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

• URL: http://arxiv.org/abs/2206.02887v1
• Date: Mon, 6 Jun 2022 20:25:20 GMT
• Title: Sample Complexity of Nonparametric Off-Policy Evaluation on Low-Dimensional Manifolds using Deep Networks
• Authors: Xiang Ji, Minshuo Chen, Mengdi Wang, Tuo Zhao
• Abstract summary: We consider the off-policy evaluation problem of reinforcement learning using deep neural networks. We show that, by choosing network size appropriately, one can leverage the low-dimensional manifold structure in the Markov decision process.
• Score: 71.95722100511627
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the off-policy evaluation problem of reinforcement learning using deep 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 the low-dimensional manifold structure in the Markov decision process and obtain a sample-efficient estimator without suffering from the curse of high representation dimensionality. Specifically, we establish a sharp error bound for the fitted Q-evaluation that depends on the intrinsic low dimension, the smoothness of the state-action space, 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. Numerical experiments are provided to support our theoretical analysis.

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