A Sharp Analysis of Model-based Reinforcement Learning with Self-Play

Model-based algorithms -- algorithms that explore the environment through building and utilizing an estimated model -- are widely used in reinforcement learning practice and theoretically shown to achieve optimal sample efficiency for single-agent reinforcement learning in Markov Decision Processes (MDPs). However, for multi-agent reinforcement learning in Markov games, the current best known sample complexity for model-based algorithms is rather suboptimal and compares unfavorably against recent model-free approaches. In this paper, we present a sharp analysis of model-based self-play algorithms for multi-agent Markov games. We design an algorithm -- Optimistic Nash Value Iteration (Nash-VI) for two-player zero-sum Markov games that is able to output an ε-approximate Nash policy in tildeO(H3SAB/ε2) episodes of game playing, where S is the number of states, A,B are the number of actions for the two players respectively, and H is the horizon length. This significantly improves over the best known model-based guarantee of tildeO(H4S2AB/ε2), and is the first that matches the information-theoretic lower bound Omega(H3S(A+B)/ε2) except for a minA,B factor. In addition, our guarantee compares favorably against the best known model-free algorithm if min A,B=o(H3), and outputs a single Markov policy while existing sample-efficient model-free algorithms output a nested mixture of Markov policies that is in general non-Markov and rather inconvenient to store and execute. We further adapt our analysis to designing a provably efficient task-agnostic algorithm for zero-sum Markov games, and designing the first line of provably sample-efficient algorithms for multi-player general-sum Markov games.