Optimal Sample Complexity for Average Reward Markov Decision Processes

We settle the sample complexity of policy learning for the maximization of the long run average reward associated with a uniformly ergodic Markov decision process (MDP), assuming a generative model. In this context, the existing literature provides a sample complexity upper bound of $\widetilde O(|S||A|t_{\text{mix}}^2 \epsilon^{-2})$ and a lower bound of $\Omega(|S||A|t_{\text{mix}} \epsilon^{-2})$. In these expressions, $|S|$ and $|A|$ denote the cardinalities of the state and action spaces respectively, $t_{\text{mix}}$ serves as a uniform upper limit for the total variation mixing times, and $\epsilon$ signifies the error tolerance. Therefore, a notable gap of $t_{\text{mix}}$ still remains to be bridged. Our primary contribution is to establish an estimator for the optimal policy of average reward MDPs with a sample complexity of $\widetilde O(|S||A|t_{\text{mix}}\epsilon^{-2})$, effectively reaching the lower bound in the literature. This is achieved by combining algorithmic ideas in Jin and Sidford (2021) with those of Li et al. (2020)