Last-Iterate Convergence in Adaptive Regret Minimization for Approximate Extensive-Form Perfect Equilibrium

Hang Ren; Jiajia Zhang; Tianzi Ma; Xiaozhen Sun; Xuan Wang

REVIEW 3 major objections 3 minor 38 references

Last-Iterate Convergence in Adaptive Regret Minimization for Approximate Extensive-Form Perfect Equilibrium

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper's central claim is that RTCFR, a reward-transformed counterfactual regret minimizer, attains last-iterate convergence to an approximate extensive-form perfect equilibrium in two-player zero-sum games, and that an adaptive informat

desk verdict RTCFR is a plausible algorithm with a nice adaptive idea, but the central last-iterate theorem isn't proven as printed: Lemma 16's key algebra step is invalid. read the letter →

arxiv 2508.07699 v1 pith:VVQJD24G submitted 2025-08-11 cs.GT

Hang Ren , Xiaozhen Sun , Tianzi Ma , Jiajia Zhang , Xuan Wang This is my paper · ORCID

classification cs.GT MSC 91A0591A1891A26

keywords last-iterateconvergencecounterfactualregretminimizationextensive-formperfectequilibriumrewardtransformationadaptiveperturbationinformation-settwo-playerzero-sumgamestrembling-handperfection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish a last-iterate guarantee for a refinement of Nash equilibrium in two-player zero-sum extensive-form games. It combines reward transformation—a regularization trick that nudges a player's plan toward a reference plan during play—with counterfactual regret minimization on a deliberately perturbed game. The claimed result is that the final strategy produced by the algorithm, RTCFR, converges asymptotically to an $\epsilon$-extensive-form perfect equilibrium, and that a new information-set regret metric can shrink the perturbation adaptively so that equilibrium accuracy improves as training proceeds. If this holds, equilibrium solvers no longer need to keep and average a long history of strategies, and off-equilibrium robustness is obtained without the fixed-perturbation trade-off between accuracy and convergence speed.

What carries the argument

RT-BSPP: the $n$-th reward-transformed bilinear saddle-point problem, in which the payoff in the perturbed game is augmented by $\mu(\pi_i - \pi_i^{ref})$ for a reference strategy; its saddle point becomes the next reference. Affine basis matrices $B(I)$ map behavioral strategies between the original and the $\epsilon$-perturbed space, enforcing the minimum action probability. The ISNE metric, defined as the largest information-set regret (counterfactual regret renormalized to unit opponent reach probability), serves as the adaptive stopping threshold that decides when to decay $\epsilon$ by a factor $\gamma$.

What would settle it

Re-derive inequality (35) in Lemma 16 from equation (34); if the step implies $\|\pi^* - \pi^{ref}\|^2 \le 0$ for distinct strategies, the contraction argument collapses. Separately, on a small game such as Kuhn Poker, solve each RT-BSPP to high precision and plot $\|\pi^* - \pi_{*n}^*\|$ across $n$; if the distances do not decrease monotonically, the claimed last-iterate convergence mechanism is falsified.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 11: in an $\epsilon$-perturbed extensive-form game, the last strategy produced by RTCFR converges asymptotically to an $\epsilon$-EFPE as both the number of reward-transformed bilinear saddle-point subproblems (RT-BSPPs) and the number of regret-matching iterations inside each subproblem go to infinity. The argument runs through a sequence of saddle points $\pi_{*n}^*$: each RT-BSPP is the perturbed game with reward $\tilde{u}_i = u_i + \mu(\pi_i - \pi_i^{ref})$, so its saddle point is pulled toward the Nash equilibrium of the perturbed game, and the updated reference is the previous saddle point. The paper also introduces the Informatio

Load-bearing premise

The load-bearing premise is that each reward-transformed saddle point is strictly closer to the perturbed game's Nash equilibrium than the reference strategy was; the paper's printed proof of that contraction (Lemma 16) contains an algebraic step that, taken literally, would force a squared distance to be non-positive, so the guarantee is not yet established.

Editorial extensions

If this is right

If Theorem 11 holds, average-strategy computation can be discarded for perturbed-game solving: the very last strategy already converges, removing the memory and bias costs of averaging.
The adaptive ISNE schedule dissolves the fixed-perturbation dilemma: large early $\epsilon$ gives fast refinement, while geometric decay preserves NE approximation accuracy later.
As $\epsilon \to 0$, the same algorithm recovers Nash-equilibrium computation, and the paper's experiments report practical exploitability decay near $O(1/T)$ although the proven bound is only $O(1/\sqrt{T})$.
The ISNE metric provides a local certificate: it detects poorly converged branches with low reach probability, something global exploitability alone cannot pinpoint.
Because RTCFR needs one tree traversal per iteration (unlike EGT's three and CFR+'s two in the paper's accounting), the method scales to comparatively large instances such as Liar's Dice with six dice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A repaired proof of the contraction lemma would likely extend the same template to quasi-perfect equilibria or to multi-player settings, since the reward transformation is applied locally at information sets rather than to global strategies.
The ISNE metric could serve as a practical stopping rule in deep reinforcement learning for imperfect-information games, where global exploitability is expensive but per-information-set regrets are directly available.
A testable corollary of the paper's mechanism: in games with a unique strict Nash equilibrium, the reward-transformation term becomes irrelevant (the paper itself notes this for Liar's Dice), so a purely adaptive version of plain CFR+ should match RTCFR's performance; the adaptive schedule's value should show up mainly in games with multiple equilibria or deep low-reach subgames.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real attempt at an important problem, and the algorithm and experiments are worth a look, but the central theorem is not supported by the submitted text. The claimed last-iterate convergence to ε-EFPE rests on Lemma 16, and Lemma 16's proof is demonstrably wrong as written. Starting from (34), they drop a nonnegative exploitability term and get (35): ⟨x*_n − x_ref, x* − x*_n⟩ ≥ ⟨x*_n − x_ref, x* − x_ref⟩. Since x* − x_ref = (x*_n − x_ref) + (x* − x*_n), this rearranges to ||x*_n − x_ref||² ≤ 0. So the printed (35) can only hold when the reference equals the saddle point; the later “adding” step then derives the contraction from a false premise. That's load-bearing: Theorem 11 and the adaptive perturbation claims in Section 5 depend on it. If the lemma has a correct proof, it's not in this version.

What's actually new and good: Meng et al. proposed RTCFR+ for perturbed games without convergence theory; this paper supplies a convergence theorem (modulo the broken lemma) and an adaptive perturbation heuristic, ISNE, which is a reasonable novelty. ISNE as an information-set-regret threshold is a sensible way to decide when to shrink ε, and the experiments cover standard benchmarks with code released. The paper also honestly says the finite-T guarantee is best-iterate (Theorem 10) and that T must go to infinity per block for the strict last-iterate statement; that honesty is good, but it means the practical claim is weaker than the title suggests.

Soft spots beyond the lemma: the adaptive method has no formal convergence proof of its own—it relies on the unproven Theorem 11. The experiments tune many hyperparameters per game, and in Liar's Dice they set T=1, which makes RTCFR identical to plain CFR+; those runs don't test RTCFR. The comparison to Bernasconi et al. could also be fairer, because their RegOMWU(adp) is modified substantially from the original and the tuning is not symmetric. Citation practice is otherwise honest: Meng et al. and Bernasconi et al. are credited properly.

Bottom line: this deserves a serious referee because the problem is central and the proof may be repairable, but in the current form I wouldn't trust the main theorem. I'd send it back for major revision with a request to fix Lemma 16 and re-check the adaptive guarantee.

Referee Report

3 major / 3 minor

Summary. The paper studies computation of approximate extensive-form perfect equilibria (EFPE) in two-player zero-sum extensive-form games. It introduces RTCFR, a reward-transformation variant of CFR that operates on epsilon-perturbed sequence-form strategy spaces, and claims asymptotic last-iterate convergence to an epsilon-EFPE (Theorem 11), supported by a contraction lemma (Lemma 16). It also defines an information-set regret metric (ISNE) and an adaptive schedule that reduces the perturbation when the metric falls below a threshold. Experiments on Kuhn Poker, Leduc Poker, Goofspiel, and Liar's Dice compare RTCFR+ against CFR+, EGT, and RegOMWU variants.

Significance. The problem is well motivated: computing EFPE efficiently with last-iterate convergence and adaptive perturbations would be a genuine advance over average-iterate CFR-based methods. The paper proposes a concrete algorithm with available code and benchmark comparisons. However, the theoretical core is not currently established: the proof of the key contraction lemma contains a demonstrable algebraic error, and the paper's own caveat about finite T reducing the guarantee to best-iterate convergence contradicts the last-iterate claim. The experimental results are suggestive but cannot compensate for the missing theoretical support.

major comments (3)

[Appendix B, Lemma 16, Eq. (35)] The claimed simplification from Eq. (34) to Eq. (35) is algebraically incorrect. From (34) the correct inequality is ⟨x*_n − x_ref, x* − x*_n⟩ ≥ 0. The printed (35), however, is equivalent to ⟨x*_n − x_ref, x* − x*_n⟩ ≥ ⟨x*_n − x_ref, x* − x_ref⟩, which reduces to −‖x*_n − x_ref‖² ≥ 0 and hence contradicts the assumption x*_n ≠ x_ref. The subsequent 'adding' step and the derivation of (32)–(33) are therefore not justified as written. Since Theorem 11 and the adaptive perturbation guarantees in Section 5 rely on this contraction, the central convergence proof is unsupported.
[Section 4, Theorem 11 and preceding paragraph] The paper states that a strict theoretical guarantee requires T → ∞ in each RT-BSPP and that with finite T only best-iterate convergence (Theorem 10) is achieved, yet it concludes 'thus fulfilling Theorem 11.' This is a non sequitur: Theorem 10 guarantees the existence of some t ≤ T with small error, not that the last iterate x_T^{n,T} is close. Algorithm 1 sets the reference strategy to the last inner iterate, so the asymptotic last-iterate claim does not follow from the stated theorems. The authors need either a genuine last-iterate guarantee for the inner loop or a revised claim (e.g., best-iterate convergence of the output sequence).
[Section 5.2, Algorithm 1] The adaptive schedule changes the perturbed game Γ_ϵ whenever rmax falls below δ. The paper claims that this preserves convergence to an EFPE as ϵ → 0, but Theorems 9–11 concern a fixed perturbed game. No theorem or continuity argument tracks the convergence of the RTCFR iterates as the target saddle point changes with ϵ. This is a load-bearing gap because the adaptive perturbation is a central contribution; the statement 'ensuring asymptotic convergence to the EFPE as ϵ → 0' is not proven by the supplied results.

minor comments (3)

[Throughout] Several equations and symbols are corrupted or missing in the provided manuscript (e.g., the RT-BSPP definitions and the statement of Lemma 14), making verification difficult. The authors should ensure the final PDF renders all notation correctly.
[Section 5.2, Definition 12] The information set regret notation is inconsistent: q'_i(I) is used both as a regret vector and later as a scalar maximum, and the relationship to counterfactual regret via Bayes' rule is stated without derivation.
[Section 6, Liar's Dice experiments] For Liar's Dice, the paper sets T = 1, which nullifies the RT term and makes RTCFR equivalent to CFR+. Thus those experiments do not test the proposed reward-transformation mechanism, weakening the empirical claim of broad superiority.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence claims are derived from internal lemmas and external prior work; the ISNE metric is an evaluation criterion, not a fitted input.

full rationale

The paper's central derivation (Theorem 11) is self-contained in the sense that it rests on saddle-point properties of the reward-transformed bilinear saddle-point problems (Lemmas 14 and 16) and on regret bounds (Theorem 10), not on the experimental results or on fitting any parameter to the target quantity. The reward transformation and behaviorally-constrained perturbation framework are imported from prior work (Abe et al. [1,2], Meng et al. [22], Farina et al. [9], Kroer et al. [16]) by citation, but these are external sources and are not authored by the present paper's authors, so no self-citation chain is load-bearing. The ISNE metric (Definition 13) is explicitly defined as a maximum information-set regret condition and is used in Algorithm 1 as a threshold to decay the perturbation; using the same quantity for both the adaptive schedule and the EFPE evaluation is a design choice, not a logical circularity, because Theorem 11's proof does not assume δ-ISNE convergence to derive EFPE convergence. The possible algebraic error in Lemma 16's proof (inequality (35)) would be a correctness or soundness defect, not a circularity: the proof step does not reduce the theorem to its assumptions by definition or fit. The manuscript contains no self-citation that carries the derivation, no ansatz smuggled in solely through the authors' own prior work, and no renaming of a known result presented as a first-principles prediction. Accordingly, no circular step is identified.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claim rests on a contraction property of the RT-BSPP saddle points (Lemma 16), which is not established soundly, plus per-game hyperparameters. No new physical or game-theoretic entities are introduced; the ISNE metric is a derived quantity, not an entity.

free parameters (5)

RT weight mu = Varies per game: 0.01, 0.0001, 0.001, etc.
Mu controls the strength of the reward transformation term. It is tuned via grid search for each game and directly affects convergence.
Inner iterations T per RT-BSPP = 5 for Kuhn, 200 for Leduc, 30 for Goofspiel, 1 for Liar's Dice
T determines how many CFR iterations run inside each RT-BSPP. In Liar's Dice, T=1 reduces the method to plain CFR+.
Initial perturbation epsilon_0 = 0.1 for all games
Starting perturbation size, chosen by hand from the set {0.1}. Not tuned extensively.
ISNE threshold delta = 1, 0.02, 0.5, 0.5 depending on game
Threshold for reducing epsilon. Tuned via grid search; controls when the perturbation shrinks.
Decay factor gamma = 0.5, 0.1, 0.95, 0.9, 0.5 depending on game
Multiplicative decay for epsilon and delta. Tuned via grid search; impacts the trade-off between NE accuracy and EFPE convergence.

assumptions (3)

domain assumption Two-player zero-sum extensive-form games with perfect recall
The paper explicitly assumes this throughout (Section 3.2). The algorithm and theory rely on the sequence-form representation and regret matching properties valid in this class.
domain assumption Lemma 14: there exists a constant C > 0 such that C * ||x - x*|| <= Exp(x) for the perturbed game
Invoked in the proof of Lemma 16 (Appendix B). This is a local strong convexity-like bound on exploitability near the equilibrium, not proven in the paper.
ad hoc to paper The sequence of RT-BSPP saddle points converges to the NE of the perturbed game
This is the content of Lemma 16, but the lemma's proof appears flawed. The convergence of the reference strategy to the epsilon-EFPE depends on this.

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Cite this review

Pith. "Pith review of Last-Iterate Convergence in Adaptive Regret Minimization for Approximate Extensive-Form Perfect Equilibrium." pith.science (2026). https://pith.science/paper/VVQJD24G

@misc{pith2026250807699,
  author       = {Pith},
  title        = {Pith review of: Last-Iterate Convergence in Adaptive Regret Minimization for Approximate Extensive-Form Perfect Equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVQJD24G}},
  note         = {Machine review of arXiv:2508.07699}
}

read the original abstract

The Nash Equilibrium (NE) assumes rational play in imperfect-information Extensive-Form Games (EFGs) but fails to ensure optimal strategies for off-equilibrium branches of the game tree, potentially leading to suboptimal outcomes in practical settings. To address this, the Extensive-Form Perfect Equilibrium (EFPE), a refinement of NE, introduces controlled perturbations to model potential player errors. However, existing EFPE-finding algorithms, which typically rely on average strategy convergence and fixed perturbations, face significant limitations: computing average strategies incurs high computational costs and approximation errors, while fixed perturbations create a trade-off between NE approximation accuracy and the convergence rate of NE refinements. To tackle these challenges, we propose an efficient adaptive regret minimization algorithm for computing approximate EFPE, achieving last-iterate convergence in two-player zero-sum EFGs. Our approach introduces Reward Transformation Counterfactual Regret Minimization (RTCFR) to solve perturbed games and defines a novel metric, the Information Set Nash Equilibrium (ISNE), to dynamically adjust perturbations. Theoretical analysis confirms convergence to EFPE, and experimental results demonstrate that our method significantly outperforms state-of-the-art algorithms in both NE and EFPE-finding tasks.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

K. Abe, M. Sakamoto, and A. Iwasaki. Mutation-driven follow the reg- ularized leader for last-iterate convergence in zero-sum games. In Un- certainty in Artificial Intelligence, pages 1–10, 2022

work page 2022

[2] [2]

K. Abe, K. Ariu, M. Sakamoto, K. Toyoshima, and A. Iwasaki. Last- iterate convergence with full and noisy feedback in two-player zero- sum games. In International Conference on Artificial Intelligence and Statistics, volume 206, pages 7999–8028, 2023

work page 2023

[3] [3]

Anagnostides, I

I. Anagnostides, I. Panageas, G. Farina, and T. Sandholm. On last- iterate convergence beyond zero-sum games. In International Confer- ence on Machine Learning, pages 536–581, 2022

work page 2022

[4] [4]

Bernasconi, A

M. Bernasconi, A. Marchesi, and F. Trovò. Learning extensive-form perfect equilibria in two-player zero-sum sequential games. In Interna- tional Conference on Artificial Intelligence and Statistics, pages 2152– 2160, 2024

work page 2024

[5] [5]

Brown and T

N. Brown and T. Sandholm. Solving imperfect-information games via discounted regret minimization. InProceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 1829–1836, 2019

work page 2019

[6] [6]

Brown, A

N. Brown, A. Lerer, S. Gross, and T. Sandholm. Deep counterfactual regret minimization. In International conference on machine learning, pages 793–802, 2019

work page 2019

[7] [7]

Y . Cai, G. Farina, J. Grand-Clément, C. Kroer, C. Lee, H. Luo, and W. Zheng. Last-iterate convergence properties of regret-matching algo- rithms in games. In The Thirteenth International Conference on Learn- ing Representations, 2025

work page 2025

[8] [8]

Farina and N

G. Farina and N. Gatti. Extensive-form perfect equilibrium computa- tion in two-player games. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 31, 2017

work page 2017