Improving Sample Efficiency of Model-Free Algorithms for Zero-Sum Markov Games
This work addresses a key bottleneck in multi-agent reinforcement learning by enabling model-free methods to achieve optimal sample efficiency, which is incremental but important for theoretical and practical applications in competitive environments.
The paper tackles the problem of sample inefficiency in model-free algorithms for two-player zero-sum Markov games by proposing a stage-based Q-learning algorithm that achieves an optimal sample complexity of O(H^3SAB/ε^2), matching the best model-based algorithms for the first time.
The problem of two-player zero-sum Markov games has recently attracted increasing interests in theoretical studies of multi-agent reinforcement learning (RL). In particular, for finite-horizon episodic Markov decision processes (MDPs), it has been shown that model-based algorithms can find an $ε$-optimal Nash Equilibrium (NE) with the sample complexity of $O(H^3SAB/ε^2)$, which is optimal in the dependence of the horizon $H$ and the number of states $S$ (where $A$ and $B$ denote the number of actions of the two players, respectively). However, none of the existing model-free algorithms can achieve such an optimality. In this work, we propose a model-free stage-based Q-learning algorithm and show that it achieves the same sample complexity as the best model-based algorithm, and hence for the first time demonstrate that model-free algorithms can enjoy the same optimality in the $H$ dependence as model-based algorithms. The main improvement of the dependency on $H$ arises by leveraging the popular variance reduction technique based on the reference-advantage decomposition previously used only for single-agent RL. However, such a technique relies on a critical monotonicity property of the value function, which does not hold in Markov games due to the update of the policy via the coarse correlated equilibrium (CCE) oracle. Thus, to extend such a technique to Markov games, our algorithm features a key novel design of updating the reference value functions as the pair of optimistic and pessimistic value functions whose value difference is the smallest in the history in order to achieve the desired improvement in the sample efficiency.