Robust Average-Reward Markov Decision Processes: Minimax-Optimal Learning via Plug-in Reductions

Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty. We study how many samples are necessary and sufficient to learn an varepsilon-optimal robust policy under the average-reward criterion. A generative model provides samples from the nominal transition kernel, whereas policy performance is evaluated over (s,a)-rectangular total-variation uncertainty sets of radius at most σ. Let H0 and Hσ denote the nominal and robust optimal bias spans, respectively. We identify σ H0 as the perturbation scale separating high- and low-tolerance regimes. Our matching upper and lower bounds show that, up to logarithmic factors, the minimax total sample complexity is NSA asymp fracSAvarepsilon2begincases minH0,Hσ, & varepsilongtrsimσ H0, minH0,Hσ+σ Hσ2, & varepsilonlesssimσ H0. endcases Here S and A are the numbers of states and actions, and N is the number of samples per state-action pair. The sample complexity consists of a linear-span term that resembles the nominal AMDP results and a robustness-specific term that appears only in the low-tolerance regime. We attain these rates using reduction-based plug-in procedures that select the reduction---nominal or robust---and its discount factor: a span-informed procedure that makes these choices using known span parameters, and a span-agnostic procedure that calibrates both choices from data.