Distributionally Robust Average-Reward Reinforcement Learning: Finite-Sample Guarantees under Weak Communication

We study distributionally robust reinforcement learning (DR-RL) in the average-reward setting under weak communication. Our main result provides finite-sample guarantees for estimating the robust optimal average reward and learning a near-optimal policy, covering both SA-rectangular and S-rectangular structures with divergence-based and distance-based uncertainty sets. Specifically, for Kullback--Leibler and fk-divergence balls, we establish explicit radius conditions under which the robust average-reward Bellman equation admits a constant-gain solution, while for total variation and Wasserstein balls, any positive radius suffices without requiring the nominal MDP to be weakly communicating. Our algorithm is prior-knowledge-free and achieves sample complexities of widetilde O(|S||A|pwedge-1operatornameSpan2(uδast)ε-2) for estimating the robust optimal average reward and widetilde O(|S||A|pwedge-2operatornameSpan2(uδast)ε-2) for learning an ε-optimal policy. Here, pwedge is the smallest positive nominal transition probability and uδast is a robust optimal bias function. We further provide an almost-tight explicit upper bound on operatornameSpan(uδast). Finally, we validate the predicted n-1/2 convergence rate through numerical experiments.