Sample Complexity of Nonparametric Off-Policy Evaluation on Low-Dimensional Manifolds using Deep NetworksDownload PDF

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22 Sept 2022, 12:30 (modified: 18 Nov 2022, 23:08)ICLR 2023 Conference Blind SubmissionReaders: Everyone
Keywords: RL theory, deep off-policy evaluation, neural network function approximation, manifold data
Abstract: We consider the off-policy evaluation problem of reinforcement learning using deep convolutional neural networks. We analyze the deep fitted Q-evaluation method for estimating the expected cumulative reward of a target policy, when the data are generated from an unknown behavior policy. We show that, by choosing network size appropriately, one can leverage any low-dimensional manifold structure in the Markov decision process and obtain a sample-efficient estimator without suffering from the curse of high data ambient dimensionality. Specifically, we establish a sharp error bound for fitted Q-evaluation, which depends on the intrinsic dimension of the state-action space, the smoothness of Bellman operator, and a function class-restricted $\chi^2$-divergence. It is noteworthy that the restricted $\chi^2$-divergence measures the behavior and target policies' {\it mismatch in the function space}, which can be small even if the two policies are not close to each other in their tabular forms. We also develop a novel approximation result for convolutional neural networks in Q-function estimation. Numerical experiments are provided to support our theoretical analysis.
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