  In this paper, we study the non-asymptotic and asymptotic performances of the
optimal robust policy and value function of robust Markov Decision
Processes(MDPs), where the optimal robust policy and value function are solved
only from a generative model. While prior work focusing on non-asymptotic
performances of robust MDPs is restricted in the setting of the KL uncertainty
set and $(s,a)$-rectangular assumption, we improve their results and also
consider other uncertainty sets, including $L_1$ and $\chi^2$ balls. Our
results show that when we assume $(s,a)$-rectangular on uncertainty sets, the
sample 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 with
the choice of uncertainty sets and is generally larger than the case under
$(s,a)$-rectangular assumption. Moreover, we also show that the optimal robust
value function is asymptotic normal with a typical rate $\sqrt{n}$ under
$(s,a)$ and $s$-rectangular assumptions from both theoretical and empirical
perspectives.
