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 of T, an O(T^{-1}) average duality gap, and an 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 norm rho_t. Coupling this bound across the two players yields recurrences whose effective coefficient scales as 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.
We introduce Multiplicatively Optimistic Regret Matching (MORM), an uncoupled learning rule for finite general-sum games. Under simultaneous full-information self-play, every player achieves external regret O(nlogd) uniformly over all horizons, using only one-step optimism. The analysis combines a potential-based regret-matching argument with multiplicative stability and Hellinger control of strategy movement. A learning-rate safeguard additionally gives O(Tlogd) regret in the face of adversarial utilities.
We study the problem of learning minimax policies in zero-sum matrix games. Fiegel et al. (2025) recently showed that achieving last-iterate convergence in this setting is harder when the players are uncoupled, by proving a lower bound on the exploitability gap of Omega(t^{-1/4}). Some online mirror descent algorithms were proposed in the literature for this problem, but none have truly attained this rate yet. We show that the use of a log-barrier regularization, along with a dual-focused analysis, allows this O-tilde(t^{-1/4}) convergence with high-probability. We additionally extend our idea to the setting of extensive-form games, proving a bound with the same rate.
Come Fiegel, Pierre Menard, Tadashi Kozuno +2
ENSAE Paris – CREST, France · ENS Lyon, France · Isara Labs +2
In this paper, we study regret minimization in repeated games with adaptive opponents who can respond based on histories of play. The standard metric of external regret in online learning is known to fail to capture such adaptivity. To account for players' counterfactual reasoning, we introduce {\tt Repeated Policy Regret (RP-Regret)}, a game-theoretic metric that measures the difference between the realized and the best-in-hindsight accumulated utility when all players can respond to the history of play. Compared to existing regret notions in this setting, ours is native to repeated game playing, enabling stronger comparators and opponents with fewer constraints, while maintaining the possibility of finding better equilibria when all players minimize it. We first identify necessary conditions for obtaining {\tt RP-Regret} sublinear in time, on the variation of the player's comparator strategies in the regret definition and on the memories of both the comparator and opponents' strategies. We then study additional conditions and provable algorithms to minimize {\tt RP-Regret}, which is by definition non-convex in the strategy space. To address this challenge, we propose three algorithms: (i) one based on an optimization oracle, as assumed in some prior work in online non-convex learning; (ii) one that minimizes a convex and linearized surrogate of {\tt RP-Regret} at each iteration; (iii) one that directly minimizes {\tt RP-Regret} when opponents change strategies slowly. Furthermore, when all players can run algorithms to minimize the {\tt RP-Regret} (or its linearized variant), certain subgame perfect equilibria of the repeated game can be learned. We also provide experiments showing that minimizing our regret notions can lead to more cooperative solutions with higher utility in games such as Stag-Hunt.
Mingyang Liu, Asuman Ozdaglar, Tiancheng Yu +1
Massachusetts Institute of Technology · OpenAI · University of Maryland, College Park