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In this paper, we study the non-asymptotic and asymptotic performances of theoptimal robust policy and value function of robust Markov DecisionProcesses(MDPs), where the optimal robust policy and value function are solvedonly from a generative model. While prior work focusing on non-asymptoticperformances of robust MDPs is restricted in the setting of the KL uncertaintyset and $(s,a)$-rectangular assumption, we improve their results and alsoconsider other uncertainty sets, including $L_1$ and $\chi^2$ balls. Ourresults show that when we assume $(s,a)$-rectangular on uncertainty sets, thesample complexity is about$\widetilde{O}\left(\frac{|\mathcal{S}|^2|\mathcal{A}|}{\varepsilon^2\rho^2(1-\gamma)^4}\right)$.In addition, we extend our results from $(s,a)$-rectangular assumption to$s$-rectangular assumption. In this scenario, the sample complexity varies withthe choice of uncertainty sets and is generally larger than the case under$(s,a)$-rectangular assumption. Moreover, we also show that the optimal robustvalue function is asymptotic normal with a typical rate $\sqrt{n}$ under$(s,a)$ and $s$-rectangular assumptions from both theoretical and empiricalperspectives.

Towards Theoretical Understandings of Robust Markov Decision Processes: Sample Complexity and Asymptotics