Distributionally Robust Average-Reward Reinforcement Learning: Finite-Sample Guarantees under Weakly Communicating

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 $f_k$-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 is weakly communicating. Our algorithm is prior-knowledge-free and achieves sample complexities of $\tilde{O}(|\mathcal{S}||\mathcal{A}|p_{\wedge }^{-1}{\mathrm{Span}}^2(u_{\delta }^{\ast }){ϵ}^{-2})$ for estimating the robust optimal average reward and $\tilde{O}(|\mathcal{S}||\mathcal{A}|p_{\wedge }^{-2}{\mathrm{Span}}^2(u_{\delta }^{\ast }){ϵ}^{-2})$ for learning an $ϵ$-optimal policy. Here, $p_{\wedge }$ is the smallest positive nominal transition probability and $u_{\delta }^{\ast }$ is a robust optimal bias function. We further provide an almost-tight explicit upper bound on $\mathrm{Span}(u_{\delta }^{\ast })$. Finally, we validate the predicted $n^{-1/2}$ convergence rate through numerical experiments.