A Bayesian Learning Algorithm for Unknown Zero-sum Stochastic Games with an Arbitrary Opponent
Mehdi Jafarnia-Jahromi, Rahul Jain, Ashutosh Nayyar
TL;DR
This work tackles online learning in infinite-horizon two-player zero-sum stochastic games with an unknown transition model and an arbitrary time-adaptive opponent under the average-reward criterion. It introduces PSRL-ZSG, a posterior sampling-based algorithm that iteratively samples transition kernels from the posterior and computes maximin policies via the Bellman equation to guide play, with episode-based exploration driven by doubling criteria. The main contribution is a Bayesian regret bound of $\widetilde{O}(HS\sqrt{AT})$, improving over the prior $\widetilde{O}(\sqrt[3]{DS^2AT^2})$ bound achieved by UCSG under the same assumptions and matching the $T$-dependence lower bound up to logarithms. This approach also handles time-adaptive, history-dependent opponents without strong ergodicity assumptions, offering a simpler, near-optimal alternative to optimism-based methods in stochastic games.
Abstract
In this paper, we propose Posterior Sampling Reinforcement Learning for Zero-sum Stochastic Games (PSRL-ZSG), the first online learning algorithm that achieves Bayesian regret bound of $O(HS\sqrt{AT})$ in the infinite-horizon zero-sum stochastic games with average-reward criterion. Here $H$ is an upper bound on the span of the bias function, $S$ is the number of states, $A$ is the number of joint actions and $T$ is the horizon. We consider the online setting where the opponent can not be controlled and can take any arbitrary time-adaptive history-dependent strategy. Our regret bound improves on the best existing regret bound of $O(\sqrt[3]{DS^2AT^2})$ by Wei et al. (2017) under the same assumption and matches the theoretical lower bound in $T$.