Distributionally Robust Markov Decision Processes and their Connection to Risk Measures

TL;DR

This paper develops a framework for robust Markov Decision Processes with unbounded costs, connecting them to risk measures, and provides algorithms and existence results for optimal policies under various assumptions.

Contribution

It introduces a novel robust MDP formulation with a Stackelberg game perspective, derives value iteration algorithms, and links robust optimization to coherent risk measures.

Findings

Existence of deterministic optimal policies for both players.

Robust cost iteration and value iteration algorithms are derived.

Under convexity, interchange of supremum and infimum is justified.

Abstract

We consider robust Markov Decision Processes with Borel state and action spaces, unbounded cost and finite time horizon. Our formulation leads to a Stackelberg game against nature. Under integrability, continuity and compactness assumptions we derive a robust cost iteration for a fixed policy of the decision maker and a value iteration for the robust optimization problem. Moreover, we show the existence of deterministic optimal policies for both players. This is in contrast to classical zero-sum games. In case the state space is the real line we show under some convexity assumptions that the interchange of supremum and infimum is possible with the help of Sion's minimax Theorem. Further, we consider the problem with special ambiguity sets. In particular we are able to derive some cases where the robust optimization problem coincides with the minimization of a coherent risk measure. In the final section we discuss two applications: A robust LQ problem and a robust problem for managing regenerative energy.