We study the sample complexity of learning an \varepsilon -optimal policy in an average-reward Markov decision process (MDP) under a generative model. For weakly communicating MDPs, we establish the complexity bound \widetilde{O}\left(SA\frac{\mathsf{H}}{\varepsilon 2} \right), where \mathsf{H} is the span of the bias function of the optimal policy and SA is the cardinality of the state-action space. Our result is the first that is minimax optimal (up to log factors) in all parameters S,A,\mathsf{H}, and \varepsilon, improving on existing work that either assumes uniformly bounded mixing times for all policies or has suboptimal dependence on the parameters. We also initiate the study of sample complexity in general (multichain) average-reward MDPs. Both results are based on reducing the average-reward MDP to a discounted MDP, which requires new ideas in the general setting.

We study the sample complexity of learning an $\varepsilon$-optimal policy in an average-reward Markov decision process (MDP) under a generative model. We establish the complexity bound $\widetilde{O}\left(SA\frac{H}{\varepsilon^2} \right)$, where $H$ is the span of the bias function of the optimal policy and $SA$ is the cardinality of the state-action space. Our result is the first that is minimax optimal (up to log factors) in all parameters $S,A,H$ and $\varepsilon$, improving on existing work that either assumes uniformly bounded mixing times for all policies or has suboptimal dependence on the parameters. Our result is based on reducing the average-reward MDP to a discounted MDP. To establish the optimality of this reduction, we develop improved bounds for $\gamma$-discounted MDPs, showing that $\widetilde{O}\left(SA\frac{H}{(1-\gamma)^2\varepsilon^2} \right)$ samples suffice to learn a $\varepsilon$-optimal policy in weakly communicating MDPs under the regime that $\gamma \geq 1 - \frac{1}{H}$, circumventing the well-known lower bound of $\widetilde{\Omega}\left(SA\frac{1}{(1-\gamma)^3\varepsilon^2} \right)$ for general $\gamma$-discounted MDPs. Our analysis develops upper bounds on certain instance-dependent variance parameters in terms of the span parameter. These bounds are tighter than those based on the mixing time or diameter of the MDP and may be of broader use.