  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.
