Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs
This work addresses the challenge of reinforcement learning in continuous spaces for researchers and practitioners, bridging gaps between discretization and regression approaches, but it is incremental as it builds on existing methods with a novel projection technique.
The paper tackles the problem of learning optimal policies in continuous-space Markov decision processes (MDPs) with smooth Bellman operators, achieving a rate-optimal sample complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-2-d/( u+1)})$ using a perturbed least-squares value iteration method with trigonometric polynomials.
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.