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

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/√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.