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

• URL: http://arxiv.org/abs/2405.06363v1
• Date: Fri, 10 May 2024 09:58:47 GMT
• Title: Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs
• Authors: Davide Maran, Alberto Maria Metelli, Matteo Papini, Marcello Restelli,
• Abstract summary: 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. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.
• Score: 56.237917407785545
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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