Optimal nonparametric inference via deep neural network

Document Type

Article

Publication Date

1-15-2022

Abstract

Deep neural network is a state-of-art method in modern science and technology. Much statistical literature have been devoted to understanding its performance in nonparametric estimation, whereas the results are suboptimal due to a redundant logarithmic sacrifice. In this paper, we show that such log-factors are not necessary. We derive upper bounds for the L2 minimax risk in nonparametric estimation. Sufficient conditions on network architectures are provided such that the upper bounds become optimal (without log-sacrifice). Our proof relies on an explicitly constructed network estimator based on tensor product B-splines. We also derive asymptotic distributions for the constructed network and a relating hypothesis testing procedure. The testing procedure is further proved as minimax optimal under suitable network architectures.

Identifier

85112785555 (Scopus)

Publication Title

Journal of Mathematical Analysis and Applications

External Full Text Location

https://doi.org/10.1016/j.jmaa.2021.125561

e-ISSN

10960813

ISSN

0022247X

Issue

2

Volume

505

Grant

DMS-1764280

Fund Ref

National Science Foundation

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