Faster Optimistic Online Mirror Descent for Extensive-Form Games

Online Mirror Descent (OMD) is a kind of regret minimization algorithms for Online Convex Optimization (OCO). Recently, they are applied to solve Extensive-Form Games (EFGs) for approximating Nash equilibrium. Especially, optimistic variants of OMD are developed, which have a better theoretical convergence rate compared to common regret minimization algorithms, e.g., Counterfactual Regret Minimization (CFR), for EFGs. However, despite the theoretical advantage, existing OMD and their optimistic variants have been shown to converge to a Nash equilibrium slower than the state-of-the-art (SOTA) CFR variants in practice. The reason for the inferior performance may be that they usually use constant regularizers whose parameters have to be chosen at the beginning. Inspired by the adaptive nature of CFRs, in this paper, an adaptive method is presented to speed up the optimistic variants of OMD. Based on this method, Adaptive Optimistic OMD (Ada-OOMD) for EFGs is proposed. In this algorithm, the regularizers can adapt to real-time regrets, thus the algorithm may converge faster in practice. Experimental results show that Ada-OOMD is at least two orders of magnitude faster than existing optimistic OMD algorithms. In some extensive-form games, such as Kuhn poker and Goofspiel, the convergence speed of Ada-OOMD even exceeds the SOTA CFRs. https://github.com/github-jhc/ada-oomd