Last-Iterate Convergence Properties of Regret-Matching Algorithms in Games

Yang Cai, Gabriele Farina, Julien Grand-Clément, Christian Kroer, Chung-Wei Lee, Haipeng Luo, Weiqiang Zheng

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Abstract

We study last-iterate convergence properties of algorithms for solving two-player zero-sum games based on Regret Matching+ (RM+). Despite their widespread use for solving real games, virtually nothing is known about their last-iterate convergence. A major obstacle to analyzing RM-type dynamics is that their regret operators lack Lipschitzness and (pseudo)monotonicity. We start by showing numerically that several variants used in practice, such as RM+, predictive RM+ and alternating RM+, all lack last-iterate convergence guarantees even on a simple 3 × 3 matrix game. We then prove that recent variants of these algorithms based on a smoothing technique, extragradient RM+ and smooth Predictive RM+, enjoy asymptotic last-iterate convergence (without a rate), $1/\sqrt{t}$ best-iterate convergence, and when combined with restarting, linear-rate last-iterate convergence. Our analysis builds on a new characterization of the geometric structure of the limit points of our algorithms, marking a significant departure from most of the literature on last-iterate convergence. We believe that our analysis may be of independent interest and offers a fresh perspective for studying last-iterate convergence in algorithms based on non-monotone operators.