Policy Gradient Algorithms for Robust MDPs with Non-Rectangular Uncertainty Sets

TL;DR

This paper introduces new policy gradient algorithms for robust MDPs with non-rectangular uncertainty sets, providing the first complete solution scheme with global guarantees and demonstrating favorable numerical performance.

Contribution

It develops the first comprehensive algorithms for robust MDPs with non-rectangular uncertainty sets, including evaluation, policy gradient, and actor-critic methods with theoretical guarantees.

Findings

Algorithms outperform state-of-the-art methods in experiments

Proposed methods provide global optimality guarantees

Approximation error scales with non-rectangularity measure

Abstract

We propose policy gradient algorithms for robust infinite-horizon Markov decision processes (MDPs) with non-rectangular uncertainty sets, thereby addressing an open challenge in the robust MDP literature. Indeed, uncertainty sets that display statistical optimality properties and make optimal use of limited data often fail to be rectangular. Unfortunately, the corresponding robust MDPs cannot be solved with dynamic programming techniques and are in fact provably intractable. We first present a randomized projected Langevin dynamics algorithm that solves the robust policy evaluation problem to global optimality but is inefficient. We also propose a deterministic policy gradient method that is efficient but solves the robust policy evaluation problem only approximately, and we prove that the approximation error scales with a new measure of non-rectangularity of the uncertainty set. Finally, we describe an actor-critic algorithm that finds an $\epsilon$-optimal solution for the robust policy improvement problem in $\mathcal{O}(1/\epsilon^4)$ iterations. We thus present the first complete solution scheme for robust MDPs with non-rectangular uncertainty sets offering global optimality guarantees. Numerical experiments show that our algorithms compare favorably against state-of-the-art methods.