Efficient Policy Iteration for Robust Markov Decision Processes via Regularization

Robust Markov decision processes (MDPs) provide a general framework to model decision problems where the system dynamics are changing or only partially known. Recent work established the equivalence between \texttt{s} rectangular $L_p$ robust MDPs and regularized MDPs, and derived a regularized policy iteration scheme that enjoys the same level of efficiency as standard MDPs. However, there lacks a clear understanding of the policy improvement step. For example, we know the greedy policy can be stochastic but have little clue how each action affects this greedy policy. In this work, we focus on the policy improvement step and derive concrete forms for the greedy policy and the optimal robust Bellman operators. We find that the greedy policy is closely related to some combination of the top $k$ actions, which provides a novel characterization of its stochasticity. The exact nature of the combination depends on the shape of the uncertainty set. Furthermore, our results allow us to efficiently compute the policy improvement step by a simple binary search, without turning to an external optimization subroutine. Moreover, for $L_1, L_2$, and $L_\infty$ robust MDPs, we can even get rid of the binary search and evaluate the optimal robust Bellman operators exactly. Our work greatly extends existing results on solving \texttt{s}-rectangular $L_p$ robust MDPs via regularized policy iteration and can be readily adapted to sample-based model-free algorithms.