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Modern Stochastics: Theory and Applications


A quantitative functional central limit theorem for shallow neural networks
Volume 11, Issue 1 (2024), pp. 85–108
Pub. online: 28 November 2023      Type: Research Article      Open accessOpen Access
Received
5 July 2023
Revised
23 October 2023
Accepted
15 November 2023
Published
28 November 2023

Abstract

We prove a quantitative functional central limit theorem for one-hidden-layer neural networks with generic activation function. Our rates of convergence depend heavily on the smoothness of the activation function, and they range from logarithmic for nondifferentiable nonlinearities such as the ReLu to $\sqrt{n}$ for highly regular activations. Our main tools are based on functional versions of the Stein–Malliavin method; in particular, we rely on a quantitative functional central limit theorem which has been recently established by Bourguin and Campese [Electron. J. Probab. 25 (2020), 150].

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© 2024 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Quantitative functional central limit theorem Wiener-chaos expansions neural networks Gaussian processes

MSC2010
60F17 68T07 60G60

Funding
The work was partially supported by the MUR Excellence Department Project MatMod@TOV awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. We also acknowledge financial support from the MUR 2022 PRIN project GRAFIA, project code 202284Z9E4, the INdAM group GNAMPA and the PNRR CN1 High Performance Computing, Spoke 3.

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