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Convergence of Regret Matching in Potential Games and Constrained Optimization

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 7.0Download Report PDF
regret matchingno-regret learningpotential gamesconstrained optimization
Abstract:

Regret matching (RM)---and its modern variants---is a foundational online algorithm that has been at the heart of many AI breakthrough results in solving benchmark zero-sum games, such as poker. Yet, surprisingly little is known so far in theory about its convergence beyond two-player zero-sum games. For example, whether regret matching converges to Nash equilibria in potential games has been an open problem for two decades. Even beyond games, one could try to use RM variants for general constrained optimization problems. Recent empirical evidence suggests that they---particularly regret matching+^+ (RM+^+)---attain strong performance on benchmark constrained optimization problems, outperforming traditional gradient descent-type algorithms.

We show that alternating RM+^+ converges to an ϵ\epsilon-KKT point after Oϵ(1/ϵ4)O_\epsilon(1/\epsilon^4) iterations, establishing for the first time that it is a sound and fast first-order optimizer. Our argument relates the KKT gap to the accumulated regret, two quantities that are entirely disparate in general but interact in an intriguing way in our setting, so much so that when regrets are bounded, our complexity bound improves all the way to Oϵ(1/ϵ2)O_\epsilon(1/\epsilon^2). From a technical standpoint, while RM+^+ does not have the usual one-step improvement property in general, we show that it does in a certain region that the algorithm will quickly reach and remain in thereafter. In sharp contrast, our second main result establishes a lower bound: RM, with or without alternation, can take an exponential number of iterations to reach a crude approximate solution even in two-player potential games. This represents the first worst-case separation between RM and RM+^+. Our lower bound shows that convergence to coarse correlated equilibria in potential games is exponentially faster than convergence to Nash equilibria.

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Overview

Overall Novelty Assessment

The paper establishes convergence guarantees for regret matching variants (particularly RM+) in constrained optimization and potential games, proving O(1/ε⁴) iteration complexity to ε-KKT points. Within the taxonomy, it resides in the 'Convergence in Potential Games' leaf under 'Theoretical Convergence Analysis', sharing this leaf with only one sibling paper. This represents a relatively sparse research direction focused on formal convergence proofs in potential game settings, contrasting with the more populated application-oriented branches of the taxonomy tree.

The taxonomy reveals neighboring work in 'Convergence in General Game Settings' (addressing broader game classes like Stackelberg games) and 'Constrained Optimization via Game-Theoretic Methods' (formulating optimization as games). The paper bridges these areas by treating constrained optimization through regret-based dynamics while maintaining focus on potential game structure. The taxonomy's scope notes clarify that pure game-theoretic learning without potential structure belongs elsewhere, positioning this work at the intersection of optimization theory and equilibrium computation in structured games.

Among 18 candidates examined, the contribution on RM+ convergence rates to KKT points shows one refutable candidate from 8 examined, while the connection between regret accumulation and Nash convergence shows two refutable candidates from 10 examined. The exponential separation between RM and RM+ received no examination in the limited search. These statistics suggest that while some prior work addresses related convergence questions, the specific complexity bounds and the regret-KKT gap relationship may represent less-explored territory within the examined literature.

Based on the top-18 semantic matches and citation expansion, the work appears to occupy a theoretically-focused niche with limited direct precedent in the examined candidates. The sparse population of its taxonomy leaf and the modest refutation rates suggest potential novelty, though the limited search scope means substantial related work may exist beyond these candidates, particularly in broader optimization or game theory venues not captured by this domain-specific search.

Taxonomy

Core-task Taxonomy Papers
11
3
Claimed Contributions
18
Contribution Candidate Papers Compared
3
Refutable Paper

Research Landscape Overview

Core task: convergence of regret matching algorithms in potential games and constrained optimization. The field structure divides naturally into three main branches. Theoretical Convergence Analysis examines the mathematical foundations and guarantees for regret-based learning dynamics, often focusing on potential games where convergence to equilibria can be rigorously established, as seen in foundational continuous-time formulations like Regret-based continuous-time dynamics[8]. Constrained Optimization via Game-Theoretic Methods reframes optimization problems as games, leveraging regret minimization to solve resource allocation and decision-making challenges under constraints, with applications ranging from UAV task allocation (Task Allocation in UAV[4]) to non-cooperative settings (Two-player games for efficient[1]). Algorithm Design and Applications emphasizes practical implementations, including distributed variants (Distributed regret matching algorithm[5], Neural regret-matching for distributed[3]) and domain-specific adaptations in networking (Cognitive forwarding control in[9]) and imperfect-information games (Decision making under imperfect[2]). These branches are interconnected: theoretical insights guide algorithm design, while applications motivate new convergence questions in constrained or multi-agent settings. Particularly active lines of work explore the tension between theoretical guarantees and computational tractability. Some studies pursue unifying frameworks that connect regret matching to broader classes of learning dynamics (A unifying framework for[6]), while others benchmark hybrid approaches that blend game-theoretic methods with optimization heuristics (Benchmarking hybrid algorithms for[7]). The original paper, Convergence of Regret Matching[0], sits squarely within the Theoretical Convergence Analysis branch, specifically addressing convergence properties in potential games. Its emphasis on rigorous convergence proofs aligns closely with the continuous-time perspective of Regret-based continuous-time dynamics[8], yet it also engages with distributed and practical concerns that bridge toward works like Distributed regret matching algorithm[5]. Compared to application-driven studies such as Neural regret-matching for distributed[3], Convergence of Regret Matching[0] prioritizes foundational guarantees, contributing to the theoretical backbone that supports the field's algorithmic and applied extensions.

Claimed Contributions

Convergence rate of RM+ to KKT points in constrained optimization

The authors prove that regret matching+ (RM+) converges to approximate Karush-Kuhn-Tucker points in constrained optimization problems over product of simplices with a polynomial rate. This establishes RM+ as a theoretically sound first-order optimization algorithm with complexity bounds that improve when regrets are bounded.

8 retrieved papers
Can Refute
Exponential separation between RM and RM+ in potential games

The authors construct two-player identical-interest games where standard regret matching (RM) requires exponentially many iterations to converge to approximate Nash equilibria, while RM+ converges polynomially fast. This is the first proven worst-case separation between these two algorithms.

0 retrieved papers
Connection between regret accumulation and convergence rate to Nash equilibria

The authors establish a novel theoretical connection showing that the rate of convergence to Nash equilibria in potential games is directly governed by the regret accumulation rate. Specifically, they prove that if regret grows as T^α for α in [0,1/2], then RM+ converges to ε-KKT points in O_ε(1/ε^(2/(1-α))) iterations.

10 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Convergence rate of RM+ to KKT points in constrained optimization

The authors prove that regret matching+ (RM+) converges to approximate Karush-Kuhn-Tucker points in constrained optimization problems over product of simplices with a polynomial rate. This establishes RM+ as a theoretically sound first-order optimization algorithm with complexity bounds that improve when regrets are bounded.

Contribution

Exponential separation between RM and RM+ in potential games

The authors construct two-player identical-interest games where standard regret matching (RM) requires exponentially many iterations to converge to approximate Nash equilibria, while RM+ converges polynomially fast. This is the first proven worst-case separation between these two algorithms.

Contribution

Connection between regret accumulation and convergence rate to Nash equilibria

The authors establish a novel theoretical connection showing that the rate of convergence to Nash equilibria in potential games is directly governed by the regret accumulation rate. Specifically, they prove that if regret grows as T^α for α in [0,1/2], then RM+ converges to ε-KKT points in O_ε(1/ε^(2/(1-α))) iterations.

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