We revisit the identification of an $\varepsilon$-optimal policy in average-reward Markov Decision Processes (MDP). In such MDPs, two measures of complexity have appeared in the literature: the diameter, $D$, and the optimal bias span, $H$, which satisfy $H\leq D$. Prior work have studied the complexity of $\varepsilon$-optimal policy identification only when a generative model is available. In this case, it is known that there exists an MDP with $D \simeq H$ for which the sample complexity to output an $\varepsilon$-optimal policy is $\Omega(SAD/\varepsilon^2)$ where $S$ and $A$ are the sizes of the state and action spaces. Recently, an algorithm with a sample complexity of order $SAH/\varepsilon^2$ has been proposed, but it requires the knowledge of $H$.

We revisit the identification of an \varepsilon -optimal policy in average-reward Markov Decision Processes (MDP). In such MDPs, two measures of complexity have appeared in the literature: the diameter, D, and the optimal bias span, H, which satisfy H\leq D . Prior work have studied the complexity of \varepsilon -optimal policy identification only when a generative model is available. In this case, it is known that there exists an MDP with D \simeq H for which the sample complexity to output an \varepsilon -optimal policy is \Omega(SAD/\varepsilon 2) where S and A are the sizes of the state and action spaces. Recently, an algorithm with a sample complexity of order SAH/\varepsilon 2 has been proposed, but it requires the knowledge of H .

We study the problem of average-reward Markov decision processes (AMDPs) and develop novel first-order methods with strong theoretical guarantees for both policy evaluation and optimization. Existing on-policy evaluation methods suffer from sub-optimal convergence rates as well as failure in handling insufficiently random policies, e.g., deterministic policies, for lack of exploration. To remedy these issues, we develop a novel variance-reduced temporal difference (VRTD) method with linear function approximation for randomized policies along with sharp convergence guarantees, and an exploratory variance-reduced temporal difference (EVRTD) method for insufficiently random policies with comparable convergence guarantees. We further establish linear convergence rate on the bias of policy evaluation, which is essential for improving the overall sample complexity of policy optimization. On the other hand, compared with intensive research interest in finite sample analysis of policy gradient methods for discounted MDPs, existing studies on policy gradient methods for AMDPs mostly focus on regret bounds under restrictive assumptions on the underlying Markov processes (see, e.g., Abbasi-Yadkori et al., 2019), and they often lack guarantees on the overall sample complexities. Towards this end, we develop an average-reward variant of the stochastic policy mirror descent (SPMD) (Lan, 2022). We establish the first $\widetilde{\mathcal{O}}(\epsilon^{-2})$ sample complexity for solving AMDPs with policy gradient method under both the generative model (with unichain assumption) and Markovian noise model (with ergodic assumption). This bound can be further improved to $\widetilde{\mathcal{O}}(\epsilon^{-1})$ for solving regularized AMDPs. Our theoretical advantages are corroborated by numerical experiments.