Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910442060775424 |
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| author | Maran, Davide Metelli, Alberto Maria Papini, Matteo Restelli, Marcello |
| author_facet | Maran, Davide Metelli, Alberto Maria Papini, Matteo Restelli, Marcello |
| contents | 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~$\widetilde{\mathcal{O}}(ε^{-2-d/(ν+1)})$ sample complexity, where $d$ is the dimension of the state-action space and $ν$ the order of smoothness, recovers the state-of-the-art result of discretization approaches for the special case of Lipschitz MDPs $(ν=0)$. At the same time, for $ν\to\infty$, it recovers and greatly generalizes the $\mathcal{O}(ε^{-2})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs Maran, Davide Metelli, Alberto Maria Papini, Matteo Restelli, Marcello Machine Learning Artificial Intelligence 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~$\widetilde{\mathcal{O}}(ε^{-2-d/(ν+1)})$ sample complexity, where $d$ is the dimension of the state-action space and $ν$ the order of smoothness, recovers the state-of-the-art result of discretization approaches for the special case of Lipschitz MDPs $(ν=0)$. At the same time, for $ν\to\infty$, it recovers and greatly generalizes the $\mathcal{O}(ε^{-2})$ 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. |
| title | Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2405.06363 |