The Power of Regularization in Solving Extensive-Form Games

In this paper, we investigate the power of {\it regularization}, a common technique in reinforcement learning and optimization, in solving extensive-form games (EFGs). We propose a series of new algorithms based on regularizing the payoff functions of the game, and establish a set of convergence results that strictly improve over the existing ones, with either weaker assumptions or stronger convergence guarantees. In particular, we first show that dilated optimistic mirror descent (DOMD), an efficient variant of OMD for solving EFGs, with adaptive regularization can achieve a fast $\tilde O(1/T)$ {last-iterate convergence rate for the output of the algorithm} in terms of duality gap and distance to the set of Nash equilibrium (NE) without uniqueness assumption of the NE. Second, we show that regularized counterfactual regret minimization (\texttt{Reg-CFR}), with a variant of optimistic mirror descent algorithm as regret-minimizer, can achieve $O(1/T^{1/4})$ best-iterate, and $O(1/T^{3/4})$ average-iterate convergence rate for finding NE in EFGs. Finally, we show that \texttt{Reg-CFR} can achieve asymptotic last-iterate convergence, and optimal $O(1/T)$ average-iterate convergence rate, for finding the NE of perturbed EFGs, which is useful for finding approximate extensive-form perfect equilibria (EFPE). To the best of our knowledge, they constitute the first last-iterate convergence results for CFR-type algorithms, while matching the state-of-the-art average-iterate convergence rate in finding NE for non-perturbed EFGs. We also provide numerical results to corroborate the advantages of our algorithms.