Instance-dependent $\ell_\infty$-bounds for policy evaluation in tabular reinforcement learning

TL;DR

This paper derives instance-dependent and data-dependent finite-sample bounds for policy evaluation in tabular reinforcement learning, providing minimax-optimal guarantees in the l_ norm.

Contribution

It introduces novel non-asymptotic bounds for l_ policy evaluation, including a robust variant, with analysis tailored to the specific MRP instance.

Findings

Bounds are minimax-optimal up to constants.

Data-dependent bounds can be computed from observations.

The leave-one-out decoupling technique is a key analytical tool.

Abstract

Markov reward processes (MRPs) are used to model stochastic phenomena arising in operations research, control engineering, robotics, and artificial intelligence, as well as communication and transportation networks. In many of these cases, such as in the policy evaluation problem encountered in reinforcement learning, the goal is to estimate the long-term value function of such a process without access to the underlying population transition and reward functions. Working with samples generated under the synchronous model, we study the problem of estimating the value function of an infinite-horizon, discounted MRP on finitely many states in the $\ell_\infty$-norm. We analyze both the standard plug-in approach to this problem and a more robust variant, and establish non-asymptotic bounds that depend on the (unknown) problem instance, as well as data-dependent bounds that can be evaluated based on the observations of state-transitions and rewards. We show that these approaches are minimax-optimal up to constant factors over natural sub-classes of MRPs. Our analysis makes use of a leave-one-out decoupling argument tailored to the policy evaluation problem, one which may be of independent interest.