Tractable Robust Markov Decision Processes

TL;DR

This paper characterizes the conditions under which robust Markov Decision Processes are computationally tractable, revealing that only certain rectangular uncertainty models are efficiently solvable without restrictions on rewards.

Contribution

It establishes that s-rectangular and sa-rectangular uncertainty sets are the only tractable models for RMDPs without reward restrictions, introducing a novel solvability property.

Findings

Only s-rectangular and sa-rectangular models are tractable.

Non-rectangular models are weakly tractable or equivalent to rectangular models under reward independence.

A new simultaneous solvability property underpins key RMDP properties.

Abstract

In this paper we investigate the tractability of robust Markov Decision Processes (RMDPs) under various structural assumptions on the uncertainty set. Surprisingly, we show that in all generality (i.e. without any assumption on the instantaneous rewards), s-rectangular and sa-rectangular uncertainty sets are the only models of uncertainty that are tractable. Our analysis also shows that existing non-rectangular models, including r-rectangular uncertainty and new generalizations, are only weakly tractable in that they require an additional structural assumption that the instantaneous rewards do not depend on the next state, and in this case they are equivalent to rectangular models, which severely undermines their significance and usefulness. Interestingly, our proof techniques rely on identifying a novel simultaneous solvability property, which we show is at the heart of several important properties of RMDPs, including the existence of stationary optimal policies and dynamic programming-based formulations. The simultaneous solvability property enables a unified approach to studying the tractability of all existing models of uncertainty, rectangular and non-rectangular alike.