_\{t\}\+m\_\{t\}\-\\gamma\_\{t\}\\mathbf\{1\}, define

qt:=\[rt\]\+−rt≥0\.q\_\{t\}:=\\left\[r\_\{t\}\\right\]\_\{\+\}\-r\_\{t\}\\geq 0\.Rearranging gives

mt=γt​𝟏\+\(\[rt\]\+−r~t\)−qt\.m\_\{t\}=\\gamma\_\{t\}\\mathbf\{1\}\+\(\\left\[r\_\{t\}\\right\]\_\{\+\}\-\\tilde\{r\}\_\{t\}\)\-q\_\{t\}\.The vectorqtq\_\{t\}is supported only on coordinates wherert<0r\_\{t\}<0, whilextx\_\{t\}is supported on coordinates where\[rt\]\+\>0\\left\[r\_\{t\}\\right\]\_\{\+\}\>0\. Hence their supports are disjoint and⟨qt,xt⟩=0\\left\\langle q\_\{t\},x\_\{t\}\\right\\rangle=0\. Also⟨𝟏,p−xt⟩=0\\left\\langle\\mathbf\{1\},p\-x\_\{t\}\\right\\rangle=0because bothppandxtx\_\{t\}lie in the simplex\. Thus

⟨mt,p−xt⟩\\displaystyle\\left\\langle m\_\{t\},p\-x\_\{t\}\\right\\rangle=⟨\[rt\]\+−r~t,p−xt⟩−⟨qt,p−xt⟩\\displaystyle=\\left\\langle\\left\[r\_\{t\}\\right\]\_\{\+\}\-\\tilde\{r\}\_\{t\},p\-x\_\{t\}\\right\\rangle\-\\left\\langle q\_\{t\},p\-x\_\{t\}\\right\\rangle=⟨\[rt\]\+−r~t,p−xt⟩−⟨qt,p⟩\\displaystyle=\\left\\langle\\left\[r\_\{t\}\\right\]\_\{\+\}\-\\tilde\{r\}\_\{t\},p\-x\_\{t\}\\right\\rangle\-\\left\\langle q\_\{t\},p\\right\\rangle≤⟨\[rt\]\+−r~t,p−xt⟩\.\\displaystyle\\leq\\left\\langle\\left\[r\_\{t\}\\right\]\_\{\+\}\-\\tilde\{r\}\_\{t\},p\-x\_\{t\}\\right\\rangle\.The last term is controlled by Cauchy–Schwarz\. Since any two simplex points are at Euclidean distance at most2\\sqrt\{2\},

\(A\.2\)⟨mt,p−xt⟩≤2​‖\[rt\]\+−r~t‖2\.\\left\\langle m\_\{t\},p\-x\_\{t\}\\right\\rangle\\leq\\sqrt\{2\}\\,\\left\\lVert\\left\[r\_\{t\}\\right\]\_\{\+\}\-\\tilde\{r\}\_\{t\}\\right\\rVert\_\{2\}\.
It remains to relate the difference of the two regret vectors to the prediction displacementxt−x~tx\_\{t\}\-\\tilde\{x\}\_\{t\}\. The prediction step preserves Euclidean norm, so‖\[rt\]\+‖2=‖r~t‖2=ρt\\left\\lVert\\left\[r\_\{t\}\\right\]\_\{\+\}\\right\\rVert\_\{2\}=\\left\\lVert\\tilde\{r\}\_\{t\}\\right\\rVert\_\{2\}=\\rho\_\{t\}\. Becausextx\_\{t\}andx~t\\tilde\{x\}\_\{t\}are theirℓ1\\ell\_\{1\}normalizations, equivalently

\[rt\]\+=ρt​xt‖xt‖2,r~t=ρt​x~t‖x~t‖2\.\\left\[r\_\{t\}\\right\]\_\{\+\}=\\rho\_\{t\}\\frac\{x\_\{t\}\}\{\\left\\lVert x\_\{t\}\\right\\rVert\_\{2\}\},\\qquad\\tilde\{r\}\_\{t\}=\\rho\_\{t\}\\frac\{\\tilde\{x\}\_\{t\}\}\{\\left\\lVert\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\}\.Leta=‖xt‖2a=\\left\\lVert x\_\{t\}\\right\\rVert\_\{2\}andb=‖x~t‖2b=\\left\\lVert\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\. Adding and subtractingρt​x~t/a\\rho\_\{t\}\\tilde\{x\}\_\{t\}/agives

‖\[rt\]\+−r~t‖2\\displaystyle\\left\\lVert\\left\[r\_\{t\}\\right\]\_\{\+\}\-\\tilde\{r\}\_\{t\}\\right\\rVert\_\{2\}=ρt​‖xta−x~tb‖2\\displaystyle=\\rho\_\{t\}\\left\\lVert\\frac\{x\_\{t\}\}\{a\}\-\\frac\{\\tilde\{x\}\_\{t\}\}\{b\}\\right\\rVert\_\{2\}≤ρta​‖xt−x~t‖2\+ρt​\|1a−1b\|​‖x~t‖2\\displaystyle\\leq\\frac\{\\rho\_\{t\}\}\{a\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\+\\rho\_\{t\}\\left\|\\frac\{1\}\{a\}\-\\frac\{1\}\{b\}\\right\|\\left\\lVert\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}=ρta​‖xt−x~t‖2\+ρt​\|b−a\|a\\displaystyle=\\frac\{\\rho\_\{t\}\}\{a\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\+\\frac\{\\rho\_\{t\}\|b\-a\|\}\{a\}≤2​ρta​‖xt−x~t‖2\.\\displaystyle\\leq\\frac\{2\\rho\_\{t\}\}\{a\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\.The last step uses the reverse triangle inequality\|b−a\|≤‖xt−x~t‖2\|b\-a\|\\leq\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\. Finally,xt∈Δkx\_\{t\}\\in\\Delta\_\{k\}impliesa=‖xt‖2≥1/ka=\\left\\lVert x\_\{t\}\\right\\rVert\_\{2\}\\geq 1/\\sqrt\{k\}, so

\(A\.3\)‖\[rt\]\+−r~t‖2≤2​k​ρt​‖xt−x~t‖2\.\\left\\lVert\\left\[r\_\{t\}\\right\]\_\{\+\}\-\\tilde\{r\}\_\{t\}\\right\\rVert\_\{2\}\\leq 2\\sqrt\{k\}\\,\\rho\_\{t\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\.Combining \([A\.2](https://arxiv.org/html/2610.19016#A1.E2)\) and \([A\.3](https://arxiv.org/html/2610.19016#A1.E3)\) gives

⟨mt,p−xt⟩≤2​2​k​ρt​‖xt−x~t‖2\.\\left\\langle m\_\{t\},p\-x\_\{t\}\\right\\rangle\\leq 2\\sqrt\{2k\}\\,\\rho\_\{t\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\.The bound is independent ofpp, so maximizing overp∈Δkp\\in\\Delta\_\{k\}proves \([A\.1](https://arxiv.org/html/2610.19016#A1.E1)\)\. ∎

### A\.3\.Proof of Theorem[3](https://arxiv.org/html/2610.19016#S6.Thmtheorem3)

###### Proof\.

We first show that the prediction and post\-update displacements, and hence the consecutive strategy movements, are square summable\. We then use the residual lemma for the predicted utility to convert these bounds into square summability of the current best\-response residuals and, hence, of the current duality gap\.

The proof of Theorem[2](https://arxiv.org/html/2610.19016#S6.Thmtheorem2)gives two facts that we will repeatedly use: the regret\-vector norms remain bounded over time, and∑tHt<∞\\sum\_\{t\}H\_\{t\}<\\infty\. Expanding the definition ofHtH\_\{t\},

∑t=1∞ρtX2​kX​‖xt−x~t‖22<∞,∑t=1∞ρtY2​kY​‖yt−y~t‖22<∞\.\\sum\_\{t=1\}^\{\\infty\}\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}<\\infty,\\qquad\\sum\_\{t=1\}^\{\\infty\}\\frac\{\\rho\_\{t\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}<\\infty\.SinceρtX,ρtY≥1\\rho\_\{t\}^\{X\},\\rho\_\{t\}^\{Y\}\\geq 1, the factorsρtX\\rho\_\{t\}^\{X\}andρtY\\rho\_\{t\}^\{Y\}can be removed directly, giving

\(A\.4\)∑t=1∞‖xt−x~t‖22<∞,∑t=1∞‖yt−y~t‖22<∞\.\\sum\_\{t=1\}^\{\\infty\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}<\\infty,\\qquad\\sum\_\{t=1\}^\{\\infty\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}<\\infty\.
The post\-update displacements are square summable for the same reason\. For example, Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)andρt\+1X≥1\\rho\_\{t\+1\}^\{X\}\\geq 1give

‖x~t\+1−xt‖22≤2​\(1\+kX\)2​\(ρt\+1X−ρtX\)\.\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq 2\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}\)\.The norm increments telescope, and the norm sequence is bounded, so summing the right\-hand side over time is finite\. The same argument applies to the column player; thus

∑t=1∞‖x~t\+1−xt‖22<∞,∑t=1∞‖y~t\+1−yt‖22<∞\.\\sum\_\{t=1\}^\{\\infty\}\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}<\\infty,\\qquad\\sum\_\{t=1\}^\{\\infty\}\\left\\lVert\\tilde\{y\}\_\{t\+1\}\-y\_\{t\}\\right\\rVert\_\{2\}^\{2\}<\\infty\.To recover the movement between consecutive played strategies, insert the intermediate strategies and use‖a\+b‖22≤2​‖a‖22\+2​‖b‖22\\left\\lVert a\+b\\right\\rVert\_\{2\}^\{2\}\\leq 2\\left\\lVert a\\right\\rVert\_\{2\}^\{2\}\+2\\left\\lVert b\\right\\rVert\_\{2\}^\{2\}\. Together with the two square\-summability statements above, this gives

\(A\.5\)∑t=2∞‖xt−xt−1‖22<∞,∑t=2∞‖yt−yt−1‖22<∞\.\\sum\_\{t=2\}^\{\\infty\}\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}<\\infty,\\qquad\\sum\_\{t=2\}^\{\\infty\}\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}<\\infty\.
We now translate these square\-summability bounds into current best\-response residuals\. Define the row and column best\-response residuals

RtX:=maxx′∈ΔkX⁡⟨A​yt,x′−xt⟩,RtY:=maxy′∈ΔkY⁡⟨−A⊤​xt,y′−yt⟩\.R\_\{t\}^\{X\}:=\\max\_\{x^\{\\prime\}\\in\\Delta\_\{k^\{X\}\}\}\\left\\langle Ay\_\{t\},x^\{\\prime\}\-x\_\{t\}\\right\\rangle,\\qquad R\_\{t\}^\{Y\}:=\\max\_\{y^\{\\prime\}\\in\\Delta\_\{k^\{Y\}\}\}\\left\\langle\-A^\{\\top\}x\_\{t\},y^\{\\prime\}\-y\_\{t\}\\right\\rangle\.Fort≥2t\\geq 2, consider first the row player, for which the previous\-utility predictor ismtX=A​yt−1m\_\{t\}^\{X\}=Ay\_\{t\-1\}\. Add and subtract this prediction inside the inner product\. For anyx′∈ΔkXx^\{\\prime\}\\in\\Delta\_\{k^\{X\}\},

⟨A​yt,x′−xt⟩\\displaystyle\\left\\langle Ay\_\{t\},x^\{\\prime\}\-x\_\{t\}\\right\\rangle=⟨mtX,x′−xt⟩\+⟨A⁡\(yt−yt−1\),x′−xt⟩\\displaystyle=\\left\\langle m\_\{t\}^\{X\},x^\{\\prime\}\-x\_\{t\}\\right\\rangle\+\\left\\langle A\(y\_\{t\}\-y\_\{t\-1\}\),x^\{\\prime\}\-x\_\{t\}\\right\\rangle≤2​2​kX​ρtX​‖xt−x~t‖2\+2​‖A‖2​‖yt−yt−1‖2\.\\displaystyle\\leq 2\\sqrt\{2k^\{X\}\}\\,\\rho\_\{t\}^\{X\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\+\\sqrt\{2\}\\left\\lVert A\\right\\rVert\_\{2\}\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}\.The first term is controlled by Lemma[1](https://arxiv.org/html/2610.19016#A1.Thmtheorem1)\. The second uses Cauchy–Schwarz, the operator\-norm bound forAA, and the simplex diameter2\\sqrt\{2\}\. Now define

CX:=2​2​kX​suptρtX\.C\_\{X\}:=2\\sqrt\{2k^\{X\}\}\\sup\_\{t\}\\rho\_\{t\}^\{X\}\.Maximizing overx′x^\{\\prime\}gives

\(A\.6\)RtX\\displaystyle R\_\{t\}^\{X\}≤CX​‖xt−x~t‖2\+2​‖A‖2​‖yt−yt−1‖2\.\\displaystyle\\leq C\_\{X\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\+\\sqrt\{2\}\\left\\lVert A\\right\\rVert\_\{2\}\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}\.For the column player, the identical calculation withmtY=−A⊤​xt−1m\_\{t\}^\{Y\}=\-A^\{\\top\}x\_\{t\-1\}gives

\(A\.7\)RtY\\displaystyle R\_\{t\}^\{Y\}≤CY​‖yt−y~t‖2\+2​‖A‖2​‖xt−xt−1‖2,\\displaystyle\\leq C\_\{Y\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}\+\\sqrt\{2\}\\left\\lVert A\\right\\rVert\_\{2\}\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\},whereCY:=2​2​kY​suptρtYC\_\{Y\}:=2\\sqrt\{2k^\{Y\}\}\\sup\_\{t\}\\rho\_\{t\}^\{Y\}\.

Each term on the right of \([A\.6](https://arxiv.org/html/2610.19016#A1.E6)\)– \([A\.7](https://arxiv.org/html/2610.19016#A1.E7)\) is square summable by \([A\.4](https://arxiv.org/html/2610.19016#A1.E4)\) and \([A\.5](https://arxiv.org/html/2610.19016#A1.E5)\)\. The initial residual terms are finite and contribute only a bounded constant\. Using\(a\+b\)2≤2​a2\+2​b2\(a\+b\)^\{2\}\\leq 2a^\{2\}\+2b^\{2\}therefore gives

∑t\(RtX\)2<∞,∑t\(RtY\)2<∞\.\\sum\_\{t\}\(R\_\{t\}^\{X\}\)^\{2\}<\\infty,\\qquad\\sum\_\{t\}\(R\_\{t\}^\{Y\}\)^\{2\}<\\infty\.The matrix\-game duality gap is exactly the sum of these two residuals,Gap⁡\(xt,yt\)=RtX\+RtY\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)=R\_\{t\}^\{X\}\+R\_\{t\}^\{Y\}\. A final application of\(a\+b\)2≤2​a2\+2​b2\(a\+b\)^\{2\}\\leq 2a^\{2\}\+2b^\{2\}yields \([22](https://arxiv.org/html/2610.19016#S6.E22)\)\.

Square summability impliesGap⁡\(xt,yt\)→0\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)\\to 0\. To prove that the distance to the Nash equilibrium set converges to zero, suppose instead that the distance does not tend to zero\. Then there areε\>0\\varepsilon\>0and a subsequence\(xtj,ytj\)\(x\_\{t\_\{j\}\},y\_\{t\_\{j\}\}\)satisfyingdist⁡\(\(xtj,ytj\),𝒵⋆\)≥ε\\operatorname\{dist\}\(\(x\_\{t\_\{j\}\},y\_\{t\_\{j\}\}\),\\mathcal\{Z\}^\{\\star\}\)\\geq\\varepsilon\. Compactness ofΔkX×ΔkY\\Delta\_\{k^\{X\}\}\\times\\Delta\_\{k^\{Y\}\}gives a convergent subsubsequence\. Its limit has zero gap by continuity and hence lies in𝒵⋆\\mathcal\{Z\}^\{\\star\}, contradicting the distance lower bound\. Thereforedist⁡\(\(xt,yt\),𝒵⋆\)→0\\operatorname\{dist\}\(\(x\_\{t\},y\_\{t\}\),\\mathcal\{Z\}^\{\\star\}\)\\to 0\.

Finally, letK:=∑t=1∞Gap⁡\(xt,yt\)2<∞K:=\\sum\_\{t=1\}^\{\\infty\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)^\{2\}<\\infty\. Since every term is nonnegative,

T​min1≤t≤T​Gap⁡\(xt,yt\)2≤∑t=1TGap⁡\(xt,yt\)2≤K\.T\\min\_\{1\\leq t\\leq T\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)^\{2\}\\leq\\sum\_\{t=1\}^\{T\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)^\{2\}\\leq K\.Taking square roots gives \([23](https://arxiv.org/html/2610.19016#S6.E23)\)\. ∎

## Appendix BProofs for the Product\-of\-Simplices Analysis

### B\.1\.Proof of Lemma[1](https://arxiv.org/html/2610.19016#S7.Thmtheorem1)

###### Proof\.

Fix a learneriiand a roundt≥2t\\geq 2\. The purpose of the calculation is to express learnerii’s norm growth in terms of nonnegative quantitiesBj,tB\_\{j,t\}, one for each learnerjj\. Previous\-utility prediction gives

ui,t−mi,t=Fi​\(xt\)−Fi​\(xt−1\)\.u\_\{i,t\}\-m\_\{i,t\}=F\_\{i\}\(x\_\{t\}\)\-F\_\{i\}\(x\_\{t\-1\}\)\.Applying the bound \([17](https://arxiv.org/html/2610.19016#S6.E17)\) and then Lipschitzness,

‖gi,t‖22\\displaystyle\\left\\lVert g\_\{i,t\}\\right\\rVert\_\{2\}^\{2\}≤\(1\+ki\)2​‖Fi​\(xt\)−Fi​\(xt−1\)‖22\\displaystyle\\leq\(1\+\\sqrt\{k\_\{i\}\}\)^\{2\}\\left\\lVert F\_\{i\}\(x\_\{t\}\)\-F\_\{i\}\(x\_\{t\-1\}\)\\right\\rVert\_\{2\}^\{2\}\(B\.1\)≤\(1\+ki\)2​Li2​‖xt−xt−1‖22\.\\displaystyle\\leq\(1\+\\sqrt\{k\_\{i\}\}\)^\{2\}L\_\{i\}^\{2\}\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\.Since we use the product Euclidean norm,

‖xt−xt−1‖22=∑j=1N‖xj,t−xj,t−1‖22\.\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}=\\sum\_\{j=1\}^\{N\}\\left\\lVert x\_\{j,t\}\-x\_\{j,t\-1\}\\right\\rVert\_\{2\}^\{2\}\.Thus it remains to control each component’s movement separately\.

Define

Bj,t:=ρj,t2​kj​‖xj,t−x~j,t‖22\+\(ρj,t−ρj,t−1\)\.B\_\{j,t\}:=\\frac\{\\rho\_\{j,t\}\}\{2k\_\{j\}\}\\left\\lVert x\_\{j,t\}\-\\tilde\{x\}\_\{j,t\}\\right\\rVert\_\{2\}^\{2\}\+\(\\rho\_\{j,t\}\-\\rho\_\{j,t\-1\}\)\.Corollary[2](https://arxiv.org/html/2610.19016#S5.Thmtheorem2)gives

‖xj,t−xj,t−1‖22\\displaystyle\\left\\lVert x\_\{j,t\}\-x\_\{j,t\-1\}\\right\\rVert\_\{2\}^\{2\}≤2​‖xj,t−x~j,t‖22\+4​\(1\+kj\)2ρj,t​\(ρj,t−ρj,t−1\)\.\\displaystyle\\leq 2\\left\\lVert x\_\{j,t\}\-\\tilde\{x\}\_\{j,t\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{4\(1\+\\sqrt\{k\_\{j\}\}\)^\{2\}\}\{\\rho\_\{j,t\}\}\(\\rho\_\{j,t\}\-\\rho\_\{j,t\-1\}\)\.We bound both terms using the same quantityBj,tB\_\{j,t\}\. Since\(1\+kj\)2≥kj\(1\+\\sqrt\{k\_\{j\}\}\)^\{2\}\\geq k\_\{j\},

2​‖xj,t−x~j,t‖22≤4​\(1\+kj\)2ρj,t​ρj,t2​kj​‖xj,t−x~j,t‖22\.2\\left\\lVert x\_\{j,t\}\-\\tilde\{x\}\_\{j,t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\frac\{4\(1\+\\sqrt\{k\_\{j\}\}\)^\{2\}\}\{\\rho\_\{j,t\}\}\\frac\{\\rho\_\{j,t\}\}\{2k\_\{j\}\}\\left\\lVert x\_\{j,t\}\-\\tilde\{x\}\_\{j,t\}\\right\\rVert\_\{2\}^\{2\}\.Therefore

\(B\.2\)‖xj,t−xj,t−1‖22≤4​\(1\+kj\)2ρj,t​Bj,t\.\\left\\lVert x\_\{j,t\}\-x\_\{j,t\-1\}\\right\\rVert\_\{2\}^\{2\}\\leq\\frac\{4\(1\+\\sqrt\{k\_\{j\}\}\)^\{2\}\}\{\\rho\_\{j,t\}\}B\_\{j,t\}\.
Substituting \([B\.2](https://arxiv.org/html/2610.19016#A2.E2)\) into the component decomposition of \([B\.1](https://arxiv.org/html/2610.19016#A2.E1)\) gives

‖gi,t‖22≤∑j=1N4​\(1\+kj\)2​\(1\+ki\)2​Li2ρj,t​Bj,t\.\\left\\lVert g\_\{i,t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\sum\_\{j=1\}^\{N\}\\frac\{4\(1\+\\sqrt\{k\_\{j\}\}\)^\{2\}\(1\+\\sqrt\{k\_\{i\}\}\)^\{2\}L\_\{i\}^\{2\}\}\{\\rho\_\{j,t\}\}B\_\{j,t\}\.Finally, the one\-step norm\-growth inequality \([16](https://arxiv.org/html/2610.19016#S6.E16)\) for learneriidivides this expression by2​ρi,t2\\rho\_\{i,t\}:

ρi,t\+1−ρi,t≤∑j=1N2​\(1\+kj\)2​\(1\+ki\)2​Li2ρi,t​ρj,t​Bj,t\.\\rho\_\{i,t\+1\}\-\\rho\_\{i,t\}\\leq\\sum\_\{j=1\}^\{N\}\\frac\{2\(1\+\\sqrt\{k\_\{j\}\}\)^\{2\}\(1\+\\sqrt\{k\_\{i\}\}\)^\{2\}L\_\{i\}^\{2\}\}\{\\rho\_\{i,t\}\\rho\_\{j,t\}\}B\_\{j,t\}\.Expanding the definition ofBj,tB\_\{j,t\}is exactly \([27](https://arxiv.org/html/2610.19016#S7.E27)\)\. ∎

### B\.2\.Current residual under weighted Minty

###### Corollary 0 \(Current residual under weighted Minty\)\.

Under the assumptions of Theorem[2](https://arxiv.org/html/2610.19016#S7.Thmtheorem2),

\(B\.3\)∑t=1∞\(∑i=1Nmaxpi∈Δki⁡⟨Fi​\(xt\),pi−xi,t⟩\)2<∞\.\\sum\_\{t=1\}^\{\\infty\}\\left\(\\sum\_\{i=1\}^\{N\}\\max\_\{p\_\{i\}\\in\\Delta\_\{k\_\{i\}\}\}\\left\\langle F\_\{i\}\(x\_\{t\}\),p\_\{i\}\-x\_\{i,t\}\\right\\rangle\\right\)^\{2\}<\\infty\.Consequently,

∑i=1Nmaxpi∈Δki⁡⟨Fi​\(xt\),pi−xi,t⟩→0,\\sum\_\{i=1\}^\{N\}\\max\_\{p\_\{i\}\\in\\Delta\_\{k\_\{i\}\}\}\\left\\langle F\_\{i\}\(x\_\{t\}\),p\_\{i\}\-x\_\{i,t\}\\right\\rangle\\to 0,and

\(B\.4\)min1≤t≤T∑i=1Nmaxpi∈Δki⟨Fi\(xt\),pi−xi,t⟩=𝒪T\(T−1/2\)\.\\min\_\{1\\leq t\\leq T\}\\sum\_\{i=1\}^\{N\}\\max\_\{p\_\{i\}\\in\\Delta\_\{k\_\{i\}\}\}\\left\\langle F\_\{i\}\(x\_\{t\}\),p\_\{i\}\-x\_\{i,t\}\\right\\rangle=\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)\.

###### Proof\.

Theorem[2](https://arxiv.org/html/2610.19016#S7.Thmtheorem2)gives local norms that remain bounded over time, and \([29](https://arxiv.org/html/2610.19016#S7.E29)\) gives a finite total stability sum:

\(B\.5\)∑t=1∞∑i=1Nρi,t2​ki​‖xi,t−x~i,t‖22<∞\.\\sum\_\{t=1\}^\{\\infty\}\\sum\_\{i=1\}^\{N\}\\frac\{\\rho\_\{i,t\}\}\{2k\_\{i\}\}\\left\\lVert x\_\{i,t\}\-\\tilde\{x\}\_\{i,t\}\\right\\rVert\_\{2\}^\{2\}<\\infty\.Sinceρi,t≥1\\rho\_\{i,t\}\\geq 1for everyii, each prediction displacement is square summable\. For its post\-update displacement, Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)gives

‖x~i,t\+1−xi,t‖22≤2​\(1\+ki\)2​\(ρi,t\+1−ρi,t\),\\left\\lVert\\tilde\{x\}\_\{i,t\+1\}\-x\_\{i,t\}\\right\\rVert\_\{2\}^\{2\}\\leq 2\(1\+\\sqrt\{k\_\{i\}\}\)^\{2\}\(\\rho\_\{i,t\+1\}\-\\rho\_\{i,t\}\),and the right\-hand side is summable because the bounded monotone norm increments telescope\. For eachii, insertingx~i,t\\tilde\{x\}\_\{i,t\}betweenxi,t−1x\_\{i,t\-1\}andxi,tx\_\{i,t\}and using‖a\+b‖22≤2​‖a‖22\+2​‖b‖22\\\|a\+b\\\|\_\{2\}^\{2\}\\leq 2\\\|a\\\|\_\{2\}^\{2\}\+2\\\|b\\\|\_\{2\}^\{2\}, then summing over theNNcomponents, yields

\(B\.6\)∑t=2∞‖xt−xt−1‖22<∞\.\\sum\_\{t=2\}^\{\\infty\}\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}<\\infty\.
Fort≥2t\\geq 2, we next bound the current residual for learnerii\. Add and subtract the previous\-utility predictionFi​\(xt−1\)F\_\{i\}\(x\_\{t\-1\}\):

maxpi∈Δki⁡⟨Fi​\(xt\),pi−xi,t⟩\\displaystyle\\max\_\{p\_\{i\}\\in\\Delta\_\{k\_\{i\}\}\}\\left\\langle F\_\{i\}\(x\_\{t\}\),p\_\{i\}\-x\_\{i,t\}\\right\\rangle≤maxpi∈Δki⁡⟨Fi​\(xt−1\),pi−xi,t⟩\\displaystyle\\quad\\leq\\max\_\{p\_\{i\}\\in\\Delta\_\{k\_\{i\}\}\}\\left\\langle F\_\{i\}\(x\_\{t\-1\}\),p\_\{i\}\-x\_\{i,t\}\\right\\rangle\+maxpi∈Δki⁡⟨Fi​\(xt\)−Fi​\(xt−1\),pi−xi,t⟩\.\\displaystyle\\qquad\+\\max\_\{p\_\{i\}\\in\\Delta\_\{k\_\{i\}\}\}\\left\\langle F\_\{i\}\(x\_\{t\}\)\-F\_\{i\}\(x\_\{t\-1\}\),p\_\{i\}\-x\_\{i,t\}\\right\\rangle\.Lemma[1](https://arxiv.org/html/2610.19016#A1.Thmtheorem1)bounds the first term\. For the second, Cauchy–Schwarz, Lipschitzness, and‖pi−xi,t‖2≤2\\left\\lVert p\_\{i\}\-x\_\{i,t\}\\right\\rVert\_\{2\}\\leq\\sqrt\{2\}give

maxpi⁡⟨Fi​\(xt\)−Fi​\(xt−1\),pi−xi,t⟩≤2​Li​‖xt−xt−1‖2\.\\max\_\{p\_\{i\}\}\\left\\langle F\_\{i\}\(x\_\{t\}\)\-F\_\{i\}\(x\_\{t\-1\}\),p\_\{i\}\-x\_\{i,t\}\\right\\rangle\\leq\\sqrt\{2\}L\_\{i\}\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}\.Therefore

maxpi∈Δki⁡⟨Fi​\(xt\),pi−xi,t⟩\\displaystyle\\max\_\{p\_\{i\}\\in\\Delta\_\{k\_\{i\}\}\}\\left\\langle F\_\{i\}\(x\_\{t\}\),p\_\{i\}\-x\_\{i,t\}\\right\\rangle≤2​2​ki​ρi,t​‖xi,t−x~i,t‖2\\displaystyle\\leq 2\\sqrt\{2k\_\{i\}\}\\,\\rho\_\{i,t\}\\left\\lVert x\_\{i,t\}\-\\tilde\{x\}\_\{i,t\}\\right\\rVert\_\{2\}\(B\.7\)\+2​Li​‖xt−xt−1‖2\.\\displaystyle\\quad\+\\sqrt\{2\}L\_\{i\}\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}\.Both sequences on the right are square summable: the first by \([B\.5](https://arxiv.org/html/2610.19016#A2.E5)\) and the time\-independent norm bound, and the second by \([B\.6](https://arxiv.org/html/2610.19016#A2.E6)\)\. Hence the local residual is square summable fort≥2t\\geq 2and the single, first iteration residual is finite\.

Letai,ta\_\{i,t\}denote the nonnegative local residual in \([B\.7](https://arxiv.org/html/2610.19016#A2.E7)\)\. Since there areNNlocal residuals, Cauchy–Schwarz inℝN\\mathbb\{R\}^\{N\}gives

\(∑i=1Nai,t\)2≤N​∑i=1Nai,t2\.\\left\(\\sum\_\{i=1\}^\{N\}a\_\{i,t\}\\right\)^\{2\}\\leq N\\sum\_\{i=1\}^\{N\}a\_\{i,t\}^\{2\}\.Summing overttand using the square summability of everyai,ta\_\{i,t\}proves \([B\.3](https://arxiv.org/html/2610.19016#A2.E3)\)\. In particular the summed residual converges to zero\.

Finally, ifK:=∑t\(∑iai,t\)2<∞K:=\\sum\_\{t\}\(\\sum\_\{i\}a\_\{i,t\}\)^\{2\}<\\infty, then

T​min1≤t≤T​\(∑iai,t\)2≤∑t=1T\(∑iai,t\)2≤K\.T\\min\_\{1\\leq t\\leq T\}\\left\(\\sum\_\{i\}a\_\{i,t\}\\right\)^\{2\}\\leq\\sum\_\{t=1\}^\{T\}\\left\(\\sum\_\{i\}a\_\{i,t\}\\right\)^\{2\}\\leq K\.Taking square roots yields \([B\.4](https://arxiv.org/html/2610.19016#A2.E4)\)\. ∎

## Appendix CProofs for Simultaneous IR\-PCFR\+

ForI∈ℐXI\\in\\mathcal\{I\}\_\{X\}, define the local counterfactual regret

RegIX⁡\(T\):=max⁡∑t=1Tp∈Δ\|A⁡\(I\)\|⁡⟨FIX​\(xt,yt\),p−xI,t⟩,\\operatorname\{Reg\}\_\{I\}^\{X\}\(T\):=\\max\_\{p\\in\\Delta\_\{\|A\(I\)\|\}\}\\sum\_\{t=1\}^\{T\}\\left\\langle F\_\{I\}^\{X\}\(x\_\{t\},y\_\{t\}\),p\-x\_\{I,t\}\\right\\rangle,and defineRegJY⁡\(T\)\\operatorname\{Reg\}\_\{J\}^\{Y\}\(T\)analogously\. The sequence\-form external regrets are

RegX𝒬⁡\(T\):=max⁡∑t=1Tq^X∈𝒬X⁡⟨Aseq​qtY,q^X−qtX⟩,\\operatorname\{Reg\}\_\{X\}^\{\\mathcal\{Q\}\}\(T\):=\\max\_\{\\widehat\{q\}^\{X\}\\in\\mathcal\{Q\}\_\{X\}\}\\sum\_\{t=1\}^\{T\}\\left\\langle A\_\{\\rm seq\}q\_\{t\}^\{Y\},\\widehat\{q\}^\{X\}\-q\_\{t\}^\{X\}\\right\\rangle,withRegY𝒬⁡\(T\)\\operatorname\{Reg\}\_\{Y\}^\{\\mathcal\{Q\}\}\(T\)defined analogously\.

### C\.1\.Standard CFR decomposition

The counterfactual utility vectors are defined so that a global sequence\-form deviation decomposes into weighted local behavioral deviations\. The following pointwise identity is the standard local\-to\-global relation underlying the CFR regret decomposition\([Zinkevich et al\., 2007](https://arxiv.org/html/2610.19016#bib.bib12);[Farina et al\., 2019](https://arxiv.org/html/2610.19016#bib.bib13);[Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)\.

###### Fact 2 \(Standard pointwise CFR decomposition\)\.

Let\(x,y\)\(x,y\)induce\(qX,qY\)\(q^\{X\},q^\{Y\}\)\. Letx^\\widehat\{x\}be any comparator behavioral strategy inducingq^X\\widehat\{q\}^\{X\}\. Then

\(C\.1\)⟨Aseq​qY,q^X−qX⟩=∑I∈ℐXq^X​\[pI\]​⟨FIX​\(x,y\),x^I−xI⟩\.\\left\\langle A\_\{\\rm seq\}q^\{Y\},\\widehat\{q\}^\{X\}\-q^\{X\}\\right\\rangle=\\sum\_\{I\\in\\mathcal\{I\}\_\{X\}\}\\widehat\{q\}^\{X\}\[p\_\{I\}\]\\left\\langle F\_\{I\}^\{X\}\(x,y\),\\widehat\{x\}\_\{I\}\-x\_\{I\}\\right\\rangle\.Similarly, for any comparatory^\\widehat\{y\}inducingq^Y\\widehat\{q\}^\{Y\},

\(C\.2\)⟨−Aseq⊤​qX,q^Y−qY⟩=∑J∈ℐYq^Y​\[pJ\]​⟨FJY​\(x,y\),y^J−yJ⟩\.\\left\\langle\-A\_\{\\rm seq\}^\{\\top\}q^\{X\},\\widehat\{q\}^\{Y\}\-q^\{Y\}\\right\\rangle=\\sum\_\{J\\in\\mathcal\{I\}\_\{Y\}\}\\widehat\{q\}^\{Y\}\[p\_\{J\}\]\\left\\langle F\_\{J\}^\{Y\}\(x,y\),\\widehat\{y\}\_\{J\}\-y\_\{J\}\\right\\rangle\.

A behavioral profile has*full reach*if each player’s realization plan assigns strictly positive probability to the parent sequence of every information set of that player\. Suppose\(qX,⋆,qY,⋆\)\(q^\{X,\\star\},q^\{Y,\\star\}\)is the realization\-plan pair induced by a full\-reach Nash equilibrium, with corresponding behavioral strategies\(x⋆,y⋆\)\(x^\{\\star\},y^\{\\star\}\)\. Applying the pointwise decomposition with these equilibrium comparators gives

∑I∈ℐXqX,⋆​\[pI\]​⟨FIX​\(x,y\),xI⋆−xI⟩\\displaystyle\\sum\_\{I\\in\\mathcal\{I\}\_\{X\}\}q^\{X,\\star\}\[p\_\{I\}\]\\left\\langle F\_\{I\}^\{X\}\(x,y\),x\_\{I\}^\{\\star\}\-x\_\{I\}\\right\\rangle\+∑J∈ℐYqY,⋆\[pJ\]⟨FJY\(x,y\),yJ⋆−yJ⟩\\displaystyle\\quad\+\\sum\_\{J\\in\\mathcal\{I\}\_\{Y\}\}q^\{Y,\\star\}\[p\_\{J\}\]\\left\\langle F\_\{J\}^\{Y\}\(x,y\),y\_\{J\}^\{\\star\}\-y\_\{J\}\\right\\rangle\(C\.3\)=U⁡\(qX,⋆,qY\)−U⁡\(qX,qY,⋆\)≥0\.\\displaystyle\\qquad=U\(q^\{X,\\star\},q^\{Y\}\)\-U\(q^\{X\},q^\{Y,\\star\}\)\\geq 0\.The inequality is saddle\-point optimality\. Under the full\-reach condition, every parent\-sequence weightqX,⋆​\[pI\]q^\{X,\\star\}\[p\_\{I\}\]andqY,⋆​\[pJ\]q^\{Y,\\star\}\[p\_\{J\}\]is strictly positive\. Hence the CFR counterfactual\-utility operator satisfies the weighted Minty condition used in Section[7\.2](https://arxiv.org/html/2610.19016#S7.SS2); this is the full\-reach CFR reduction of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)\.

Summing \([C\.1](https://arxiv.org/html/2610.19016#A3.E1)\)–\([C\.2](https://arxiv.org/html/2610.19016#A3.E2)\) over time and upper\-bounding each local cumulative deviation by the corresponding local external regret gives the standard cumulative CFR regret decomposition; cf\. Proposition 6\.1 of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)\. At a single iteration, the same pointwise identities transfer bounds on local residuals to the sequence\-form residuals\.

### C\.2\.Proof of Theorem[3](https://arxiv.org/html/2610.19016#S7.Thmtheorem3)

###### Proof\.

Equation \([C\.3](https://arxiv.org/html/2610.19016#A3.E3)\) supplies the weighted Minty condition, while the boundedness and Lipschitzness of the counterfactual\-utility operator stated in Section[7\.3](https://arxiv.org/html/2610.19016#S7.SS3)supply the remaining assumptions of Theorem[2](https://arxiv.org/html/2610.19016#S7.Thmtheorem2)\. Hence every local counterfactual regret is𝒪T​\(1\)\\mathcal\{O\}\_\{T\}\(1\)\.We now transfer those local bounds to the sequence\-form external regrets\.

Fix a sequence\-form comparatorq^X\\widehat\{q\}^\{X\}induced by a behavioral strategyx^\\widehat\{x\}\. Summing the pointwise CFR decomposition \([C\.1](https://arxiv.org/html/2610.19016#A3.E1)\) overttgives

∑t=1T⟨Aseq​qtY,q^X−qtX⟩=∑I∈ℐXq^X​\[pI\]​∑t=1T⟨FIX​\(xt,yt\),x^I−xI,t⟩\.\\sum\_\{t=1\}^\{T\}\\left\\langle A\_\{\\rm seq\}q\_\{t\}^\{Y\},\\widehat\{q\}^\{X\}\-q\_\{t\}^\{X\}\\right\\rangle=\\sum\_\{I\\in\\mathcal\{I\}\_\{X\}\}\\widehat\{q\}^\{X\}\[p\_\{I\}\]\\sum\_\{t=1\}^\{T\}\\left\\langle F\_\{I\}^\{X\}\(x\_\{t\},y\_\{t\}\),\\widehat\{x\}\_\{I\}\-x\_\{I,t\}\\right\\rangle\.For each information setII, the inner sum is one particular fixed\-action comparison and is therefore at most the local external regretRegIX⁡\(T\)\\operatorname\{Reg\}\_\{I\}^\{X\}\(T\)\. Since0≤q^X​\[pI\]≤10\\leq\\widehat\{q\}^\{X\}\[p\_\{I\}\]\\leq 1,

∑t=1T⟨Aseq​qtY,q^X−qtX⟩\\displaystyle\\sum\_\{t=1\}^\{T\}\\left\\langle A\_\{\\rm seq\}q\_\{t\}^\{Y\},\\widehat\{q\}^\{X\}\-q\_\{t\}^\{X\}\\right\\rangle≤∑Iq^X​\[pI\]​RegIX⁡\(T\)\\displaystyle\\leq\\sum\_\{I\}\\widehat\{q\}^\{X\}\[p\_\{I\}\]\\operatorname\{Reg\}\_\{I\}^\{X\}\(T\)≤∑I\[RegIX⁡\(T\)\]\+=𝒪T​\(1\)\.\\displaystyle\\leq\\sum\_\{I\}\\left\[\\operatorname\{Reg\}\_\{I\}^\{X\}\(T\)\\right\]\_\{\+\}=\\mathcal\{O\}\_\{T\}\(1\)\.The bound is uniform in the comparatorq^X\\widehat\{q\}^\{X\}, so maximizing over all sequence\-form comparators givesRegX𝒬⁡\(T\)=𝒪T​\(1\)\\operatorname\{Reg\}\_\{X\}^\{\\mathcal\{Q\}\}\(T\)=\\mathcal\{O\}\_\{T\}\(1\)\. The same argument for the column player givesRegY𝒬⁡\(T\)=𝒪T​\(1\)\\operatorname\{Reg\}\_\{Y\}^\{\\mathcal\{Q\}\}\(T\)=\\mathcal\{O\}\_\{T\}\(1\)\.

For completeness, we now make the regret\-to\-gap identity explicit\. By linearity in the comparator,

RegX𝒬⁡\(T\)\\displaystyle\\operatorname\{Reg\}\_\{X\}^\{\\mathcal\{Q\}\}\(T\)=T​maxq^X​⟨Aseq​q¯TY,q^X⟩−∑t=1TU⁡\(qtX,qtY\),\\displaystyle=T\\max\_\{\\widehat\{q\}^\{X\}\}\\left\\langle A\_\{\\rm seq\}\\bar\{q\}\_\{T\}^\{Y\},\\widehat\{q\}^\{X\}\\right\\rangle\-\\sum\_\{t=1\}^\{T\}U\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\),RegY𝒬⁡\(T\)\\displaystyle\\operatorname\{Reg\}\_\{Y\}^\{\\mathcal\{Q\}\}\(T\)=∑t=1TU⁡\(qtX,qtY\)−T​minq^Y​⟨Aseq​q^Y,q¯TX⟩\.\\displaystyle=\\sum\_\{t=1\}^\{T\}U\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\-T\\min\_\{\\widehat\{q\}^\{Y\}\}\\left\\langle A\_\{\\rm seq\}\\widehat\{q\}^\{Y\},\\bar\{q\}\_\{T\}^\{X\}\\right\\rangle\.The realized\-payoff sums cancel when the two equations are added\. The remaining two optimization terms are exactly the sequence\-form duality gap, so

\(C\.4\)RegX𝒬⁡\(T\)\+RegY𝒬⁡\(T\)=T​Gap𝒬⁡\(q¯TX,q¯TY\)\.\\operatorname\{Reg\}\_\{X\}^\{\\mathcal\{Q\}\}\(T\)\+\\operatorname\{Reg\}\_\{Y\}^\{\\mathcal\{Q\}\}\(T\)=T\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(\\bar\{q\}\_\{T\}^\{X\},\\bar\{q\}\_\{T\}^\{Y\}\)\.Since the left side is bounded independently ofTT, division byTTgivesGap𝒬⁡\(q¯TX,q¯TY\)=𝒪T​\(T−1\)\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(\\bar\{q\}\_\{T\}^\{X\},\\bar\{q\}\_\{T\}^\{Y\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)\. ∎

### C\.3\.Proof of Theorem[4](https://arxiv.org/html/2610.19016#S7.Thmtheorem4)

###### Proof\.

The product\-of\-simplices corollary already gives square summability of the sum of the local current residuals\. The only remaining step is to show that the sequence\-form duality gap is no larger than that local sum\.

Write

RI,tX\\displaystyle R\_\{I,t\}^\{X\}:=maxp∈Δ\|A⁡\(I\)\|⁡⟨FIX​\(xt,yt\),p−xI,t⟩,\\displaystyle:=\\max\_\{p\\in\\Delta\_\{\|A\(I\)\|\}\}\\left\\langle F\_\{I\}^\{X\}\(x\_\{t\},y\_\{t\}\),p\-x\_\{I,t\}\\right\\rangle,RJ,tY\\displaystyle R\_\{J,t\}^\{Y\}:=maxp∈Δ\|A⁡\(J\)\|⁡⟨FJY​\(xt,yt\),p−yJ,t⟩,\\displaystyle:=\\max\_\{p\\in\\Delta\_\{\|A\(J\)\|\}\}\\left\\langle F\_\{J\}^\{Y\}\(x\_\{t\},y\_\{t\}\),p\-y\_\{J,t\}\\right\\rangle,and set

Rt:=∑I∈ℐXRI,tX\+∑J∈ℐYRJ,tY\.R\_\{t\}:=\\sum\_\{I\\in\\mathcal\{I\}\_\{X\}\}R\_\{I,t\}^\{X\}\+\\sum\_\{J\\in\\mathcal\{I\}\_\{Y\}\}R\_\{J,t\}^\{Y\}\.Corollary[1](https://arxiv.org/html/2610.19016#A2.Thmtheorem1)gives

\(C\.5\)∑t=1∞Rt2<∞\.\\sum\_\{t=1\}^\{\\infty\}R\_\{t\}^\{2\}<\\infty\.
Fix a roundttand a sequence\-form row comparatorq^X\\widehat\{q\}^\{X\}induced byx^\\widehat\{x\}\. The pointwise decomposition gives

⟨Aseq​qtY,q^X−qtX⟩=∑I∈ℐXq^X​\[pI\]​⟨FIX​\(xt,yt\),x^I−xI,t⟩\.\\left\\langle A\_\{\\rm seq\}q\_\{t\}^\{Y\},\\widehat\{q\}^\{X\}\-q\_\{t\}^\{X\}\\right\\rangle=\\sum\_\{I\\in\\mathcal\{I\}\_\{X\}\}\\widehat\{q\}^\{X\}\[p\_\{I\}\]\\left\\langle F\_\{I\}^\{X\}\(x\_\{t\},y\_\{t\}\),\\widehat\{x\}\_\{I\}\-x\_\{I,t\}\\right\\rangle\.For everyII, the local inner product is at mostRI,tXR\_\{I,t\}^\{X\}, and the realization weight satisfies0≤q^X​\[pI\]≤10\\leq\\widehat\{q\}^\{X\}\[p\_\{I\}\]\\leq 1\. Therefore

⟨Aseq​qtY,q^X−qtX⟩≤∑I∈ℐXRI,tX\.\\left\\langle A\_\{\\rm seq\}q\_\{t\}^\{Y\},\\widehat\{q\}^\{X\}\-q\_\{t\}^\{X\}\\right\\rangle\\leq\\sum\_\{I\\in\\mathcal\{I\}\_\{X\}\}R\_\{I,t\}^\{X\}\.The right side does not depend on the comparator, so maximizing overq^X\\widehat\{q\}^\{X\}yields

maxq^X⁡⟨Aseq​qtY,q^X−qtX⟩≤∑I∈ℐXRI,tX\.\\max\_\{\\widehat\{q\}^\{X\}\}\\left\\langle A\_\{\\rm seq\}q\_\{t\}^\{Y\},\\widehat\{q\}^\{X\}\-q\_\{t\}^\{X\}\\right\\rangle\\leq\\sum\_\{I\\in\\mathcal\{I\}\_\{X\}\}R\_\{I,t\}^\{X\}\.The same argument for the column player gives

maxq^Y⁡⟨−Aseq⊤​qtX,q^Y−qtY⟩≤∑J∈ℐYRJ,tY\.\\max\_\{\\widehat\{q\}^\{Y\}\}\\left\\langle\-A\_\{\\rm seq\}^\{\\top\}q\_\{t\}^\{X\},\\widehat\{q\}^\{Y\}\-q\_\{t\}^\{Y\}\\right\\rangle\\leq\\sum\_\{J\\in\\mathcal\{I\}\_\{Y\}\}R\_\{J,t\}^\{Y\}\.The sequence\-form duality gap is the sum of these two sequence\-form residuals\. Hence

0≤Gap𝒬⁡\(qtX,qtY\)≤Rt\.0\\leq\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\\leq R\_\{t\}\.Squaring and summing, \([C\.5](https://arxiv.org/html/2610.19016#A3.E5)\) gives \([30](https://arxiv.org/html/2610.19016#S7.E30)\)\.

It follows that the current sequence\-form gap converges to zero\. For the distance conclusion, suppose instead that

dist⁡\(\(qtX,qtY\),𝒵𝒬⋆\)\\operatorname\{dist\}\\bigl\(\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\),\\mathcal\{Z\}\_\{\\mathcal\{Q\}\}^\{\\star\}\\bigr\)did not tend to zero\. Then someε\>0\\varepsilon\>0and subsequence would remain at leastε\\varepsilonaway from the sequence\-form Nash equilibrium set\. The realization\- plan polytopes are compact, so this subsequence has a convergent subsubsequence\. Continuity ofGap𝒬\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}makes the limit have zero gap, hence the limit belongs to𝒵𝒬⋆\\mathcal\{Z\}\_\{\\mathcal\{Q\}\}^\{\\star\}, a contradiction\.

Finally, ifK:=∑tGap𝒬⁡\(qtX,qtY\)2<∞K:=\\sum\_\{t\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)^\{2\}<\\infty, then

T​min1≤t≤T​Gap𝒬​\(qtX,qtY\)2≤K\.T\\min\_\{1\\leq t\\leq T\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)^\{2\}\\leq K\.Taking square roots gives \([31](https://arxiv.org/html/2610.19016#S7.E31)\)\. ∎

## Appendix DProofs for Alternating IR\-PRM\+

We use the alternating timing from Section[8](https://arxiv.org/html/2610.19016#S8), includingy0=y~1y\_\{0\}=\\tilde\{y\}\_\{1\}and the first column update at\(x1,y0\)\(x\_\{1\},y\_\{0\}\)\. For the row player,x~t\\tilde\{x\}\_\{t\}is the strategy induced by its regret vector before the prediction producingxtx\_\{t\}\. For the column player, the indexing is shifted by one half\-step:yt−1y\_\{t\-1\}is the strategy present when−A⊤​xt\-A^\{\\top\}x\_\{t\}is observed, the update produces the next stored strategyy~t\+1\\tilde\{y\}\_\{t\+1\}and normρt\+1Y\\rho\_\{t\+1\}^\{Y\}, and the prediction based on−A⊤​xt\-A^\{\\top\}x\_\{t\}then producesyty\_\{t\}\.

###### Lemma 0 \(One\-step inequality with the updated prediction\)\.

Consider onekk\-action IR\-PRM\+ learner\. After it playsxtx\_\{t\}, observesutu\_\{t\}, and updates its regret vector, suppose that the next prediction ismt\+1=utm\_\{t\+1\}=u\_\{t\}\. Ifxt\+1x\_\{t\+1\}is the strategy produced by this prediction, then

\(D\.1\)⟨ut,xt\+1−xt⟩≥−\(ρt\+1−ρt\)\.\\left\\langle u\_\{t\},x\_\{t\+1\}\-x\_\{t\}\\right\\rangle\\geq\-\(\\rho\_\{t\+1\}\-\\rho\_\{t\}\)\.

###### Proof\.

By \([5](https://arxiv.org/html/2610.19016#S3.E5)\),ρt\>0\\rho\_\{t\}\>0\. Writeqt=\[rt\]\+q\_\{t\}=\[r\_\{t\}\]\_\{\+\}andQt=‖qt‖1Q\_\{t\}=\\left\\lVert q\_\{t\}\\right\\rVert\_\{1\}, soxt=qt/Qtx\_\{t\}=q\_\{t\}/Q\_\{t\}and‖qt‖2=ρt\\left\\lVert q\_\{t\}\\right\\rVert\_\{2\}=\\rho\_\{t\}\. Letqt\+1q\_\{t\+1\}be the positive\-part vector after applying the predictionmt\+1=utm\_\{t\+1\}=u\_\{t\}to the updated regret vectorr~t\+1\\tilde\{r\}\_\{t\+1\}\. Thus, for the prediction shiftγt\+1\\gamma\_\{t\+1\},

qt\+1=\[r~t\+1\+ut−γt\+1​𝟏\]\+,‖qt\+1‖2=ρt\+1\.q\_\{t\+1\}=\[\\tilde\{r\}\_\{t\+1\}\+u\_\{t\}\-\\gamma\_\{t\+1\}\\mathbf\{1\}\]\_\{\+\},\\qquad\\left\\lVert q\_\{t\+1\}\\right\\rVert\_\{2\}=\\rho\_\{t\+1\}\.This prediction preserves Euclidean norm, which gives the second equality\.

We first compare the prediction shiftγt\+1\\gamma\_\{t\+1\}with the payoff of the previously played strategyxtx\_\{t\}\. Since\[v\]\+≥v\[v\]\_\{\+\}\\geq vcoordinatewise andqt≥0q\_\{t\}\\geq 0,

⟨qt,qt\+1⟩≥⟨qt,r~t\+1\+ut−γt\+1​𝟏⟩\.\\left\\langle q\_\{t\},q\_\{t\+1\}\\right\\rangle\\geq\\left\\langle q\_\{t\},\\tilde\{r\}\_\{t\+1\}\+u\_\{t\}\-\\gamma\_\{t\+1\}\\mathbf\{1\}\\right\\rangle\.Using⟨qt,ut⟩=Qt​⟨ut,xt⟩\\left\\langle q\_\{t\},u\_\{t\}\\right\\rangle=Q\_\{t\}\\left\\langle u\_\{t\},x\_\{t\}\\right\\rangleand⟨qt,𝟏⟩=Qt\\left\\langle q\_\{t\},\\mathbf\{1\}\\right\\rangle=Q\_\{t\}, rearrangement gives

\(D\.2\)Qt​\(γt\+1−⟨ut,xt⟩\)≥⟨qt,r~t\+1⟩−⟨qt,qt\+1⟩\.Q\_\{t\}\\bigl\(\\gamma\_\{t\+1\}\-\\left\\langle u\_\{t\},x\_\{t\}\\right\\rangle\\bigr\)\\geq\\left\\langle q\_\{t\},\\tilde\{r\}\_\{t\+1\}\\right\\rangle\-\\left\\langle q\_\{t\},q\_\{t\+1\}\\right\\rangle\.The first inner product on the right is at leastρt2\\rho\_\{t\}^\{2\}by the same argument as \([10](https://arxiv.org/html/2610.19016#S5.E10)\), since the update vector satisfies⟨gt,qt⟩=0\\left\\langle g\_\{t\},q\_\{t\}\\right\\rangle=0\. The second is at mostρt​ρt\+1\\rho\_\{t\}\\rho\_\{t\+1\}by Cauchy–Schwarz\. Therefore

Qt​\(γt\+1−⟨ut,xt⟩\)≥ρt2−ρt​ρt\+1=−ρt​\(ρt\+1−ρt\)\.Q\_\{t\}\\bigl\(\\gamma\_\{t\+1\}\-\\left\\langle u\_\{t\},x\_\{t\}\\right\\rangle\\bigr\)\\geq\\rho\_\{t\}^\{2\}\-\\rho\_\{t\}\\rho\_\{t\+1\}=\-\\rho\_\{t\}\(\\rho\_\{t\+1\}\-\\rho\_\{t\}\)\.BecauseQt=‖qt‖1≥‖qt‖2=ρtQ\_\{t\}=\\left\\lVert q\_\{t\}\\right\\rVert\_\{1\}\\geq\\left\\lVert q\_\{t\}\\right\\rVert\_\{2\}=\\rho\_\{t\}, division byQtQ\_\{t\}yields

\(D\.3\)γt\+1−⟨ut,xt⟩≥−\(ρt\+1−ρt\)\.\\gamma\_\{t\+1\}\-\\left\\langle u\_\{t\},x\_\{t\}\\right\\rangle\\geq\-\(\\rho\_\{t\+1\}\-\\rho\_\{t\}\)\.
Next we show that the strategy produced by the updated prediction,xt\+1x\_\{t\+1\}, has payoff at leastγt\+1\\gamma\_\{t\+1\}\. Define

h:=qt\+1−\(r~t\+1\+ut−γt\+1​𝟏\)≥0\.h:=q\_\{t\+1\}\-\(\\tilde\{r\}\_\{t\+1\}\+u\_\{t\}\-\\gamma\_\{t\+1\}\\mathbf\{1\}\)\\geq 0\.The residualhhis nonzero only where the unclipped vector is negative, whereasqt\+1q\_\{t\+1\}is zero there\. Thushhandqt\+1q\_\{t\+1\}have disjoint support and⟨h,xt\+1⟩=0\\left\\langle h,x\_\{t\+1\}\\right\\rangle=0\. WithQt\+1:=‖qt\+1‖1Q\_\{t\+1\}:=\\left\\lVert q\_\{t\+1\}\\right\\rVert\_\{1\}andxt\+1=qt\+1/Qt\+1x\_\{t\+1\}=q\_\{t\+1\}/Q\_\{t\+1\}, dottingqt\+1=r~t\+1\+ut−γt\+1​𝟏\+hq\_\{t\+1\}=\\tilde\{r\}\_\{t\+1\}\+u\_\{t\}\-\\gamma\_\{t\+1\}\\mathbf\{1\}\+hwithqt\+1q\_\{t\+1\}gives

⟨ut,xt\+1⟩−γt\+1\\displaystyle\\left\\langle u\_\{t\},x\_\{t\+1\}\\right\\rangle\-\\gamma\_\{t\+1\}=‖qt\+1‖22−⟨r~t\+1,qt\+1⟩Qt\+1\\displaystyle=\\frac\{\\left\\lVert q\_\{t\+1\}\\right\\rVert\_\{2\}^\{2\}\-\\left\\langle\\tilde\{r\}\_\{t\+1\},q\_\{t\+1\}\\right\\rangle\}\{Q\_\{t\+1\}\}≥0\.\\displaystyle\\geq 0\.The last inequality follows from Cauchy–Schwarz and‖r~t\+1‖2=‖qt\+1‖2=ρt\+1\\left\\lVert\\tilde\{r\}\_\{t\+1\}\\right\\rVert\_\{2\}=\\left\\lVert q\_\{t\+1\}\\right\\rVert\_\{2\}=\\rho\_\{t\+1\}\. Adding this nonnegative quantity to \([D\.3](https://arxiv.org/html/2610.19016#A4.E3)\) gives

⟨ut,xt\+1−xt⟩=\(⟨ut,xt\+1⟩−γt\+1\)\+\(γt\+1−⟨ut,xt⟩\)≥−\(ρt\+1−ρt\),\\left\\langle u\_\{t\},x\_\{t\+1\}\-x\_\{t\}\\right\\rangle=\\bigl\(\\left\\langle u\_\{t\},x\_\{t\+1\}\\right\\rangle\-\\gamma\_\{t\+1\}\\bigr\)\+\\bigl\(\\gamma\_\{t\+1\}\-\\left\\langle u\_\{t\},x\_\{t\}\\right\\rangle\\bigr\)\\geq\-\(\\rho\_\{t\+1\}\-\\rho\_\{t\}\),which proves \([D\.1](https://arxiv.org/html/2610.19016#A4.E1)\)\.

∎

We next record the two staggered norm\-growth inequalities, which use the same constants as Lemma[1](https://arxiv.org/html/2610.19016#S6.Thmtheorem1); no additional game parameter is required\.

###### Lemma 0 \(Alternating regret\-vector norm growth\)\.

For everyt≥2t\\geq 2, the alternating updates satisfy

ρt\+1Y−ρtY\\displaystyle\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}≤2​\(1\+kX\)2​\(1\+kY\)2​‖A‖22ρtY​ρtX\\displaystyle\\leq\\frac\{2\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{\\rho\_\{t\}^\{Y\}\\rho\_\{t\}^\{X\}\}\(D\.4\)×\[ρtX2​kX​‖xt−x~t‖22\+\(ρtX−ρt−1X\)\],\\displaystyle\\quad\\times\\left\[\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\(\\rho\_\{t\}^\{X\}\-\\rho\_\{t\-1\}^\{X\}\)\\right\],ρt\+1X−ρtX\\displaystyle\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}≤2​\(1\+kY\)2​\(1\+kX\)2​‖A‖22ρtX​ρt\+1Y\\displaystyle\\leq\\frac\{2\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{\\rho\_\{t\}^\{X\}\\rho\_\{t\+1\}^\{Y\}\}\(D\.5\)×\[ρt\+1Y2​kY​‖yt−y~t\+1‖22\+\(ρt\+1Y−ρtY\)\]\.\\displaystyle\\quad\\times\\left\[\\frac\{\\rho\_\{t\+1\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\+1\}\\right\\rVert\_\{2\}^\{2\}\+\(\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}\)\\right\]\.

###### Proof\.

The derivation is the same two\-step conversion as in Lemma[1](https://arxiv.org/html/2610.19016#S6.Thmtheorem1), but the column player is half a step ahead of the row player\. We therefore derive the two recurrences separately so that the shifted indices are visible\.

For the column player, the utility observed at its current half\-step is−A⊤​xt\-A^\{\\top\}x\_\{t\}, while its previous\-utility prediction was−A⊤​xt−1\-A^\{\\top\}x\_\{t\-1\}\. Thus its prediction error is−A⊤​\(xt−xt−1\)\-A^\{\\top\}\(x\_\{t\}\-x\_\{t\-1\}\)\. Applying the centering bound \([17](https://arxiv.org/html/2610.19016#S6.E17)\), the operator\-norm inequality, and then \([16](https://arxiv.org/html/2610.19016#S6.E16)\) gives

ρt\+1Y−ρtY\\displaystyle\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}≤‖gtY‖222​ρtY\\displaystyle\\leq\\frac\{\\left\\lVert g\_\{t\}^\{Y\}\\right\\rVert\_\{2\}^\{2\}\}\{2\\rho\_\{t\}^\{Y\}\}≤\(1\+kY\)2​‖A‖222​ρtY​‖xt−xt−1‖22\.\\displaystyle\\leq\\frac\{\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{2\\rho\_\{t\}^\{Y\}\}\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\.The row player’s consecutive movement is controlled by Corollary[2](https://arxiv.org/html/2610.19016#S5.Thmtheorem2):

‖xt−xt−1‖22≤2​‖xt−x~t‖22\+4​\(1\+kX\)2ρtX​\(ρtX−ρt−1X\)\.\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\\leq 2\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{4\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\}\{\\rho\_\{t\}^\{X\}\}\(\\rho\_\{t\}^\{X\}\-\\rho\_\{t\-1\}^\{X\}\)\.Substitution gives

ρt\+1Y−ρtY\\displaystyle\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}≤\(1\+kY\)2​‖A‖22ρtY​‖xt−x~t‖22\\displaystyle\\leq\\frac\{\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{\\rho\_\{t\}^\{Y\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+2​\(1\+kX\)2​\(1\+kY\)2​‖A‖22ρtY​ρtX​\(ρtX−ρt−1X\)\.\\displaystyle\\quad\+\\frac\{2\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{\\rho\_\{t\}^\{Y\}\\rho\_\{t\}^\{X\}\}\(\\rho\_\{t\}^\{X\}\-\\rho\_\{t\-1\}^\{X\}\)\.As in the simultaneous proof,\(1\+kX\)2/kX≥1\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}/k^\{X\}\\geq 1, so

\(1\+kY\)2​‖A‖22ρtY≤2​\(1\+kX\)2​\(1\+kY\)2​‖A‖22ρtY​ρtX​ρtX2​kX\.\\frac\{\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{\\rho\_\{t\}^\{Y\}\}\\leq\\frac\{2\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{\\rho\_\{t\}^\{Y\}\\rho\_\{t\}^\{X\}\}\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\.Putting the two terms under this common coefficient yields \([D\.4](https://arxiv.org/html/2610.19016#A4.E4)\)\.

For the row player, the prediction that producedxtx\_\{t\}usedA​yt−1Ay\_\{t\-1\}, whereas the utility observed after the column update isA​ytAy\_\{t\}\. Hence the row prediction error isA⁡\(yt−yt−1\)A\(y\_\{t\}\-y\_\{t\-1\}\), and the same argument gives

ρt\+1X−ρtX≤\(1\+kX\)2​‖A‖222​ρtX​‖yt−yt−1‖22\.\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}\\leq\\frac\{\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\}\{2\\rho\_\{t\}^\{X\}\}\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\.The movementyt−1→yty\_\{t\-1\}\\to y\_\{t\}is one complete update of the column player under this half\-step indexing\. Therefore Corollary[2](https://arxiv.org/html/2610.19016#S5.Thmtheorem2)applies with the newly produced normρt\+1Y\\rho\_\{t\+1\}^\{Y\}:

‖yt−yt−1‖22≤2​‖yt−y~t\+1‖22\+4​\(1\+kY\)2ρt\+1Y​\(ρt\+1Y−ρtY\)\.\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\\leq 2\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\+1\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{4\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\}\{\\rho\_\{t\+1\}^\{Y\}\}\(\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}\)\.Substituting and using\(1\+kY\)2/kY≥1\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}/k^\{Y\}\\geq 1gives \([D\.5](https://arxiv.org/html/2610.19016#A4.E5)\)\.

The different column denominators in the two recurrences are therefore not an extra assumption: they record which column norm exists at the relevant point under the staggered indexing\. The column recurrence usesρtY\\rho\_\{t\}^\{Y\}before the column update, while the row recurrence usesρt\+1Y\\rho\_\{t\+1\}^\{Y\}after that update\. All denominators are bounded below by one under the initialization \([2](https://arxiv.org/html/2610.19016#S3.E2)\)\. ∎

### D\.1\.Proof of Theorem[1](https://arxiv.org/html/2610.19016#S8.Thmtheorem1)

###### Proof\.

Use the constantκA\\kappa\_\{A\}defined in Lemma[1](https://arxiv.org/html/2610.19016#S6.Thmtheorem1), namelyκA=2​\(1\+kX\)2​\(1\+kY\)2​‖A‖22\\kappa\_\{A\}=2\(1\+\\sqrt\{k^\{X\}\}\)^\{2\}\(1\+\\sqrt\{k^\{Y\}\}\)^\{2\}\\left\\lVert A\\right\\rVert\_\{2\}^\{2\}\. The proof again splits according to whether the norm product is below or above the threshold, but the two players now use staggered indices\. By \([2](https://arxiv.org/html/2610.19016#S3.E2)\) and monotonicity, both norms are at least one throughout\. Suppose first that

\(D\.6\)ρtX​ρtY<4​κA\\rho\_\{t\}^\{X\}\\rho\_\{t\}^\{Y\}<4\\kappa\_\{A\}for everytt\. Since both norms are at least one, this product bound directly gives

ρtX≤4​κA,ρtY≤4​κA,\\rho\_\{t\}^\{X\}\\leq 4\\kappa\_\{A\},\\qquad\\rho\_\{t\}^\{Y\}\\leq 4\\kappa\_\{A\},so both norms are already bounded over all rounds\.

Now suppose the threshold is reached\. Letτ≥2\\tau\\geq 2be the first iteration from which it holds; if it already holds at iteration11, takeτ=2\\tau=2and absorb the first iteration into the constants\. As in the simultaneous proof, the first crossing can overshoot by only a bounded amount: positive\-part projection is nonexpansive, soρt\+1−ρt≤‖gt‖2\\rho\_\{t\+1\}\-\\rho\_\{t\}\\leq\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}, and matrix\-game prediction errors are bounded over all rounds\. We therefore absorb the finite crossing and all earlier rounds into constants and analyze the suffixt≥τt\\geq\\tau\.

Fort≥τt\\geq\\tau, both coefficients in \([D\.4](https://arxiv.org/html/2610.19016#A4.E4)\)–\([D\.5](https://arxiv.org/html/2610.19016#A4.E5)\) are at most1/41/4\. For the row recurrence, the denominator containsρt\+1Y\\rho\_\{t\+1\}^\{Y\}, and monotonicity givesρt\+1Y≥ρtY\\rho\_\{t\+1\}^\{Y\}\\geq\\rho\_\{t\}^\{Y\}, so the same threshold is sufficient\. Hence

\(D\.7\)ρt\+1Y−ρtY\\displaystyle\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}≤14​\[ρtX2​kX​‖xt−x~t‖22\+\(ρtX−ρt−1X\)\],\\displaystyle\\leq\\frac\{1\}\{4\}\\left\[\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\(\\rho\_\{t\}^\{X\}\-\\rho\_\{t\-1\}^\{X\}\)\\right\],\(D\.8\)ρt\+1X−ρtX\\displaystyle\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}≤14​\[ρt\+1Y2​kY​‖yt−y~t\+1‖22\+\(ρt\+1Y−ρtY\)\]\.\\displaystyle\\leq\\frac\{1\}\{4\}\\left\[\\frac\{\\rho\_\{t\+1\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\+1\}\\right\\rVert\_\{2\}^\{2\}\+\(\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}\)\\right\]\.
We now sum these two staggered recurrences carefully\. Define

PT\\displaystyle P\_\{T\}:=∑t=τTρtX2​kX​‖xt−x~t‖22,\\displaystyle:=\\sum\_\{t=\\tau\}^\{T\}\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\},QT\\displaystyle Q\_\{T\}:=∑t=τTρt\+1Y2​kY​‖yt−y~t\+1‖22,\\displaystyle:=\\sum\_\{t=\\tau\}^\{T\}\\frac\{\\rho\_\{t\+1\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\+1\}\\right\\rVert\_\{2\}^\{2\},XT\\displaystyle X\_\{T\}:=ρT\+1X−ρτX,YT:=ρT\+1Y−ρτY,\\displaystyle:=\\rho\_\{T\+1\}^\{X\}\-\\rho\_\{\\tau\}^\{X\},\\qquad Y\_\{T\}:=\\rho\_\{T\+1\}^\{Y\}\-\\rho\_\{\\tau\}^\{Y\},bτ\\displaystyle b\_\{\\tau\}:=ρτX−ρτ−1X\.\\displaystyle:=\\rho\_\{\\tau\}^\{X\}\-\\rho\_\{\\tau\-1\}^\{X\}\.Summing \([D\.7](https://arxiv.org/html/2610.19016#A4.E7)\) gives

\(D\.9\)YT≤14​PT\+14​XT\+14​bτ,Y\_\{T\}\\leq\\frac\{1\}\{4\}P\_\{T\}\+\\frac\{1\}\{4\}X\_\{T\}\+\\frac\{1\}\{4\}b\_\{\\tau\},where the extrabτb\_\{\\tau\}comes from the one\-index shift in the row norm increments\. Summing \([D\.8](https://arxiv.org/html/2610.19016#A4.E8)\) gives

\(D\.10\)XT≤14​QT\+14​YT\.X\_\{T\}\\leq\\frac\{1\}\{4\}Q\_\{T\}\+\\frac\{1\}\{4\}Y\_\{T\}\.Substituting \([D\.9](https://arxiv.org/html/2610.19016#A4.E9)\) into \([D\.10](https://arxiv.org/html/2610.19016#A4.E10)\) gives

XT≤14​QT\+116​PT\+116​XT\+116​bτ\.X\_\{T\}\\leq\\frac\{1\}\{4\}Q\_\{T\}\+\\frac\{1\}\{16\}P\_\{T\}\+\\frac\{1\}\{16\}X\_\{T\}\+\\frac\{1\}\{16\}b\_\{\\tau\}\.Thus

XT≤415​QT\+115​PT\+115​bτ\.X\_\{T\}\\leq\\frac\{4\}\{15\}Q\_\{T\}\+\\frac\{1\}\{15\}P\_\{T\}\+\\frac\{1\}\{15\}b\_\{\\tau\}\.Putting this bound back into \([D\.9](https://arxiv.org/html/2610.19016#A4.E9)\) gives

YT≤415​PT\+115​QT\+415​bτ\.Y\_\{T\}\\leq\\frac\{4\}\{15\}P\_\{T\}\+\\frac\{1\}\{15\}Q\_\{T\}\+\\frac\{4\}\{15\}b\_\{\\tau\}\.Adding the last two displays yields the clean combined bound

\(D\.11\)XT\+YT≤13​\(PT\+QT\+bτ\)\.X\_\{T\}\+Y\_\{T\}\\leq\\frac\{1\}\{3\}\(P\_\{T\}\+Q\_\{T\}\+b\_\{\\tau\}\)\.This is where the factor1/31/3in the alternating proof comes from\.

Reindex the shifted column sum inQTQ\_\{T\}bys=t\+1s=t\+1\. All interior terms become the corresponding shifted prediction displacement termsρsY2​kY​‖ys−1−y~s‖22\\frac\{\\rho\_\{s\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{s\-1\}\-\\tilde\{y\}\_\{s\}\\right\\rVert\_\{2\}^\{2\}; only the terminal term ats=T\+1s=T\+1remains outside the usual sum\. Absorbing the finite prefix andbτb\_\{\\tau\}into a constantCaltC\_\{\\rm alt\}gives

ρT\+1X\+ρT\+1Y\\displaystyle\\rho\_\{T\+1\}^\{X\}\+\\rho\_\{T\+1\}^\{Y\}≤Calt\+13​∑t=1T\[ρtX2​kX​‖xt−x~t‖22\+ρtY2​kY​‖yt−1−y~t‖22\]\\displaystyle\\leq C\_\{\\rm alt\}\+\\frac\{1\}\{3\}\\sum\_\{t=1\}^\{T\}\\left\[\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{\\rho\_\{t\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\-1\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\right\]\(D\.12\)\+13​ρT\+1Y2​kY​‖yT−y~T\+1‖22\.\\displaystyle\\quad\+\\frac\{1\}\{3\}\\frac\{\\rho\_\{T\+1\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{T\}\-\\tilde\{y\}\_\{T\+1\}\\right\\rVert\_\{2\}^\{2\}\.
We next bound the prediction displacement sum on the right\. Define the column player’s external regret under the half\-step indexing by

RegYhalf⁡\(T\):=max⁡∑t=1Ty′∈ΔkY⁡⟨−A⊤​xt,y′−yt−1⟩\.\\operatorname\{Reg\}\_\{Y\}^\{\\rm half\}\(T\):=\\max\_\{y^\{\\prime\}\\in\\Delta\_\{k^\{Y\}\}\}\\sum\_\{t=1\}^\{T\}\\left\\langle\-A^\{\\top\}x\_\{t\},y^\{\\prime\}\-y\_\{t\-1\}\\right\\rangle\.To see the relation with the duality gap, expand the two regrets\. The row regret is

RegX⁡\(T\)=T​maxx′∈ΔkX​⟨A​y¯T,x′⟩−∑t=1Txt⊤​A​yt,\\operatorname\{Reg\}\_\{X\}\(T\)=T\\max\_\{x^\{\\prime\}\\in\\Delta\_\{k^\{X\}\}\}\\left\\langle A\\bar\{y\}\_\{T\},x^\{\\prime\}\\right\\rangle\-\\sum\_\{t=1\}^\{T\}x\_\{t\}^\{\\top\}Ay\_\{t\},while the half\-step column external regret is

RegYhalf⁡\(T\)=∑t=1Txt⊤​A​yt−1−T​miny′∈ΔkY​⟨A​y′,x¯T⟩\.\\operatorname\{Reg\}\_\{Y\}^\{\\rm half\}\(T\)=\\sum\_\{t=1\}^\{T\}x\_\{t\}^\{\\top\}Ay\_\{t\-1\}\-T\\min\_\{y^\{\\prime\}\\in\\Delta\_\{k^\{Y\}\}\}\\left\\langle Ay^\{\\prime\},\\bar\{x\}\_\{T\}\\right\\rangle\.Adding them and inserting and subtracting∑txt⊤​A​yt\\sum\_\{t\}x\_\{t\}^\{\\top\}Ay\_\{t\}gives

\(D\.13\)RegX⁡\(T\)\+RegYhalf⁡\(T\)=T​Gap⁡\(x¯T,y¯T\)\+∑t=1T⟨−A⊤​xt,yt−yt−1⟩\.\\operatorname\{Reg\}\_\{X\}\(T\)\+\\operatorname\{Reg\}\_\{Y\}^\{\\rm half\}\(T\)=T\\operatorname\{Gap\}\(\\bar\{x\}\_\{T\},\\bar\{y\}\_\{T\}\)\+\\sum\_\{t=1\}^\{T\}\\left\\langle\-A^\{\\top\}x\_\{t\},y\_\{t\}\-y\_\{t\-1\}\\right\\rangle\.Thus the only difference from the usual zero\-sum identity is the payoff change caused by the column update\.

Lemma[1](https://arxiv.org/html/2610.19016#A4.Thmtheorem1), applied to the column player under the half\-step indexing, bounds each such payoff change from below:

⟨−A⊤​xt,yt−yt−1⟩≥−\(ρt\+1Y−ρtY\)\.\\left\\langle\-A^\{\\top\}x\_\{t\},y\_\{t\}\-y\_\{t\-1\}\\right\\rangle\\geq\-\(\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}\)\.Summing telescopes, andρ1Y=1\\rho\_\{1\}^\{Y\}=1gives a payoff\-shift lower bound−\(ρT\+1Y−ρ1Y\)≥−ρT\+1Y\-\(\\rho\_\{T\+1\}^\{Y\}\-\\rho\_\{1\}^\{Y\}\)\\geq\-\\rho\_\{T\+1\}^\{Y\}\. Since the gap in \([D\.13](https://arxiv.org/html/2610.19016#A4.E13)\) is nonnegative,

\(D\.14\)RegX⁡\(T\)\+RegYhalf⁡\(T\)≥−ρT\+1Y\.\\operatorname\{Reg\}\_\{X\}\(T\)\+\\operatorname\{Reg\}\_\{Y\}^\{\\rm half\}\(T\)\\geq\-\\rho\_\{T\+1\}^\{Y\}\.
Fact[1](https://arxiv.org/html/2610.19016#Thmfact1), applied to the row and column updates at their respective indices, gives an upper bound on the same regret sum:

RegX⁡\(T\)\+RegYhalf⁡\(T\)\\displaystyle\\operatorname\{Reg\}\_\{X\}\(T\)\+\\operatorname\{Reg\}\_\{Y\}^\{\\rm half\}\(T\)≤ρT\+1X\+ρT\+1Y\\displaystyle\\leq\\rho\_\{T\+1\}^\{X\}\+\\rho\_\{T\+1\}^\{Y\}−∑t=1TρtX2​kX‖xt−x~t‖22\\displaystyle\\quad\-\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}−∑t=1TρtY2​kY‖yt−1−y~t‖22\.\\displaystyle\\quad\-\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\-1\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\.Let

HTalt:=∑t=1T\[ρtX2​kX​‖xt−x~t‖22\+ρtY2​kY​‖yt−1−y~t‖22\]\.H\_\{T\}^\{\\rm alt\}:=\\sum\_\{t=1\}^\{T\}\\left\[\\frac\{\\rho\_\{t\}^\{X\}\}\{2k^\{X\}\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\+\\frac\{\\rho\_\{t\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{t\-1\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\right\]\.Combining the upper bound above with \([D\.14](https://arxiv.org/html/2610.19016#A4.E14)\) gives

−ρT\+1Y≤ρT\+1X\+ρT\+1Y−HTalt\.\-\\rho\_\{T\+1\}^\{Y\}\\leq\\rho\_\{T\+1\}^\{X\}\+\\rho\_\{T\+1\}^\{Y\}\-H\_\{T\}^\{\\rm alt\}\.Moving the prediction displacement terms to the left and the lower\-bound term to the right gives

\(D\.15\)HTalt≤ρT\+1X\+2​ρT\+1Y\.H\_\{T\}^\{\\rm alt\}\\leq\\rho\_\{T\+1\}^\{X\}\+2\\rho\_\{T\+1\}^\{Y\}\.
The remaining terminal prediction displacement in \([D\.12](https://arxiv.org/html/2610.19016#A4.E12)\) is also controlled by terminal norm\. IfkY=1k^\{Y\}=1, the column simplex is a singleton and the term is zero\. IfkY≥2k^\{Y\}\\geq 2, the squared simplex diameter is at most22, so

ρT\+1Y2​kY​‖yT−y~T\+1‖22≤ρT\+1YkY≤12​ρT\+1Y≤12​\(ρT\+1X\+ρT\+1Y\)\.\\frac\{\\rho\_\{T\+1\}^\{Y\}\}\{2k^\{Y\}\}\\left\\lVert y\_\{T\}\-\\tilde\{y\}\_\{T\+1\}\\right\\rVert\_\{2\}^\{2\}\\leq\\frac\{\\rho\_\{T\+1\}^\{Y\}\}\{k^\{Y\}\}\\leq\\frac\{1\}\{2\}\\rho\_\{T\+1\}^\{Y\}\\leq\\frac\{1\}\{2\}\(\\rho\_\{T\+1\}^\{X\}\+\\rho\_\{T\+1\}^\{Y\}\)\.LetST\+1:=ρT\+1X\+ρT\+1YS\_\{T\+1\}:=\\rho\_\{T\+1\}^\{X\}\+\\rho\_\{T\+1\}^\{Y\}\. From \([D\.15](https://arxiv.org/html/2610.19016#A4.E15)\), the prediction displacement contribution in \([D\.12](https://arxiv.org/html/2610.19016#A4.E12)\) contributes at most13​\(ρT\+1X\+2​ρT\+1Y\)≤23​ST\+1\\frac\{1\}\{3\}\(\\rho\_\{T\+1\}^\{X\}\+2\\rho\_\{T\+1\}^\{Y\}\)\\leq\\frac\{2\}\{3\}S\_\{T\+1\}, while the terminal prediction displacement contributes at most16​ST\+1\\frac\{1\}\{6\}S\_\{T\+1\}\. Hence

ST\+1≤Calt\+\(23\+16\)​ST\+1=Calt\+56​ST\+1\.S\_\{T\+1\}\\leq C\_\{\\rm alt\}\+\\left\(\\frac\{2\}\{3\}\+\\frac\{1\}\{6\}\\right\)S\_\{T\+1\}=C\_\{\\rm alt\}\+\\frac\{5\}\{6\}S\_\{T\+1\}\.Moving the last term to the left givesST\+1≤6​CaltS\_\{T\+1\}\\leq 6C\_\{\\rm alt\}, so both norm sequences remain bounded over time\.

Fact[1](https://arxiv.org/html/2610.19016#Thmfact1)now givesRegX⁡\(T\)=𝒪T​\(1\)\\operatorname\{Reg\}\_\{X\}\(T\)=\\mathcal\{O\}\_\{T\}\(1\)andRegYhalf⁡\(T\)=𝒪T​\(1\)\\operatorname\{Reg\}\_\{Y\}^\{\\rm half\}\(T\)=\\mathcal\{O\}\_\{T\}\(1\)\. To relate the half\-step external regret to the full\-profile external regret, note that the difference between the two comparator objectives is independent of the comparator:

RegY⁡\(T\)\\displaystyle\\operatorname\{Reg\}\_\{Y\}\(T\)=RegYhalf⁡\(T\)−∑t=1T⟨−A⊤​xt,yt−yt−1⟩\\displaystyle=\\operatorname\{Reg\}\_\{Y\}^\{\\rm half\}\(T\)\-\\sum\_\{t=1\}^\{T\}\\left\\langle\-A^\{\\top\}x\_\{t\},y\_\{t\}\-y\_\{t\-1\}\\right\\rangle≤RegYhalf⁡\(T\)\+ρT\+1Y=𝒪T​\(1\),\\displaystyle\\leq\\operatorname\{Reg\}\_\{Y\}^\{\\rm half\}\(T\)\+\\rho\_\{T\+1\}^\{Y\}=\\mathcal\{O\}\_\{T\}\(1\),where the inequality again uses Lemma[1](https://arxiv.org/html/2610.19016#A4.Thmtheorem1)and telescoping\. Thus both full\-profile external regrets are𝒪T​\(1\)\\mathcal\{O\}\_\{T\}\(1\)\. The usual zero\-sum regret\-to\-gap identity then givesGap⁡\(x¯T,y¯T\)=𝒪T​\(T−1\)\\operatorname\{Gap\}\(\\bar\{x\}\_\{T\},\\bar\{y\}\_\{T\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)\. ∎

### D\.2\.Proof of Theorem[2](https://arxiv.org/html/2610.19016#S8.Thmtheorem2)

###### Proof\.

The previous proof gives regret\-vector norms that remain bounded over time and the prediction displacement bound \([D\.15](https://arxiv.org/html/2610.19016#A4.E15)\)\. Since every player starts with norm one, its norm is bounded below by a fixed positive constant\. Therefore \([D\.15](https://arxiv.org/html/2610.19016#A4.E15)\) implies square summability of the row prediction displacements‖xt−x~t‖2\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}and the shifted column prediction displacements‖yt−1−y~t‖2\\left\\lVert y\_\{t\-1\}\-\\tilde\{y\}\_\{t\}\\right\\rVert\_\{2\}\.

Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)gives the corresponding post\-update displacement bounds\. Specifically,

‖x~t\+1−xt‖22≤CX​\(ρt\+1X−ρtX\),\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq C\_\{X\}\(\\rho\_\{t\+1\}^\{X\}\-\\rho\_\{t\}^\{X\}\),and, under the half\-step column indexing,

‖y~t\+1−yt−1‖22≤CY​\(ρt\+1Y−ρtY\),\\left\\lVert\\tilde\{y\}\_\{t\+1\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\\leq C\_\{Y\}\(\\rho\_\{t\+1\}^\{Y\}\-\\rho\_\{t\}^\{Y\}\),for fixed finite constantsCX,CYC\_\{X\},C\_\{Y\}\. The right\-hand sides are summable because the bounded monotone norm increments telescope\. Inserting these updated strategies between consecutive played strategies therefore gives

∑t‖xt−xt−1‖22<∞,∑t‖yt−yt−1‖22<∞\.\\sum\_\{t\}\\left\\lVert x\_\{t\}\-x\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}<\\infty,\\qquad\\sum\_\{t\}\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}<\\infty\.
We now bound the two current best\-response residuals\. The column strategyyty\_\{t\}is produced using−A⊤​xt\-A^\{\\top\}x\_\{t\}as its prediction, which is also the current column utility\. Lemma[1](https://arxiv.org/html/2610.19016#A1.Thmtheorem1)therefore gives directly

\(D\.16\)maxy′∈ΔkY⁡⟨−A⊤​xt,y′−yt⟩≤2​2​kY​ρt\+1Y​‖yt−y~t\+1‖2\.\\max\_\{y^\{\\prime\}\\in\\Delta\_\{k^\{Y\}\}\}\\left\\langle\-A^\{\\top\}x\_\{t\},y^\{\\prime\}\-y\_\{t\}\\right\\rangle\\leq 2\\sqrt\{2k^\{Y\}\}\\,\\rho\_\{t\+1\}^\{Y\}\\left\\lVert y\_\{t\}\-\\tilde\{y\}\_\{t\+1\}\\right\\rVert\_\{2\}\.Fort≥2t\\geq 2, the row strategyxtx\_\{t\}was produced from the predictionA​yt−1Ay\_\{t\-1\}, whereas its current utility isA​ytAy\_\{t\}\. Adding and subtractingA​yt−1Ay\_\{t\-1\}gives, for everyx′∈ΔkXx^\{\\prime\}\\in\\Delta\_\{k^\{X\}\},

⟨A​yt,x′−xt⟩\\displaystyle\\left\\langle Ay\_\{t\},x^\{\\prime\}\-x\_\{t\}\\right\\rangle=⟨A​yt−1,x′−xt⟩\+⟨A⁡\(yt−yt−1\),x′−xt⟩\\displaystyle=\\left\\langle Ay\_\{t\-1\},x^\{\\prime\}\-x\_\{t\}\\right\\rangle\+\\left\\langle A\(y\_\{t\}\-y\_\{t\-1\}\),x^\{\\prime\}\-x\_\{t\}\\right\\rangle≤2​2​kX​ρtX​‖xt−x~t‖2\+2​‖A‖2​‖yt−yt−1‖2\.\\displaystyle\\leq 2\\sqrt\{2k^\{X\}\}\\,\\rho\_\{t\}^\{X\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\+\\sqrt\{2\}\\left\\lVert A\\right\\rVert\_\{2\}\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}\.Maximizing overx′x^\{\\prime\}yields

maxx′∈ΔkX⁡⟨A​yt,x′−xt⟩\\displaystyle\\max\_\{x^\{\\prime\}\\in\\Delta\_\{k^\{X\}\}\}\\left\\langle Ay\_\{t\},x^\{\\prime\}\-x\_\{t\}\\right\\rangle≤2​2​kX​ρtX​‖xt−x~t‖2\\displaystyle\\leq 2\\sqrt\{2k^\{X\}\}\\,\\rho\_\{t\}^\{X\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}\(D\.17\)\+2​‖A‖2​‖yt−yt−1‖2\.\\displaystyle\\quad\+\\sqrt\{2\}\\left\\lVert A\\right\\rVert\_\{2\}\\left\\lVert y\_\{t\}\-y\_\{t\-1\}\\right\\rVert\_\{2\}\.The factor2\\sqrt\{2\}in the second term is the Euclidean diameter bound‖x′−xt‖2≤2\\left\\lVert x^\{\\prime\}\-x\_\{t\}\\right\\rVert\_\{2\}\\leq\\sqrt\{2\}\.

The norm sequences are bounded, and every prediction displacement and consecutive strategy movement on the right sides of \([D\.16](https://arxiv.org/html/2610.19016#A4.E16)\)–\([D\.17](https://arxiv.org/html/2610.19016#A4.E17)\) is square summable\. Thus both current best\-response residuals are square summable\. The row display is used fort≥2t\\geq 2; its iteration\-11residual is finite and contributes only one bounded term\. Since the matrix\-game gap is the sum of the two residuals,

∑t=1∞Gap⁡\(xt,yt\)2<∞\.\\sum\_\{t=1\}^\{\\infty\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)^\{2\}<\\infty\.ConsequentlyGap⁡\(xt,yt\)→0\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)\\to 0\.

LetK:=∑tGap⁡\(xt,yt\)2<∞K:=\\sum\_\{t\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)^\{2\}<\\infty\. Then

T​min1≤t≤T​Gap⁡\(xt,yt\)2≤K,T\\min\_\{1\\leq t\\leq T\}\\operatorname\{Gap\}\(x\_\{t\},y\_\{t\}\)^\{2\}\\leq K,so the best\-current\-iterate gap is𝒪T\(T−1/2\)\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)\. Finally, if the distance to𝒵⋆\\mathcal\{Z\}^\{\\star\}did not tend to zero, a subsequence would stay a fixed positive distance away\. Compactness would give a convergent subsubsequence, and continuity of the gap would make its limit a zero\-gap point, hence a Nash equilibrium, a contradiction\. Thereforedist⁡\(\(xt,yt\),𝒵⋆\)→0\\operatorname\{dist\}\(\(x\_\{t\},y\_\{t\}\),\\mathcal\{Z\}^\{\\star\}\)\\to 0\. ∎

## Appendix EProofs for Alternating IR\-PCFR\+

### E\.1\.Proof of Theorem[3](https://arxiv.org/html/2610.19016#S8.Thmtheorem3)

###### Proof\.

Let

ht=\(xt,yt−1\),zt=\(xt,yt\)\.h\_\{t\}=\(x\_\{t\},y\_\{t\-1\}\),\\qquad z\_\{t\}=\(x\_\{t\},y\_\{t\}\)\.The proof has the same structure as the alternating matrix\-game proof, but we must track finitely many information sets\. We first make the local norm\-growth recurrences explicit, then sum them with a threshold argument, and finally use the full\-reach equilibrium weights to control the prediction displacement terms\.

For each information setHH, letkH:=\|A⁡\(H\)\|k\_\{H\}:=\|A\(H\)\|\. ForI∈ℐXI\\in\\mathcal\{I\}\_\{X\}andJ∈ℐYJ\\in\\mathcal\{I\}\_\{Y\}, letLIXL\_\{I\}^\{X\}andLJYL\_\{J\}^\{Y\}denote finite Euclidean Lipschitz constants for the respective counterfactual\-utility mapsFIXF\_\{I\}^\{X\}andFJYF\_\{J\}^\{Y\}on the product of behavioral simplices\.

Fort≥2t\\geq 2, the local learner at column\-player information setJJcompares the new half\-step profilehth\_\{t\}withht−1h\_\{t\-1\}, while the local learner at row\-player information setIIcompares the new full profileztz\_\{t\}withzt−1z\_\{t\-1\}\. Therefore \([17](https://arxiv.org/html/2610.19016#S6.E17)\), Lipschitzness, and \([16](https://arxiv.org/html/2610.19016#S6.E16)\) give

\(E\.1\)ρJ,t\+1Y−ρJ,tY\\displaystyle\\rho^\{Y\}\_\{J,t\+1\}\-\\rho^\{Y\}\_\{J,t\}≤\(1\+kJ\)2​\(LJY\)22​ρJ,tY​‖ht−ht−1‖22,\\displaystyle\\leq\\frac\{\(1\+\\sqrt\{k\_\{J\}\}\)^\{2\}\(L\_\{J\}^\{Y\}\)^\{2\}\}\{2\\rho^\{Y\}\_\{J,t\}\}\\left\\lVert h\_\{t\}\-h\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\},\(E\.2\)ρI,t\+1X−ρI,tX\\displaystyle\\rho^\{X\}\_\{I,t\+1\}\-\\rho^\{X\}\_\{I,t\}≤\(1\+kI\)2​\(LIX\)22​ρI,tX​‖zt−zt−1‖22\.\\displaystyle\\leq\\frac\{\(1\+\\sqrt\{k\_\{I\}\}\)^\{2\}\(L\_\{I\}^\{X\}\)^\{2\}\}\{2\\rho^\{X\}\_\{I,t\}\}\\left\\lVert z\_\{t\}\-z\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}\.We now expand the two profile movements over information sets:

‖ht−ht−1‖22\\displaystyle\\left\\lVert h\_\{t\}\-h\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}=∑K∈ℐX‖xK,t−xK,t−1‖22\+∑L∈ℐY‖yL,t−1−yL,t−2‖22,\\displaystyle=\\sum\_\{K\\in\\mathcal\{I\}\_\{X\}\}\\left\\lVert x\_\{K,t\}\-x\_\{K,t\-1\}\\right\\rVert\_\{2\}^\{2\}\+\\sum\_\{L\\in\\mathcal\{I\}\_\{Y\}\}\\left\\lVert y\_\{L,t\-1\}\-y\_\{L,t\-2\}\\right\\rVert\_\{2\}^\{2\},‖zt−zt−1‖22\\displaystyle\\left\\lVert z\_\{t\}\-z\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}=∑K∈ℐX‖xK,t−xK,t−1‖22\+∑L∈ℐY‖yL,t−yL,t−1‖22\.\\displaystyle=\\sum\_\{K\\in\\mathcal\{I\}\_\{X\}\}\\left\\lVert x\_\{K,t\}\-x\_\{K,t\-1\}\\right\\rVert\_\{2\}^\{2\}\+\\sum\_\{L\\in\\mathcal\{I\}\_\{Y\}\}\\left\\lVert y\_\{L,t\}\-y\_\{L,t\-1\}\\right\\rVert\_\{2\}^\{2\}\.The row\-player local utilities are evaluated under full\-profile indexing, while the two column movements differ by one half\-step\.

Define the nonnegative quantities

BK,tX\\displaystyle B^\{X\}\_\{K,t\}:=ρK,tX2​kK​‖xK,t−x~K,t‖22\+\(ρK,tX−ρK,t−1X\),\\displaystyle:=\\frac\{\\rho^\{X\}\_\{K,t\}\}\{2k\_\{K\}\}\\left\\lVert x\_\{K,t\}\-\\tilde\{x\}\_\{K,t\}\\right\\rVert\_\{2\}^\{2\}\+\(\\rho^\{X\}\_\{K,t\}\-\\rho^\{X\}\_\{K,t\-1\}\),BL,tY\\displaystyle B^\{Y\}\_\{L,t\}:=ρL,tY2​kL​‖yL,t−1−y~L,t‖22\+\(ρL,tY−ρL,t−1Y\)\.\\displaystyle:=\\frac\{\\rho^\{Y\}\_\{L,t\}\}\{2k\_\{L\}\}\\left\\lVert y\_\{L,t\-1\}\-\\tilde\{y\}\_\{L,t\}\\right\\rVert\_\{2\}^\{2\}\+\(\\rho^\{Y\}\_\{L,t\}\-\\rho^\{Y\}\_\{L,t\-1\}\)\.Corollary[2](https://arxiv.org/html/2610.19016#S5.Thmtheorem2), with the same algebra as in \([B\.2](https://arxiv.org/html/2610.19016#A2.E2)\), gives

‖xK,t−xK,t−1‖22\\displaystyle\\left\\lVert x\_\{K,t\}\-x\_\{K,t\-1\}\\right\\rVert\_\{2\}^\{2\}≤4​\(1\+kK\)2ρK,tX​BK,tX,\\displaystyle\\leq\\frac\{4\(1\+\\sqrt\{k\_\{K\}\}\)^\{2\}\}\{\\rho^\{X\}\_\{K,t\}\}B^\{X\}\_\{K,t\},‖yL,t−1−yL,t−2‖22\\displaystyle\\left\\lVert y\_\{L,t\-1\}\-y\_\{L,t\-2\}\\right\\rVert\_\{2\}^\{2\}≤4​\(1\+kL\)2ρL,tY​BL,tY,\\displaystyle\\leq\\frac\{4\(1\+\\sqrt\{k\_\{L\}\}\)^\{2\}\}\{\\rho^\{Y\}\_\{L,t\}\}B^\{Y\}\_\{L,t\},‖yL,t−yL,t−1‖22\\displaystyle\\left\\lVert y\_\{L,t\}\-y\_\{L,t\-1\}\\right\\rVert\_\{2\}^\{2\}≤4​\(1\+kL\)2ρL,t\+1Y​BL,t\+1Y\.\\displaystyle\\leq\\frac\{4\(1\+\\sqrt\{k\_\{L\}\}\)^\{2\}\}\{\\rho^\{Y\}\_\{L,t\+1\}\}B^\{Y\}\_\{L,t\+1\}\.
Substituting these information\-set bounds into \([E\.1](https://arxiv.org/html/2610.19016#A5.E1)\)–\([E\.2](https://arxiv.org/html/2610.19016#A5.E2)\) exposes the same two reciprocal norms as in the simultaneous product proof\. To write the coefficients compactly, for any local information setH∈ℐX∪ℐYH\\in\\mathcal\{I\}\_\{X\}\\cup\\mathcal\{I\}\_\{Y\}define

cJ,HY\\displaystyle c^\{Y\}\_\{J,H\}:=2​\(1\+kJ\)2​\(LJY\)2​\(1\+kH\)2,\\displaystyle:=2\(1\+\\sqrt\{k\_\{J\}\}\)^\{2\}\(L\_\{J\}^\{Y\}\)^\{2\}\(1\+\\sqrt\{k\_\{H\}\}\)^\{2\},cI,HX\\displaystyle c^\{X\}\_\{I,H\}:=2​\(1\+kI\)2​\(LIX\)2​\(1\+kH\)2\.\\displaystyle:=2\(1\+\\sqrt\{k\_\{I\}\}\)^\{2\}\(L\_\{I\}^\{X\}\)^\{2\}\(1\+\\sqrt\{k\_\{H\}\}\)^\{2\}\.Then the column recurrence is

\(E\.3\)ρJ,t\+1Y−ρJ,tY\\displaystyle\\rho^\{Y\}\_\{J,t\+1\}\-\\rho^\{Y\}\_\{J,t\}≤∑K∈ℐXcJ,KYρJ,tY​ρK,tX​BK,tX\+∑L∈ℐYcJ,LYρJ,tY​ρL,tY​BL,tY\.\\displaystyle\\leq\\sum\_\{K\\in\\mathcal\{I\}\_\{X\}\}\\frac\{c^\{Y\}\_\{J,K\}\}\{\\rho^\{Y\}\_\{J,t\}\\rho^\{X\}\_\{K,t\}\}B^\{X\}\_\{K,t\}\+\\sum\_\{L\\in\\mathcal\{I\}\_\{Y\}\}\\frac\{c^\{Y\}\_\{J,L\}\}\{\\rho^\{Y\}\_\{J,t\}\\rho^\{Y\}\_\{L,t\}\}B^\{Y\}\_\{L,t\}\.
and the row recurrence is

\(E\.4\)ρI,t\+1X−ρI,tX\\displaystyle\\rho^\{X\}\_\{I,t\+1\}\-\\rho^\{X\}\_\{I,t\}≤∑K∈ℐXcI,KXρI,tX​ρK,tX​BK,tX\+∑L∈ℐYcI,LXρI,tX​ρL,t\+1Y​BL,t\+1Y\.\\displaystyle\\leq\\sum\_\{K\\in\\mathcal\{I\}\_\{X\}\}\\frac\{c^\{X\}\_\{I,K\}\}\{\\rho^\{X\}\_\{I,t\}\\rho^\{X\}\_\{K,t\}\}B^\{X\}\_\{K,t\}\+\\sum\_\{L\\in\\mathcal\{I\}\_\{Y\}\}\\frac\{c^\{X\}\_\{I,L\}\}\{\\rho^\{X\}\_\{I,t\}\\rho^\{Y\}\_\{L,t\+1\}\}B^\{Y\}\_\{L,t\+1\}\.These displays are the local analogue of \([D\.4](https://arxiv.org/html/2610.19016#A4.E4)\)–\([D\.5](https://arxiv.org/html/2610.19016#A4.E5)\)\. The only difference between the two indexings is the indext\+1t\+1on the column quantities entering the row recurrence\.

We next sum these recurrences\. LetMMbe the total number of information sets and fixε∈\(0,1/3\)\\varepsilon\\in\(0,1/3\)\. Every local norm is at least one\. Hence, exactly as in Theorem[2](https://arxiv.org/html/2610.19016#S7.Thmtheorem2), we can choose a finite threshold for each local learner such that, whenever its regret\-vector norm reaches that threshold, every coefficient in its recurrence \([E\.3](https://arxiv.org/html/2610.19016#A5.E3)\) or \([E\.4](https://arxiv.org/html/2610.19016#A5.E4)\) is at mostε/M\\varepsilon/M\. Bounded counterfactual utilities give a round\-independent one\-step bound on local norm growth, so the first crossing of each threshold has only a finite overshoot\. Put all below\-threshold growth and all crossing overshoots into a finite constantC~ε\\widetilde\{C\}\_\{\\varepsilon\}\.

Let

ST:=∑I∈ℐXρI,T\+1X\+∑J∈ℐYρJ,T\+1YS\_\{T\}:=\\sum\_\{I\\in\\mathcal\{I\}\_\{X\}\}\\rho^\{X\}\_\{I,T\+1\}\+\\sum\_\{J\\in\\mathcal\{I\}\_\{Y\}\}\\rho^\{Y\}\_\{J,T\+1\}and

𝒮T:=∑t,IρI,tX2​kI​‖xI,t−x~I,t‖22\+∑t,JρJ,tY2​kJ​‖yJ,t−1−y~J,t‖22\.\\mathcal\{S\}\_\{T\}:=\\sum\_\{t,I\}\\frac\{\\rho^\{X\}\_\{I,t\}\}\{2k\_\{I\}\}\\left\\lVert x\_\{I,t\}\-\\tilde\{x\}\_\{I,t\}\\right\\rVert\_\{2\}^\{2\}\+\\sum\_\{t,J\}\\frac\{\\rho^\{Y\}\_\{J,t\}\}\{2k\_\{J\}\}\\left\\lVert y\_\{J,t\-1\}\-\\tilde\{y\}\_\{J,t\}\\right\\rVert\_\{2\}^\{2\}\.At a fixed round, summing the column recurrences above threshold makes the total coefficient on eachBK,tXB^\{X\}\_\{K,t\}andBL,tYB^\{Y\}\_\{L,t\}at mostε\\varepsilon\. Summing the above\-threshold row recurrences likewise makes the total coefficient on eachBK,tXB^\{X\}\_\{K,t\}and shiftedBL,t\+1YB^\{Y\}\_\{L,t\+1\}at mostε\\varepsilon\. Consequently

ST≤C~ε\+ε​∑t\[2​∑KBK,tX\+∑LBL,tY\+∑LBL,t\+1Y\]\.\\displaystyle S\_\{T\}\\leq\\widetilde\{C\}\_\{\\varepsilon\}\+\\varepsilon\\sum\_\{t\}\\left\[2\\sum\_\{K\}B^\{X\}\_\{K,t\}\+\\sum\_\{L\}B^\{Y\}\_\{L,t\}\+\\sum\_\{L\}B^\{Y\}\_\{L,t\+1\}\\right\]\.The row quantities appear twice because they enter both the half\-step and full\-profile movements\. The column quantities appear once under each of the two shifted indexings\.

Expanding these quantities, the stability parts contribute at most2​ε​𝒮T2\\varepsilon\\mathcal\{S\}\_\{T\}, apart from the single terminal column stability term created by reindexingBL,t\+1YB^\{Y\}\_\{L,t\+1\}\. The norm\-increment parts telescope: the row increments appear twice and the column increments appear twice\. The terminal column prediction displacement term is at most the corresponding terminal norm because

ρL,T\+1Y2​kL​‖yL,T−y~L,T\+1‖22≤ρL,T\+1Y,\\frac\{\\rho^\{Y\}\_\{L,T\+1\}\}\{2k\_\{L\}\}\\left\\lVert y\_\{L,T\}\-\\tilde\{y\}\_\{L,T\+1\}\\right\\rVert\_\{2\}^\{2\}\\leq\\rho^\{Y\}\_\{L,T\+1\},using the squared simplex diameter bound22\. Thus all telescoping and endpoint terms together are at most3​ε​ST3\\varepsilon S\_\{T\}, after absorbing finite initial boundary terms intoC~ε\\widetilde\{C\}\_\{\\varepsilon\}\. We obtain

ST≤C~ε\+2​ε​𝒮T\+3​ε​ST\.S\_\{T\}\\leq\\widetilde\{C\}\_\{\\varepsilon\}\+2\\varepsilon\\mathcal\{S\}\_\{T\}\+3\\varepsilon S\_\{T\}\.Sinceε<1/3\\varepsilon<1/3, moving the last term to the left gives

\(E\.5\)ST≤C~ε1−3​ε\+2​ε1−3​ε​𝒮T\.S\_\{T\}\\leq\\frac\{\\widetilde\{C\}\_\{\\varepsilon\}\}\{1\-3\\varepsilon\}\+\\frac\{2\\varepsilon\}\{1\-3\\varepsilon\}\\,\\mathcal\{S\}\_\{T\}\.Thus \([E\.5](https://arxiv.org/html/2610.19016#A5.E5)\) gives the required relation between terminal norm growth and the accumulated prediction displacement terms\. The denominator1−3​ε1\-3\\varepsilonand numerator2​ε2\\varepsilonarise from the staggered prediction displacement and terminal\-norm contributions above\.

We now control𝒮T\\mathcal\{S\}\_\{T\}using the equilibrium weights supplied by the full\-reach condition\. First, apply the pointwise CFR decomposition atht=\(xt,yt−1\)h\_\{t\}=\(x\_\{t\},y\_\{t\-1\}\)withyty\_\{t\}as the column comparator\. Since the comparator inducesqtYq\_\{t\}^\{Y\}, this gives

U⁡\(qtX,qt−1Y\)−U⁡\(qtX,qtY\)=∑JqtY​\[pJ\]​⟨FJY​\(ht\),yJ,t−yJ,t−1⟩\.U\(q\_\{t\}^\{X\},q\_\{t\-1\}^\{Y\}\)\-U\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)=\\sum\_\{J\}q\_\{t\}^\{Y\}\[p\_\{J\}\]\\left\\langle F\_\{J\}^\{Y\}\(h\_\{t\}\),y\_\{J,t\}\-y\_\{J,t\-1\}\\right\\rangle\.
Lemma[1](https://arxiv.org/html/2610.19016#A4.Thmtheorem1)gives, for each column information setJJ,

⟨FJY​\(ht\),yJ,t−yJ,t−1⟩≥−\(ρJ,t\+1Y−ρJ,tY\)\.\\left\\langle F\_\{J\}^\{Y\}\(h\_\{t\}\),y\_\{J,t\}\-y\_\{J,t\-1\}\\right\\rangle\\geq\-\(\\rho^\{Y\}\_\{J,t\+1\}\-\\rho^\{Y\}\_\{J,t\}\)\.Since0≤qtY​\[pJ\]≤10\\leq q\_\{t\}^\{Y\}\[p\_\{J\}\]\\leq 1and every norm increment is nonnegative, weighting and summing preserves the lower bound

\(E\.6\)U\(qtX,qt−1Y\)−U\(qtX,qtY\)≥−∑J\(ρJ,t\+1Y−ρJ,tY\)\.U\(q\_\{t\}^\{X\},q\_\{t\-1\}^\{Y\}\)\-U\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\\geq\-\\sum\_\{J\}\(\\rho^\{Y\}\_\{J,t\+1\}\-\\rho^\{Y\}\_\{J,t\}\)\.
Next fix a full\-reach equilibrium\(qX,⋆,qY,⋆\)\(q^\{X,\\star\},q^\{Y,\\star\}\)\. Applying the pointwise CFR decomposition with the equilibrium strategies as comparators gives the row deviation term atztz\_\{t\}and the column deviation term athth\_\{t\}\. Their sum is

∑IqX,⋆​\[pI\]​⟨FIX​\(zt\),xI⋆−xI,t⟩\\displaystyle\\sum\_\{I\}q^\{X,\\star\}\[p\_\{I\}\]\\left\\langle F\_\{I\}^\{X\}\(z\_\{t\}\),x\_\{I\}^\{\\star\}\-x\_\{I,t\}\\right\\rangle\+∑JqY,⋆\[pJ\]⟨FJY\(ht\),yJ⋆−yJ,t−1⟩\\displaystyle\\quad\+\\sum\_\{J\}q^\{Y,\\star\}\[p\_\{J\}\]\\left\\langle F\_\{J\}^\{Y\}\(h\_\{t\}\),y\_\{J\}^\{\\star\}\-y\_\{J,t\-1\}\\right\\rangle=\[U⁡\(qX,⋆,qtY\)−U⁡\(qtX,qtY\)\]\\displaystyle=\\bigl\[U\(q^\{X,\\star\},q\_\{t\}^\{Y\}\)\-U\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\\bigr\]\+\[U⁡\(qtX,qt−1Y\)−U⁡\(qtX,qY,⋆\)\]\.\\displaystyle\\quad\+\\bigl\[U\(q\_\{t\}^\{X\},q\_\{t\-1\}^\{Y\}\)\-U\(q\_\{t\}^\{X\},q^\{Y,\\star\}\)\\bigr\]\.Because\(qX,⋆,qY,⋆\)\(q^\{X,\\star\},q^\{Y,\\star\}\)is a saddle point,U⁡\(qX,⋆,qtY\)≥U⁡\(qtX,qY,⋆\)U\(q^\{X,\\star\},q\_\{t\}^\{Y\}\)\\geq U\(q\_\{t\}^\{X\},q^\{Y,\\star\}\)\. Dropping their nonnegative difference yields

∑IqX,⋆​\[pI\]​⟨FIX​\(zt\),xI⋆−xI,t⟩\\displaystyle\\sum\_\{I\}q^\{X,\\star\}\[p\_\{I\}\]\\left\\langle F\_\{I\}^\{X\}\(z\_\{t\}\),x\_\{I\}^\{\\star\}\-x\_\{I,t\}\\right\\rangle\+∑JqY,⋆\[pJ\]⟨FJY\(ht\),yJ⋆−yJ,t−1⟩\\displaystyle\\quad\+\\sum\_\{J\}q^\{Y,\\star\}\[p\_\{J\}\]\\left\\langle F\_\{J\}^\{Y\}\(h\_\{t\}\),y\_\{J\}^\{\\star\}\-y\_\{J,t\-1\}\\right\\rangle\(E\.7\)≥U⁡\(qtX,qt−1Y\)−U⁡\(qtX,qtY\)\.\\displaystyle\\quad\\geq U\(q\_\{t\}^\{X\},q\_\{t\-1\}^\{Y\}\)\-U\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\.To compare the lower and upper estimates on exactly the same quantity, define

WT:=∑t=1T\[\\displaystyle W\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\Bigg\[∑IqX,⋆​\[pI\]​⟨FIX​\(zt\),xI⋆−xI,t⟩\\displaystyle\\sum\_\{I\}q^\{X,\\star\}\[p\_\{I\}\]\\left\\langle F\_\{I\}^\{X\}\(z\_\{t\}\),x\_\{I\}^\{\\star\}\-x\_\{I,t\}\\right\\rangle\+∑JqY,⋆\[pJ\]⟨FJY\(ht\),yJ⋆−yJ,t−1⟩\]\.\\displaystyle\+\\sum\_\{J\}q^\{Y,\\star\}\[p\_\{J\}\]\\left\\langle F\_\{J\}^\{Y\}\(h\_\{t\}\),y\_\{J\}^\{\\star\}\-y\_\{J,t\-1\}\\right\\rangle\\Bigg\]\.Combining \([E\.7](https://arxiv.org/html/2610.19016#A5.E7)\) with \([E\.6](https://arxiv.org/html/2610.19016#A5.E6)\) and summing overttgives

WT\\displaystyle W\_\{T\}≥−∑t=1T∑J\(ρJ,t\+1Y−ρJ,tY\)\\displaystyle\\geq\-\\sum\_\{t=1\}^\{T\}\\sum\_\{J\}\(\\rho^\{Y\}\_\{J,t\+1\}\-\\rho^\{Y\}\_\{J,t\}\)=−∑J\(ρJ,T\+1Y−ρJ,1Y\)\\displaystyle=\-\\sum\_\{J\}\(\\rho^\{Y\}\_\{J,T\+1\}\-\\rho^\{Y\}\_\{J,1\}\)≥−∑JρYJ,T\+1\.\\displaystyle\\geq\-\\sum\_\{J\}\\rho^\{Y\}\_\{J,T\+1\}\.The second line is the telescoping step; the last inequality usesρJ,1Y≥0\\rho^\{Y\}\_\{J,1\}\\geq 0\.

On the other hand, each fixed\-equilibrium local comparator sum appearing inWTW\_\{T\}is at most the corresponding local external regret\. Applying Fact[1](https://arxiv.org/html/2610.19016#Thmfact1)to those row external regrets and half\-step column external regrets gives the upper bound

WT\\displaystyle W\_\{T\}≤∑IqX,⋆​\[pI\]​ρI,T\+1X\\displaystyle\\leq\\sum\_\{I\}q^\{X,\\star\}\[p\_\{I\}\]\\rho^\{X\}\_\{I,T\+1\}\+∑JqY,⋆\[pJ\]ρJ,T\+1Y\\displaystyle\\quad\+\\sum\_\{J\}q^\{Y,\\star\}\[p\_\{J\}\]\\rho^\{Y\}\_\{J,T\+1\}−∑t,IqX,⋆\[pI\]ρI,tX2​kI‖xI,t−x~I,t‖22\\displaystyle\\quad\-\\sum\_\{t,I\}q^\{X,\\star\}\[p\_\{I\}\]\\frac\{\\rho^\{X\}\_\{I,t\}\}\{2k\_\{I\}\}\\left\\lVert x\_\{I,t\}\-\\tilde\{x\}\_\{I,t\}\\right\\rVert\_\{2\}^\{2\}−∑t,JqY,⋆\[pJ\]ρJ,tY2​kJ‖yJ,t−1−y~J,t‖22\.\\displaystyle\\quad\-\\sum\_\{t,J\}q^\{Y,\\star\}\[p\_\{J\}\]\\frac\{\\rho^\{Y\}\_\{J,t\}\}\{2k\_\{J\}\}\\left\\lVert y\_\{J,t\-1\}\-\\tilde\{y\}\_\{J,t\}\\right\\rVert\_\{2\}^\{2\}\.Putting the lower and upper bounds on the same quantity together and moving the prediction displacement terms to the left gives

∑t,IqX,⋆​\[pI\]​ρI,tX2​kI​‖xI,t−x~I,t‖22\+∑t,JqY,⋆​\[pJ\]​ρJ,tY2​kJ​‖yJ,t−1−y~J,t‖22\\displaystyle\\sum\_\{t,I\}q^\{X,\\star\}\[p\_\{I\}\]\\frac\{\\rho^\{X\}\_\{I,t\}\}\{2k\_\{I\}\}\\left\\lVert x\_\{I,t\}\-\\tilde\{x\}\_\{I,t\}\\right\\rVert\_\{2\}^\{2\}\+\\sum\_\{t,J\}q^\{Y,\\star\}\[p\_\{J\}\]\\frac\{\\rho^\{Y\}\_\{J,t\}\}\{2k\_\{J\}\}\\left\\lVert y\_\{J,t\-1\}\-\\tilde\{y\}\_\{J,t\}\\right\\rVert\_\{2\}^\{2\}\(E\.8\)≤∑IqX,⋆​\[pI\]​ρI,T\+1X\+∑JqY,⋆​\[pJ\]​ρJ,T\+1Y\+∑JρJ,T\+1Y\.\\displaystyle\\quad\\leq\\sum\_\{I\}q^\{X,\\star\}\[p\_\{I\}\]\\rho^\{X\}\_\{I,T\+1\}\+\\sum\_\{J\}q^\{Y,\\star\}\[p\_\{J\}\]\\rho^\{Y\}\_\{J,T\+1\}\+\\sum\_\{J\}\\rho^\{Y\}\_\{J,T\+1\}\.Under the full\-reach condition, every parent\-sequence weight is strictly positive\. Letwmin\>0w\_\{\\min\}\>0andwmaxw\_\{\\max\}be their minimum and maximum\. The left side of \([E\.8](https://arxiv.org/html/2610.19016#A5.E8)\) is at leastwmin​𝒮Tw\_\{\\min\}\\mathcal\{S\}\_\{T\}, while the right side is at most\(wmax\+1\)​ST\(w\_\{\\max\}\+1\)S\_\{T\}\. Hence

𝒮T≤Kw​ST,Kw:=wmax\+1wmin\.\\mathcal\{S\}\_\{T\}\\leq K\_\{w\}S\_\{T\},\\qquad K\_\{w\}:=\\frac\{w\_\{\\max\}\+1\}\{w\_\{\\min\}\}\.
Substitute this into \([E\.5](https://arxiv.org/html/2610.19016#A5.E5)\):

ST≤C~ε\+2​ε​Kw1−3​ε​ST\.S\_\{T\}\\leq\\widetilde\{C\}\_\{\\varepsilon\}\+\\frac\{2\\varepsilon K\_\{w\}\}\{1\-3\\varepsilon\}S\_\{T\}\.Chooseε<1/\(3\+2​Kw\)\\varepsilon<1/\(3\+2K\_\{w\}\)\. This inequality is exactly the condition2​ε​Kw<1−3​ε2\\varepsilon K\_\{w\}<1\-3\\varepsilon, so the coefficient ofSTS\_\{T\}on the right is strictly less than one\. Moving it to the left proves thatSTS\_\{T\}is bounded independently ofTT\.

Dropping the negative prediction displacement terms in Fact[1](https://arxiv.org/html/2610.19016#Thmfact1)now gives𝒪T​\(1\)\\mathcal\{O\}\_\{T\}\(1\)local external regret at all row information sets and𝒪T​\(1\)\\mathcal\{O\}\_\{T\}\(1\)half\-step local external regret at all column information sets\. The standard CFR decomposition transfers the row bounds directly to full\-profile sequence\-form external regret\.

For the column player, the half\-step and full\-profile sequence\-form external regrets differ only by the realized payoff shift

DT:=∑t=1T\[U⁡\(qtX,qt−1Y\)−U⁡\(qtX,qtY\)\]\.D\_\{T\}:=\\sum\_\{t=1\}^\{T\}\\bigl\[U\(q\_\{t\}^\{X\},q\_\{t\-1\}^\{Y\}\)\-U\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\\bigr\]\.Summing \([E\.6](https://arxiv.org/html/2610.19016#A5.E6)\) gives

DT≥−∑JρYJ,T\+1≥−CY,D\_\{T\}\\geq\-\\sum\_\{J\}\\rho^\{Y\}\_\{J,T\+1\}\\geq\-C\_\{Y\},whereCY<∞C\_\{Y\}<\\inftyis independent ofTT, since the preceding bound onSTS\_\{T\}bounds the finite sum of column regret\-vector norms\. Thus−DT≤CY\-D\_\{T\}\\leq C\_\{Y\}\. For every fixed column comparator, replacingqt−1Yq\_\{t\-1\}^\{Y\}byqtYq\_\{t\}^\{Y\}in the realized profile subtracts exactlyDTD\_\{T\}from the cumulative regret expression\. Because this correction is independent of the comparator, the same identity holds after maximizing over comparators\. Hence the full\-profile column sequence\-form external regret is at most the half\-step sequence\-form external regret plusCYC\_\{Y\}, and is therefore𝒪T​\(1\)\\mathcal\{O\}\_\{T\}\(1\)\. Both players thus have𝒪T​\(1\)\\mathcal\{O\}\_\{T\}\(1\)full\-profile sequence\-form external regret\. The exact sequence\-form zero\-sum regret\-to\-gap identity then yieldsGap𝒬⁡\(q¯TX,q¯TY\)=𝒪T​\(T−1\)\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(\\bar\{q\}\_\{T\}^\{X\},\\bar\{q\}\_\{T\}^\{Y\}\)=\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)\. ∎

### E\.2\.Proof of Theorem[4](https://arxiv.org/html/2610.19016#S8.Thmtheorem4)

###### Proof\.

The preceding proof gives a bound onSTS\_\{T\}independent ofTTand the weighted stability estimate \([E\.8](https://arxiv.org/html/2610.19016#A5.E8)\)\. Because the positive parent\-sequence weights are bounded below bywmin\>0w\_\{\\min\}\>0, the unweighted stability sum𝒮T\\mathcal\{S\}\_\{T\}is finite over the infinite horizon\. Every local norm is at least one, so every local prediction displacement appearing in𝒮T\\mathcal\{S\}\_\{T\}is square summable\.

Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)converts bounded monotone local norm growth into square summability of the corresponding post\-update displacements, exactly as in the simultaneous product proof\. Since there are finitely many information sets, summing these bounds over information sets preserves finiteness\. In particular,

\(E\.9\)∑t‖zt−zt−1‖22<∞,∑t‖zt−ht‖22<∞\.\\sum\_\{t\}\\left\\lVert z\_\{t\}\-z\_\{t\-1\}\\right\\rVert\_\{2\}^\{2\}<\\infty,\\qquad\\sum\_\{t\}\\left\\lVert z\_\{t\}\-h\_\{t\}\\right\\rVert\_\{2\}^\{2\}<\\infty\.The first sum records full\-profile movement\. The second records only the column movement, sincezt−ht=\(0,yt−yt−1\)z\_\{t\}\-h\_\{t\}=\(0,y\_\{t\}\-y\_\{t\-1\}\)\.

We next bound the local current residuals\. Fort≥2t\\geq 2, consider first a row information setII\. ThenxI,tx\_\{I,t\}was generated using the previous full\-profile predictionFIX​\(zt−1\)F\_\{I\}^\{X\}\(z\_\{t\-1\}\), while the current utility isFIX​\(zt\)F\_\{I\}^\{X\}\(z\_\{t\}\)\. Adding and subtracting that prediction, applying Lemma[1](https://arxiv.org/html/2610.19016#A1.Thmtheorem1)to the predicted term, and applying Lipschitzness plus the simplex diameter bound to the utility difference gives

\(E\.10\)RI,tX\\displaystyle R^\{X\}\_\{I,t\}≤2​2​kI​ρI,tX​‖xI,t−x~I,t‖2\+2​LIX​‖zt−zt−1‖2\.\\displaystyle\\leq 2\\sqrt\{2k\_\{I\}\}\\,\\rho^\{X\}\_\{I,t\}\\left\\lVert x\_\{I,t\}\-\\tilde\{x\}\_\{I,t\}\\right\\rVert\_\{2\}\+\\sqrt\{2\}L\_\{I\}^\{X\}\\left\\lVert z\_\{t\}\-z\_\{t\-1\}\\right\\rVert\_\{2\}\.For a column information setJJ,yJ,ty\_\{J,t\}is generated from the half\-step predictionFJY​\(ht\)F\_\{J\}^\{Y\}\(h\_\{t\}\)\. Its current full\-profile utility isFJY​\(zt\)F\_\{J\}^\{Y\}\(z\_\{t\}\)\. The same add\-and\-subtract argument therefore gives

\(E\.11\)RJ,tY\\displaystyle R^\{Y\}\_\{J,t\}≤2​2​kJ​ρJ,t\+1Y​‖yJ,t−y~J,t\+1‖2\+2​LJY​‖zt−ht‖2\.\\displaystyle\\leq 2\\sqrt\{2k\_\{J\}\}\\,\\rho^\{Y\}\_\{J,t\+1\}\\left\\lVert y\_\{J,t\}\-\\tilde\{y\}\_\{J,t\+1\}\\right\\rVert\_\{2\}\+\\sqrt\{2\}L\_\{J\}^\{Y\}\\left\\lVert z\_\{t\}\-h\_\{t\}\\right\\rVert\_\{2\}\.The first term in each display is a residual term for the predicted utility; the second is the Lipschitz error between the profile used in the prediction and the current full profile\.

All norm sequences in the displays remain bounded over time, and every prediction displacement and profile movement term there is square summable by the preceding paragraph and \([E\.9](https://arxiv.org/html/2610.19016#A5.E9)\)\. Thus every local residual is square summable; the finitely many initial boundary residuals are finite\.

LetRtR\_\{t\}be the sum of all row and column local residuals\. Because there are finitely many information sets, Cauchy–Schwarz gives∑tRt2<∞\\sum\_\{t\}R\_\{t\}^\{2\}<\\infty\. The pointwise CFR decomposition, exactly as in the proof of Theorem[4](https://arxiv.org/html/2610.19016#S7.Thmtheorem4), gives

0≤Gap𝒬⁡\(qtX,qtY\)≤Rt\.0\\leq\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)\\leq R\_\{t\}\.Therefore

∑t=1∞Gap𝒬⁡\(qtX,qtY\)2<∞,\\sum\_\{t=1\}^\{\\infty\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)^\{2\}<\\infty,which proves \([34](https://arxiv.org/html/2610.19016#S8.E34)\)\. In particular the current gap tends to zero\.

IfK:=∑tGap𝒬⁡\(qtX,qtY\)2<∞K:=\\sum\_\{t\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)^\{2\}<\\infty, then

T​min1≤t≤T​Gap𝒬​\(qtX,qtY\)2≤K,T\\min\_\{1\\leq t\\leq T\}\\operatorname\{Gap\}\_\{\\mathcal\{Q\}\}\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\)^\{2\}\\leq K,so the best\-current\-iterate gap is𝒪T\(T−1/2\)\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)\. Finally, if the distance to𝒵𝒬⋆\\mathcal\{Z\}\_\{\\mathcal\{Q\}\}^\{\\star\}did not tend to zero, compactness of the realization\-plan polytopes would give a convergent subsequence staying a fixed positive distance away\. Continuity of the gap would make its limit a zero\-gap point, hence a sequence\-form Nash equilibrium, a contradiction\. Thusdist⁡\(\(qtX,qtY\),𝒵𝒬⋆\)→0\\operatorname\{dist\}\(\(q\_\{t\}^\{X\},q\_\{t\}^\{Y\}\),\\mathcal\{Z\}\_\{\\mathcal\{Q\}\}^\{\\star\}\)\\to 0\. ∎

## Appendix FStandard Zero Initialization

The main text usesr~1=k−1/2𝟏\\tilde\{r\}\_\{1\}=k^\{\-1/2\}\\mathbf\{1\}, so thatρ1=1\\rho\_\{1\}=1, only to avoid repeated division\-by\-zero cases\. Here we prove the same conclusions for the standard initializationr~1=0\\tilde\{r\}\_\{1\}=0\. We do not repeat the analysis forρt\>0\\rho\_\{t\}\>0; we treat only rounds withρt=0\\rho\_\{t\}=0and the first transition to positive norm\.

For zero initialization, fix an arbitrary fixed default strategyxo∈Δkx^\{o\}\\in\\Delta\_\{k\}\. Wheneverρt=0\\rho\_\{t\}=0, set

x~t=xt=xo,mt=0,\\tilde\{x\}\_\{t\}=x\_\{t\}=x^\{o\},\\qquad m\_\{t\}=0,and after observingutu\_\{t\}update

\(F\.1\)r~t\+1=\[ut−⟨ut,xo⟩​𝟏\]\+\.\\tilde\{r\}\_\{t\+1\}=\\left\[u\_\{t\}\-\\left\\langle u\_\{t\},x^\{o\}\\right\\rangle\\mathbf\{1\}\\right\]\_\{\+\}\.If this vector is zero, use the same rule on the next round\. Once the stored regret vector has positive norm, use the standard IR\-PRM\+ update and the previous\-utility predictor from the main text\.

### F\.1\.Handling zero regret vectors

The following observations are the only additional ingredients needed for zero initialization\.

###### Lemma 0 \(Zero\-initialization boundary cases\)\.

Consider a zero\-initializedkk\-action learner with the rule above\. Then:

1. \(1\)Ifρt=ρt\+1=0\\rho\_\{t\}=\\rho\_\{t\+1\}=0, the played strategy does not move and the current utility has zero best\-response residual atxox^\{o\}\.
2. \(2\)Ifρt=0<ρt\+1\\rho\_\{t\}=0<\\rho\_\{t\+1\}, the post\-update displacement bound \([9](https://arxiv.org/html/2610.19016#S5.E9)\) still holds on this first positive transition\.
3. \(3\)Ifτ:=min⁡\{t:ρt\>0\}<∞\\tau:=\\min\\\{t:\\rho\_\{t\}\>0\\\}<\\inftyandβ:=ρτ\\beta:=\\rho\_\{\\tau\}, thenβ\>0\\beta\>0andρt≥β\\rho\_\{t\}\\geq\\betafor everyt≥τt\\geq\\tau\.
4. \(4\)Ifρt=0\\rho\_\{t\}=0, the residual bound for the predicted utility \([A\.1](https://arxiv.org/html/2610.19016#A1.E1)\) holds trivially becausemt=0m\_\{t\}=0\.

###### Proof\.

If \([F\.1](https://arxiv.org/html/2610.19016#A6.E1)\) is zero, thenut,a≤⟨ut,xo⟩u\_\{t,a\}\\leq\\left\\langle u\_\{t\},x^\{o\}\\right\\ranglefor every actionaa\. Since⟨ut,xo⟩\\left\\langle u\_\{t\},x^\{o\}\\right\\rangleis a convex combination of the coordinates ofutu\_\{t\}, this forces

maxa⁡ut,a=⟨ut,xo⟩,\\max\_\{a\}u\_\{t,a\}=\\left\\langle u\_\{t\},x^\{o\}\\right\\rangle,so the best\-response residual at the fixed default strategy is zero\. The rule also keepsxt\+1=xt=xox\_\{t\+1\}=x\_\{t\}=x^\{o\}as long asr~t=0\\tilde\{r\}\_\{t\}=0\.

For the first positive transition, the right side of \([9](https://arxiv.org/html/2610.19016#S5.E9)\) becomes2​\(1\+k\)22\(1\+\\sqrt\{k\}\)^\{2\}, becauseρt=0\\rho\_\{t\}=0\. The squared Euclidean diameter ofΔk\\Delta\_\{k\}is at most22, so the same inequality covers that transition\. Afterρt\\rho\_\{t\}becomes positive, the monotonicity argument in the proof of Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)applies without change, givingρt≥ρτ=β\>0\\rho\_\{t\}\\geq\\rho\_\{\\tau\}=\\beta\>0thereafter\. Finally,mt=0m\_\{t\}=0wheneverρt=0\\rho\_\{t\}=0, so the left side of \([A\.1](https://arxiv.org/html/2610.19016#A1.E1)\) is zero\. ∎

The main proofs used the numerical lower boundρt≥1\\rho\_\{t\}\\geq 1only to keep reciprocal norm factors uniformly bounded before the threshold is reached and to remove the weightsρt\\rho\_\{t\}from square\-summability statements\. On any suffix on whichρt≥β\>0\\rho\_\{t\}\\geq\\beta\>0, the same arguments hold with11replaced byβ\\beta; only the constants change\. Thus the finitely many initial rounds withρt=0\\rho\_\{t\}=0can be separated from the argument forρt\>0\\rho\_\{t\}\>0\.

### F\.2\.Simultaneous matrix games

Consider first the two\-player simultaneous dynamics\. If one player’s regret vector remains zero forever, its strategy is fixed\. After one observation, the other player’s previous\-utility prediction is exact against that fixed strategy, sout−mt=0u\_\{t\}\-m\_\{t\}=0and hencegt=0g\_\{t\}=0and its regret\-vector norm stops growing\. Hence both regrets are bounded directly\.

Otherwise both regret\-vector norms become positive\. LetτX,τY\\tau\_\{X\},\\tau\_\{Y\}be their first positive times and let

βX:=ρτXX\>0,βY:=ρτYY\>0\.\\beta\_\{X\}:=\\rho^\{X\}\_\{\\tau\_\{X\}\}\>0,\\qquad\\beta\_\{Y\}:=\\rho^\{Y\}\_\{\\tau\_\{Y\}\}\>0\.Afterτ=max⁡\{τX,τY\}\\tau=\\max\\\{\\tau\_\{X\},\\tau\_\{Y\}\\\}, monotonicity givesρtX≥βX\\rho\_\{t\}^\{X\}\\geq\\beta\_\{X\}andρtY≥βY\\rho\_\{t\}^\{Y\}\\geq\\beta\_\{Y\}\. The coupled recurrences \([14](https://arxiv.org/html/2610.19016#S6.E14)\)–\([15](https://arxiv.org/html/2610.19016#S6.E15)\) therefore apply on this suffix\. Before the product threshold is reached, the fixed lower boundsβX,βY\\beta\_\{X\},\\beta\_\{Y\}replace the unit lower bounds used in the proof of Theorem[2](https://arxiv.org/html/2610.19016#S6.Thmtheorem2); after the threshold is crossed, the proof is identical\. The finitely many preceding rounds contribute only a finite additive constant\. This proves the same𝒪T​\(1\)\\mathcal\{O\}\_\{T\}\(1\)regret and𝒪T​\(T−1\)\\mathcal\{O\}\_\{T\}\(T^\{\-1\}\)average\-gap guarantees\.

For current\-profile convergence, Lemma[1](https://arxiv.org/html/2610.19016#A6.Thmtheorem1)handles the rounds with zero norm and the first transitions to positive norm\. Afterτ\\tau, the factorsρtX\\rho\_\{t\}^\{X\}andρtY\\rho\_\{t\}^\{Y\}in the stability sums can be removed usingβX\\beta\_\{X\}andβY\\beta\_\{Y\}instead of one\. The remainder of the proof of Theorem[3](https://arxiv.org/html/2610.19016#S6.Thmtheorem3)is unchanged\. Hence the square\-summable current gap, convergence in distance to the Nash equilibrium set, and the𝒪T\(T−1/2\)\\mathcal\{O\}\_\{T\}\(T^\{\-1/2\}\)best\-current\-iterate bound all continue to hold\.

### F\.3\.Products of simplices and simultaneous IR\-PCFR\+

For the product\-space argument, let𝒜\\mathcal\{A\}be the set of learners whose regret\-vector norms ever become positive\. A learner outside𝒜\\mathcal\{A\}maintains a fixed strategy, and the first part of Lemma[1](https://arxiv.org/html/2610.19016#A6.Thmtheorem1)shows that its instantaneous local regret and current local residual are identically zero\. In particular, its comparator term in the weighted Minty sum is nonpositive, so removing those terms can only increase that sum; the weighted Minty lower bound therefore remains valid on𝒜\\mathcal\{A\}\. For eachj∈𝒜j\\in\\mathcal\{A\}, letβj\>0\\beta\_\{j\}\>0be its first positive norm\. Since there are finitely many learners, after the last transition to positive norm, everyj∈𝒜j\\in\\mathcal\{A\}satisfiesρj,t≥βj\\rho\_\{j,t\}\\geq\\beta\_\{j\}\.

On that suffix, the proof of Lemma[1](https://arxiv.org/html/2610.19016#S7.Thmtheorem1)applies with the sum restricted toj∈𝒜j\\in\\mathcal\{A\}\. In the proof of Theorem[2](https://arxiv.org/html/2610.19016#S7.Thmtheorem2), choose the thresholdΘi\\Theta\_\{i\}so that

ci​jΘi​βj≤wmin4​N​wmax\(j∈𝒜\)\.\\frac\{c\_\{ij\}\}\{\\Theta\_\{i\}\\beta\_\{j\}\}\\leq\\frac\{w\_\{\\min\}\}\{4Nw\_\{\\max\}\}\\qquad\(j\\in\\mathcal\{A\}\)\.This is exactly the threshold argument from the main proof with the unit lower bound replaced by the fixed positive valuesβj\\beta\_\{j\}\. Each learner has at most one transition from zero to positive norm, so all initial zero\-norm rounds and threshold crossings contribute only a finite constant\. Indices outside𝒜\\mathcal\{A\}contribute neither strategy movement nor regret\. The weighted Minty summation then closes exactly as before\. The current\-residual argument is unchanged after replacing the unit lower bounds by theβj\\beta\_\{j\}values\. Consequently Theorem[2](https://arxiv.org/html/2610.19016#S7.Thmtheorem2)and Corollary[1](https://arxiv.org/html/2610.19016#A2.Thmtheorem1)also hold under zero initialization\. Applying the same CFR decomposition proves the zero\-initialized versions of Theorems[3](https://arxiv.org/html/2610.19016#S7.Thmtheorem3)and[4](https://arxiv.org/html/2610.19016#S7.Thmtheorem4)\.

### F\.4\.Alternating updates

For alternating updates, one additional boundary calculation is needed because the second player uses the newly observed utility as its prediction\. The proof of Lemma[1](https://arxiv.org/html/2610.19016#A4.Thmtheorem1)for positive regret\-vector norm already covers every later iteration\. We verify the first positive transition here\.

Supposeρt=0<ρt\+1\\rho\_\{t\}=0<\\rho\_\{t\+1\}\. Writec=⟨ut,xo⟩c=\\left\\langle u\_\{t\},x^\{o\}\\right\\rangleand

p=\[ut−c​𝟏\]\+,z=p‖p‖1\.p=\[u\_\{t\}\-c\\mathbf\{1\}\]\_\{\+\},\\qquad z=\\frac\{p\}\{\\left\\lVert p\\right\\rVert\_\{1\}\}\.On every coordinate wherepa\>0p\_\{a\}\>0,ut,a−c=pau\_\{t,a\}\-c=p\_\{a\}, so

⟨ut,z⟩−c=‖p‖22‖p‖1\>0\.\\left\\langle u\_\{t\},z\\right\\rangle\-c=\\frac\{\\left\\lVert p\\right\\rVert\_\{2\}^\{2\}\}\{\\left\\lVert p\\right\\rVert\_\{1\}\}\>0\.Now apply the prediction based onutu\_\{t\}\. Write

r=\[p\+ut−γ​𝟏\]\+,x\+=r‖r‖1,h=r−\(p\+ut−γ​𝟏\)≥0\.r=\[p\+u\_\{t\}\-\\gamma\\mathbf\{1\}\]\_\{\+\},\\qquad x^\{\+\}=\\frac\{r\}\{\\left\\lVert r\\right\\rVert\_\{1\}\},\\qquad h=r\-\(p\+u\_\{t\}\-\\gamma\\mathbf\{1\}\)\\geq 0\.Norm preservation gives‖r‖2=‖p‖2\\left\\lVert r\\right\\rVert\_\{2\}=\\left\\lVert p\\right\\rVert\_\{2\}, andhhhas disjoint support fromrr\. Usingut−γ​𝟏=r−p−hu\_\{t\}\-\\gamma\\mathbf\{1\}=r\-p\-hgives

⟨ut,x\+⟩−γ=‖r‖22−⟨p,r⟩‖r‖1,\\left\\langle u\_\{t\},x^\{\+\}\\right\\rangle\-\\gamma=\\frac\{\\left\\lVert r\\right\\rVert\_\{2\}^\{2\}\-\\left\\langle p,r\\right\\rangle\}\{\\left\\lVert r\\right\\rVert\_\{1\}\},whereas dotting the same identity withz=p/‖p‖1z=p/\\left\\lVert p\\right\\rVert\_\{1\}gives

⟨ut,z⟩−γ=⟨r,p⟩−‖p‖22‖p‖1−⟨h,z⟩\.\\left\\langle u\_\{t\},z\\right\\rangle\-\\gamma=\\frac\{\\left\\langle r,p\\right\\rangle\-\\left\\lVert p\\right\\rVert\_\{2\}^\{2\}\}\{\\left\\lVert p\\right\\rVert\_\{1\}\}\-\\left\\langle h,z\\right\\rangle\.Subtracting and using‖r‖2=‖p‖2\\left\\lVert r\\right\\rVert\_\{2\}=\\left\\lVert p\\right\\rVert\_\{2\}yields

⟨ut,x\+−z⟩\\displaystyle\\left\\langle u\_\{t\},x^\{\+\}\-z\\right\\rangle=\(‖p‖22−⟨p,r⟩\)​\(1‖r‖1\+1‖p‖1\)\+⟨h,z⟩\\displaystyle=\\bigl\(\\left\\lVert p\\right\\rVert\_\{2\}^\{2\}\-\\left\\langle p,r\\right\\rangle\\bigr\)\\left\(\\frac\{1\}\{\\left\\lVert r\\right\\rVert\_\{1\}\}\+\\frac\{1\}\{\\left\\lVert p\\right\\rVert\_\{1\}\}\\right\)\+\\left\\langle h,z\\right\\rangle=‖r−p‖222​\(1‖r‖1\+1‖p‖1\)\+⟨h,z⟩≥0\.\\displaystyle=\\frac\{\\left\\lVert r\-p\\right\\rVert\_\{2\}^\{2\}\}\{2\}\\left\(\\frac\{1\}\{\\left\\lVert r\\right\\rVert\_\{1\}\}\+\\frac\{1\}\{\\left\\lVert p\\right\\rVert\_\{1\}\}\\right\)\+\\left\\langle h,z\\right\\rangle\\geq 0\.Thus this prediction does not reduce payoff relative tozz, and⟨ut,xt\+1−xt⟩≥0\\left\\langle u\_\{t\},x\_\{t\+1\}\-x\_\{t\}\\right\\rangle\\geq 0\. This is stronger than \([D\.1](https://arxiv.org/html/2610.19016#A4.E1)\) on the first positive transition\. Ifρt\+1=0\\rho\_\{t\+1\}=0, the claim is immediate because the played strategy remainsxox^\{o\}\.

Once a matrix\-game player’s or a CFR local learner’s regret\-vector norm first becomes positive, it has a fixed positive lower bound and all subsequent staggered recurrences are exactly those proved in the main alternating analysis\. If such a regret vector never attains positive norm, its strategy does not move and its residual for the utility at which the update withρt=0\\rho\_\{t\}=0is performed is zero\. In alternating CFR, when the local regret vector at a column\-player information setJJis zero, the residual at the full profile differs from its half\-step residual only through the utility change fromhth\_\{t\}toztz\_\{t\}, and is bounded by2​LJY​‖zt−ht‖2\\sqrt\{2\}L\_\{J\}^\{Y\}\\left\\lVert z\_\{t\}\-h\_\{t\}\\right\\rVert\_\{2\}; the latter movement is already square summable in the alternating proof\. Since there are only finitely many matrix\-game players and CFR local learners, for those that eventually become positive the prefixes before their first positive norm, together with the first positive transitions themselves, contribute only finitely many bounded terms; those that remain at zero are handled above\. The proofs of Theorems[1](https://arxiv.org/html/2610.19016#S8.Thmtheorem1),[2](https://arxiv.org/html/2610.19016#S8.Thmtheorem2),[3](https://arxiv.org/html/2610.19016#S8.Thmtheorem3), and[4](https://arxiv.org/html/2610.19016#S8.Thmtheorem4)therefore carry over with constants depending on the first positive regret\-vector norms\.

###### Corollary 0 \(Zero initialization\)\.

All regret, average\-gap, current\-gap, Nash\-set convergence, and best\-current\-iterate conclusions proved in this paper remain valid when IR\-PRM\+ and IR\-PCFR\+ are initialized withr~1=0\\tilde\{r\}\_\{1\}=0and use the rule above forr~t=0\\tilde\{r\}\_\{t\}=0\.

## Appendix GNormalization Underℓ1\\ell\_\{1\}Scaling

###### Lemma 0 \(Normalization bound\)\.

Ifu,v∈ℝ\+ku,v\\in\\mathbb\{R\}\_\{\+\}^\{k\}satisfy‖u‖2=‖v‖2=1\\left\\lVert u\\right\\rVert\_\{2\}=\\left\\lVert v\\right\\rVert\_\{2\}=1, then

\(G\.1\)‖u‖u‖1−v‖v‖1‖2≤\(1\+k\)​‖u−v‖2\.\\left\\lVert\\frac\{u\}\{\\left\\lVert u\\right\\rVert\_\{1\}\}\-\\frac\{v\}\{\\left\\lVert v\\right\\rVert\_\{1\}\}\\right\\rVert\_\{2\}\\leq\(1\+\\sqrt\{k\}\)\\left\\lVert u\-v\\right\\rVert\_\{2\}\.

###### Proof\.

Leta=‖u‖1a=\\left\\lVert u\\right\\rVert\_\{1\}andb=‖v‖1b=\\left\\lVert v\\right\\rVert\_\{1\}\. Adding and subtractingv/av/agives

‖ua−vb‖2\\displaystyle\\left\\lVert\\frac\{u\}\{a\}\-\\frac\{v\}\{b\}\\right\\rVert\_\{2\}≤‖ua−va‖2\+‖va−vb‖2\\displaystyle\\leq\\left\\lVert\\frac\{u\}\{a\}\-\\frac\{v\}\{a\}\\right\\rVert\_\{2\}\+\\left\\lVert\\frac\{v\}\{a\}\-\\frac\{v\}\{b\}\\right\\rVert\_\{2\}=‖u−v‖2a\+\|a−b\|a​b​‖v‖2\.\\displaystyle=\\frac\{\\left\\lVert u\-v\\right\\rVert\_\{2\}\}\{a\}\+\\frac\{\|a\-b\|\}\{ab\}\\left\\lVert v\\right\\rVert\_\{2\}\.Becauseuuandvvare nonnegative Euclidean unit vectors,a,b≥1a,b\\geq 1and‖v‖2=1≤b\\left\\lVert v\\right\\rVert\_\{2\}=1\\leq b\. Therefore

‖ua−vb‖2≤‖u−v‖2\+\|a−b\|\.\\left\\lVert\\frac\{u\}\{a\}\-\\frac\{v\}\{b\}\\right\\rVert\_\{2\}\\leq\\left\\lVert u\-v\\right\\rVert\_\{2\}\+\|a\-b\|\.By the reverse triangle inequality and norm comparison,

\|a−b\|=\|‖u‖1−‖v‖1\|≤‖u−v‖1≤k​‖u−v‖2\.\|a\-b\|=\|\\left\\lVert u\\right\\rVert\_\{1\}\-\\left\\lVert v\\right\\rVert\_\{1\}\|\\leq\\left\\lVert u\-v\\right\\rVert\_\{1\}\\leq\\sqrt\{k\}\\left\\lVert u\-v\\right\\rVert\_\{2\}\.Combining the last two displays proves \([G\.1](https://arxiv.org/html/2610.19016#A7.E1)\)\. ∎

## Appendix HA Two\-Displacement RVU Bound for IR\-PRM\+

The single\-learner analysis of prediction and post\-update displacements also yields a strengthened RVU inequality for IR\-PRM\+\. The existing RVU bound contains a negative term for the prediction displacementxt−x~tx\_\{t\}\-\\tilde\{x\}\_\{t\}\. Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)shows that growth of the regret\-vector norm controls the post\-update displacementx~t\+1−xt\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}, allowing an explicit stability term for this second part of the update as well\. This result is not needed for the main convergence analysis, which works directly with regret\-vector norm growth, but it gives a two\-displacement RVU form that makes both the prediction and post\-update displacements explicit\.

For comparison, Theorem 3\.2 of Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)has the form

\(H\.1\)Reg⁡\(T\)≤\(ρ12\+∑t=1T‖gt‖22\)1/2−12​k​∑t=1Tρt​‖xt−x~t‖22\.\\operatorname\{Reg\}\(T\)\\leq\\left\(\\rho\_\{1\}^\{2\}\+\\sum\_\{t=1\}^\{T\}\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\right\)^\{1/2\}\-\\frac\{1\}\{2k\}\\sum\_\{t=1\}^\{T\}\\rho\_\{t\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\.
###### Theorem 1 \(Two\-displacement IR\-PRM\+ RVU bound\)\.

For akk\-action IR\-PRM\+ learner with any nonnegative initial regret vector,

Reg⁡\(T\)≤\\displaystyle\\operatorname\{Reg\}\(T\)\\leq\{\}2​\(ρ12\+∑t=1T‖gt‖22\)1/2−ρ1\\displaystyle 2\\left\(\\rho\_\{1\}^\{2\}\+\\sum\_\{t=1\}^\{T\}\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\right\)^\{1/2\}\-\\rho\_\{1\}−12​k∑t=1Tρt‖xt−x~t‖22\\displaystyle\-\\frac\{1\}\{2k\}\\sum\_\{t=1\}^\{T\}\\rho\_\{t\}\\left\\lVert x\_\{t\}\-\\tilde\{x\}\_\{t\}\\right\\rVert\_\{2\}^\{2\}\(H\.2\)−12​\(1\+k\)2∑t=1Tρt\+1‖x~t\+1−xt‖22\.\\displaystyle\-\\frac\{1\}\{2\(1\+\\sqrt\{k\}\)^\{2\}\}\\sum\_\{t=1\}^\{T\}\\rho\_\{t\+1\}\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\.

###### Proof\.

Suppose first thatρt\>0\\rho\_\{t\}\>0\. Lemma[1](https://arxiv.org/html/2610.19016#S5.Thmtheorem1)gives

‖x~t\+1−xt‖22≤2​\(1\+k\)2​\(ρt\+1−ρt\)ρt\+1\.\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\frac\{2\(1\+\\sqrt\{k\}\)^\{2\}\(\\rho\_\{t\+1\}\-\\rho\_\{t\}\)\}\{\\rho\_\{t\+1\}\}\.Multiplying byρt\+1/\[2​\(1\+k\)2\]\\rho\_\{t\+1\}/\[2\(1\+\\sqrt\{k\}\)^\{2\}\]yields

\(H\.3\)ρt\+12​\(1\+k\)2​‖x~t\+1−xt‖22≤ρt\+1−ρt\.\\frac\{\\rho\_\{t\+1\}\}\{2\(1\+\\sqrt\{k\}\)^\{2\}\}\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\rho\_\{t\+1\}\-\\rho\_\{t\}\.
The same inequality holds whenρt=0\\rho\_\{t\}=0\. Ifρt\+1=0\\rho\_\{t\+1\}=0, the rule in Appendix[F](https://arxiv.org/html/2610.19016#A6)leaves bothxtx\_\{t\}andx~t\+1\\tilde\{x\}\_\{t\+1\}equal to the fixed strategyxox^\{o\}, so the left\-hand side of \([H\.3](https://arxiv.org/html/2610.19016#A8.E3)\) is zero\. Ifρt=0<ρt\+1\\rho\_\{t\}=0<\\rho\_\{t\+1\}, thenxt,x~t\+1∈Δkx\_\{t\},\\tilde\{x\}\_\{t\+1\}\\in\\Delta\_\{k\}, and hence

‖x~t\+1−xt‖22≤2\.\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq 2\.Therefore

ρt\+12​\(1\+k\)2​‖x~t\+1−xt‖22≤ρt\+1\(1\+k\)2≤ρt\+1=ρt\+1−ρt\.\\frac\{\\rho\_\{t\+1\}\}\{2\(1\+\\sqrt\{k\}\)^\{2\}\}\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\frac\{\\rho\_\{t\+1\}\}\{\(1\+\\sqrt\{k\}\)^\{2\}\}\\leq\\rho\_\{t\+1\}=\\rho\_\{t\+1\}\-\\rho\_\{t\}\.Thus \([H\.3](https://arxiv.org/html/2610.19016#A8.E3)\) holds for every round\. Summing overttgives

\(H\.4\)∑t=1Tρt\+12​\(1\+k\)2​‖x~t\+1−xt‖22≤ρT\+1−ρ1\.\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\+1\}\}\{2\(1\+\\sqrt\{k\}\)^\{2\}\}\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\leq\\rho\_\{T\+1\}\-\\rho\_\{1\}\.
The squared\-norm increment bound for IR\-PRM\+established by Zhang et al\.\([Zhang et al\., 2026](https://arxiv.org/html/2610.19016#bib.bib11)\)gives

ρt\+12−ρt2≤‖gt‖22,\\rho\_\{t\+1\}^\{2\}\-\\rho\_\{t\}^\{2\}\\leq\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\},including whenρt=0\\rho\_\{t\}=0\. Summing yields

\(H\.5\)ρT\+1≤\(ρ12\+∑t=1T‖gt‖22\)1/2\.\\rho\_\{T\+1\}\\leq\\left\(\\rho\_\{1\}^\{2\}\+\\sum\_\{t=1\}^\{T\}\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\right\)^\{1/2\}\.Combining \([H\.4](https://arxiv.org/html/2610.19016#A8.E4)\) and \([H\.5](https://arxiv.org/html/2610.19016#A8.E5)\) gives

∑t=1Tρt\+12​\(1\+k\)2​‖x~t\+1−xt‖22\\displaystyle\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\+1\}\}\{2\(1\+\\sqrt\{k\}\)^\{2\}\}\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\(H\.6\)≤\(ρ12\+∑t=1T‖gt‖22\)1/2−ρ1\.\\displaystyle\\qquad\\leq\\left\(\\rho\_\{1\}^\{2\}\+\\sum\_\{t=1\}^\{T\}\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\right\)^\{1/2\}\-\\rho\_\{1\}\.Equivalently,

\(ρ12\+∑t=1T‖gt‖22\)1/2−ρ1\\displaystyle\\left\(\\rho\_\{1\}^\{2\}\+\\sum\_\{t=1\}^\{T\}\\left\\lVert g\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\right\)^\{1/2\}\-\\rho\_\{1\}\(H\.7\)−∑t=1Tρt\+12​\(1\+k\)2‖x~t\+1−xt‖22≥0\.\\displaystyle\\qquad\-\\sum\_\{t=1\}^\{T\}\\frac\{\\rho\_\{t\+1\}\}\{2\(1\+\\sqrt\{k\}\)^\{2\}\}\\left\\lVert\\tilde\{x\}\_\{t\+1\}\-x\_\{t\}\\right\\rVert\_\{2\}^\{2\}\\geq 0\.Adding this nonnegative quantity to the right\-hand side of \([H\.1](https://arxiv.org/html/2610.19016#A8.E1)\) gives \([H\.2](https://arxiv.org/html/2610.19016#A8.E2)\)\. ∎

## AI\-Assisted Technology Disclosure

OpenAI ChatGPT 5 and 6 was used during the development of this work in extended multi\-turn interactions for research brainstorming, including generating and refining candidate hypotheses, exploring possible proof strategies and counterexamples, identifying technical obstacles, and assisting with the presentation and organization of the resulting arguments\. It was also used for language editing, LaTeX editing, and checks for notational and expository consistency\.

For the theoretical results, AI\-generated suggestions were treated as candidate arguments rather than as authoritative results\. The authors independently selected the research directions pursued, wrote and checked the mathematical arguments presented in the paper, and take responsibility for all statements, proofs, citations, and conclusions\.

The interactions consisted of iterative prompts seeking to analyze regret\-matching and predictive regret\-matching dynamics, derive candidate inequalities and convergence arguments, search for counterexamples to proposed claims, compare alternative proof approaches, and refine the resulting theory and exposition\. Because the research process involved extended conversations across multiple sessions, not every individual prompt and model version was separately retained\. The authors therefore report the nature of the interactions here rather than reconstructing a lengthy and in some cases unavailable sequence of prompts\.

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