Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs

TL;DR

This paper introduces a novel projection technique using harmonic analysis for reinforcement learning in continuous-space MDPs, achieving rate-optimal sample complexity that bridges discretization and low-rank approaches.

Contribution

It presents a simple perturbed least-squares value iteration method with a new harmonic analysis-based projection, achieving optimal sample complexity for smooth Bellman operators in continuous MDPs.

Findings

Achieves rate-optimal sample complexity for continuous-space MDPs.

Recovers and generalizes existing rates for Lipschitz and low-rank MDPs.

Bridges the gap between discretization and low-rank approaches.

Abstract

We consider the problem of learning an $\varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Given access to a generative model, we achieve rate-optimal sample complexity by performing a simple, \emph{perturbed} version of least-squares value iteration with orthogonal trigonometric polynomials as features. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our~$\widetilde{\mathcal{O}}(\epsilon^{-2-d/(\nu+1)})$ sample complexity, where $d$ is the dimension of the state-action space and $\nu$ the order of smoothness, recovers the state-of-the-art result of discretization approaches for the special case of Lipschitz MDPs $(\nu=0)$. At the same time, for $\nu\to\infty$, it recovers and greatly generalizes the $\mathcal{O}(\epsilon^{-2})$ rate of low-rank MDPs, which are more amenable to regression approaches. In this sense, our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.