Distributionally Robust Online Markov Game with Linear Function Approximation

The sim-to-real gap, where agents trained in a simulator face significant performance degradation during testing, is a fundamental challenge in reinforcement learning. Extensive works adopt the framework of distributionally robust RL, to learn a policy that acts robustly under worst case environment shift. Within this framework, our objective is to devise algorithms that are sample efficient with interactive data collection and large state spaces. By assuming d-rectangularity of environment dynamic shift, we identify a fundamental hardness result for learning in online Markov game, and address it by adopting minimum value assumption. Then, a novel least square value iteration type algorithm, DR-CCE-LSI, with exploration bonus devised specifically for multiple agents, is proposed to find an epsilon-approximate robust Coarse Correlated Equilibrium(CCE). To obtain sample efficient learning, we find that: when the feature mapping function satisfies certain properties, our algorithm, DR-CCE-LSI, is able to achieve epsilon-approximate CCE with a regret bound of O(dH min(H, 1/ min(sigma_i)) sqrt(K)), where K is the number of interacting episodes, H is the horizon length, d is the feature dimension, and sigma_i represents the uncertainty level of player i. Our work introduces the first sample-efficient algorithm for this setting, matches the best result so far in single agent setting, and achieves minimax optimal sample complexity in terms of the feature dimension d. Meanwhile, we also conduct simulation study to validate the efficacy of our algorithm in learning a robust equilibrium.