Towards Minimax Optimality of Model-based Robust Reinforcement Learning

TL;DR

This paper establishes near-optimal sample complexity bounds for model-based robust reinforcement learning in discounted MDPs with $L_p$-ball uncertainty sets, improving previous bounds and matching non-robust lower bounds under certain conditions.

Contribution

It provides the first minimax optimal sample complexity bounds for robust RL with $L_p$-ball uncertainty sets, matching non-robust bounds when the uncertainty is small.

Findings

Sample complexity of $ ilde{O}(H^4 |S||A| / \\epsilon^2)$ for general case.

Improved sample complexity to $ ilde{O}(H^3 |S||A| / \\epsilon^2)$ when uncertainty is small.

Results recover non-robust lower bounds and establish robust lower bounds under specific conditions.

Abstract

We study the sample complexity of obtaining an $\epsilon$-optimal policy in \emph{Robust} discounted Markov Decision Processes (RMDPs), given only access to a generative model of the nominal kernel. This problem is widely studied in the non-robust case, and it is known that any planning approach applied to an empirical MDP estimated with $\tilde{\mathcal{O}}(\frac{H^3 \mid S \mid\mid A \mid}{\epsilon^2})$ samples provides an $\epsilon$-optimal policy, which is minimax optimal. Results in the robust case are much more scarce. For $sa$- (resp $s$-)rectangular uncertainty sets, the best known sample complexity is $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid^2\mid A \mid}{\epsilon^2})$ (resp. $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid^2\mid A \mid^2}{\epsilon^2})$), for specific algorithms and when the uncertainty set is based on the total variation (TV), the KL or the Chi-square divergences. In this paper, we consider uncertainty sets defined with an $L_p$-ball (recovering the TV case), and study the sample complexity of \emph{any} planning algorithm (with high accuracy guarantee on the solution) applied to an empirical RMDP estimated using the generative model. In the general case, we prove a sample complexity of $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid\mid A \mid}{\epsilon^2})$ for both the $sa$- and $s$-rectangular cases (improvements of $\mid S \mid$ and $\mid S \mid\mid A \mid$ respectively). When the size of the uncertainty is small enough, we improve the sample complexity to $\tilde{\mathcal{O}}(\frac{H^3 \mid S \mid\mid A \mid }{\epsilon^2})$, recovering the lower-bound for the non-robust case for the first time and a robust lower-bound when the size of the uncertainty is small enough.