Deep Nonparametric Regression on Approximate Manifolds: Nonasymptotic Error Bounds with Polynomial Prefactors

Guohao Shen, Jian Huang (Corresponding Author), Yuling Jiao, Yuanyuan Lin

Research output: Journal article publicationJournal articleAcademic researchpeer-review

7 Citations (Scopus)

Abstract

We study the properties of nonparametric least squares regression using deep neural networks. We derive nonasymptotic upper bounds for the excess risk of the empirical risk minimizer of feedforward deep neural regression. Our error bounds achieve minimax optimal rate and improve over the existing ones in the sense that they depend polynomially on the dimension of the predictor, instead of exponentially on dimension. We show that the neural regression estimator can circumvent the curse of dimensionality under the assumption that the predictor is supported on an approximate low-dimensional manifold or a set with low Minkowski dimension. We also establish the optimal convergence rate under the exact manifold support assumption. We investigate how the prediction error of the neural regression estimator depends on the structure of neural networks and propose a notion of network relative efficiency between two types of neural networks, which provides a quantitative measure for evaluating the relative merits of different network structures. To establish these results, we derive a novel approximation error bound for the Hölder smooth functions using ReLU activated neural networks, which may be of independent interest. Our results are derived under weaker assumptions on the data distribution and the neural network structure than those in the existing literature.
Original languageEnglish
Pages (from-to)691-716
Number of pages26
JournalAnnals of Statistics
Volume51
Issue number2
DOIs
Publication statusPublished - Apr 2023

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