Open Access
2022 Convergence rates of deep ReLU networks for multiclass classification
Thijs Bos, Johannes Schmidt-Hieber
Author Affiliations +
Electron. J. Statist. 16(1): 2724-2773 (2022). DOI: 10.1214/22-EJS2011


For classification problems, trained deep neural networks return probabilities of class memberships. In this work we study convergence of the learned probabilities to the true conditional class probabilities. More specifically we consider sparse deep ReLU network reconstructions minimizing cross-entropy loss in the multiclass classification setup. Interesting phenomena occur when the class membership probabilities are close to zero. Convergence rates are derived that depend on the near-zero behaviour via a margin-type condition.

Funding Statement

The research has been supported by the NWO/STAR grant 613.009.034b and the NWO Vidi grant VI.Vidi.192.021.


We thank the two referees and the editor for many helpful suggestions.


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Thijs Bos. Johannes Schmidt-Hieber. "Convergence rates of deep ReLU networks for multiclass classification." Electron. J. Statist. 16 (1) 2724 - 2773, 2022.


Received: 1 August 2021; Published: 2022
First available in Project Euclid: 12 April 2022

MathSciNet: MR4406243
zbMATH: 1493.62156
Digital Object Identifier: 10.1214/22-EJS2011

Primary: 62G05
Secondary: 63H30 , 68T07

Keywords: conditional class probabilities , Convergence rates , margin condition , Multiclass classification , ReLU networks

Vol.16 • No. 1 • 2022
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