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 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), 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.
View on arXiv@article{cai2025_2311.00676,
title={ Last-Iterate Convergence Properties of Regret-Matching Algorithms in Games },
author={ Yang Cai and Gabriele Farina and Julien Grand-Clément and Christian Kroer and Chung-Wei Lee and Haipeng Luo and Weiqiang Zheng },
journal={arXiv preprint arXiv:2311.00676},
year={ 2025 }
}