Minimax Sample Complexity for Turn-based Stochastic Game

The empirical success of multi-agent reinforcement learning is encouraging, while few theoretical guarantees have been revealed. In this work, we prove that the plug-in solver approach, probably the most natural reinforcement learning algorithm, achieves minimax sample complexity for turn-based stochastic game (TBSG). Specifically, we perform planning in an empirical TBSG by utilizing a `simulator' that allows sampling from arbitrary state-action pair. We show that the empirical Nash equilibrium strategy is an approximate Nash equilibrium strategy in the true TBSG and give both problem-dependent and problem-independent bound. We develop reward perturbation techniques to tackle the non-stationarity in the game and Taylor-expansion-type analysis to improve the dependence on approximation error. With these novel techniques, we prove the minimax sample complexity of turn-based stochastic game.