Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs
<|reference_start|>Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs: 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.<|reference_end|>
@article{maran2024projection,
    title={Projection by Convolution: Optimal Sample Complexity for Reinforcement
  Learning in Continuous-Space MDPs},
    author={Davide Maran, Alberto Maria Metelli, Matteo Papini and Marcello
  Restelli},
    journal={arXiv preprint arXiv:2405.06363},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.06363},
    primaryClass={cs.LG cs.AI}
}