Open Access
2023 Optimal convergence rates of deep neural networks in a classification setting
Joseph T. Meyer
Author Affiliations +
Electron. J. Statist. 17(2): 3613-3659 (2023). DOI: 10.1214/23-EJS2187


We establish optimal convergence rates up to a log factor for a class of deep neural networks in a classification setting under a restraint sometimes referred to as the Tsybakov noise condition. We construct classifiers based on empirical risk minimization in a general setting where the boundary of the Bayes rule can be approximated well by neural networks. Corresponding rates of convergence are proven with respect to the misclassification error using an additional condition that acts as a requirement for the “correct noise exponent”. It is then shown that these rates are optimal in the minimax sense. For other estimation procedures, similar convergence rates have been established. Our first main contribution is to prove that the rates are optimal under the additional condition. Secondly, our main theorem establishes almost optimal rates in a generalized setting. We use this to show optimal rates which circumvent the curse of dimensionality.


First and foremost, I would like to thank Enno Mammen for supporting me with some helpful comments and inspiring insights during the creation of this paper. Additionally, many thanks goes to Munir Hiabu for assisting with comments during the final stages of the working process.


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Joseph T. Meyer. "Optimal convergence rates of deep neural networks in a classification setting." Electron. J. Statist. 17 (2) 3613 - 3659, 2023.


Received: 1 June 2022; Published: 2023
First available in Project Euclid: 7 December 2023

Digital Object Identifier: 10.1214/23-EJS2187

Primary: 62C20 , 62G05

Keywords: ‎classification‎ , Deep neural networks , Tsybakov noise condition

Vol.17 • No. 2 • 2023
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