NEURAL NETWORK APPROXIMATION AND ESTIMATION OF CLASSIFIERS WITH CLASSIFICATION BOUNDARY IN A BARRON CLASS

• 成果类型: Article
• 署名作者: Caragea, Andrei; Petersen, Philipp; Voigtlaender, Felix
• 署名单位: University of Vienna
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1884
• 发表日期: 2023
• 页码: 3039-3079
• 关键词: dimensionality; curse; overcome; rates

摘要

We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guarantees for empirical risk minimization us-ing networks of a suitable size, depending on the number of training samples available. The obtained approximation and estimation rates are independent of the dimension of the input, showing that the curse of dimensionality can be overcome in this setting; in fact, the input dimension only enters in the form of a polynomial factor. Regarding the regularity of the target classification function, we assume the interfaces between the different classes to be locally of Barron-type. We complement our results by studying the relations between various Barron-type spaces that have been proposed in the literature. These spaces differ substantially more from each other than the current literature suggests.