Beyond discounted returns: Robust Markov decision processes with average and Blackwell optimality

TL;DR

This paper advances the theory of Robust Markov Decision Processes by exploring average and Blackwell optimality, revealing new existence results, and proposing algorithms, with implications for decision-making under uncertainty.

Contribution

It provides foundational results for RMDPs beyond discounted returns, including existence conditions for average and Blackwell optimal policies, and introduces algorithms leveraging stochastic game connections.

Findings

Average optimal policies can be stationary and deterministic for sa-rectangular RMDPs.

Average optimal policies may not exist or may need to be history-dependent for s-rectangular RMDPs.

Epsilon-Blackwell optimal policies always exist under certain conditions.

Abstract

Robust Markov Decision Processes (RMDPs) are a widely used framework for sequential decision-making under parameter uncertainty. RMDPs have been extensively studied when the objective is to maximize the discounted return, but little is known for average optimality (optimizing the long-run average of the rewards obtained over time) and Blackwell optimality (remaining discount optimal for all discount factors sufficiently close to ). In this paper, we prove several foundational results for RMDPs beyond the discounted return. We show that average optimal policies can be chosen stationary and deterministic for sa-rectangular RMDPs but, perhaps surprisingly, we show that for s-rectangular RMDPs average optimal policies may not exist, and if they exist, may need to be history-dependent (Markovian). We also study Blackwell optimality for sa-rectangular RMDPs, where we show that $\epsilon$-Blackwell optimal policies always exist, although Blackwell optimal policies may not exist. We also provide a sufficient condition for their existence, which encompasses virtually any examples from the literature. We then discuss the connection between average and Blackwell optimality, and we describe several algorithms to compute the optimal average return. Interestingly, our approach leverages the connections between RMDPs and stochastic games. Overall, our paper emphasizes the superior practical properties of distance-based sa-rectangular models over s-rectangular models for average and Blackwell optimality.