Propagation of Chaos in One-hidden-layer Neural Networks beyond Logarithmic Time

Margalit Glasgow; Denny Wu; Joan Bruna

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arXiv · 2504.13110

Propagation of Chaos in One-hidden-layer Neural Networks beyond Logarithmic Time

Margalit Glasgow·Denny Wu·Joan Bruna

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Abstract

We study the approximation gap between the dynamics of a polynomial-width neural network and its infinite-width counterpart, both trained using projected gradient descent in the mean-field scaling regime. We demonstrate how to tightly bound this approximation gap through a differential equation governed by the mean-field dynamics. A key factor influencing the growth of this ODE is the local Hessian of each particle, defined as the derivative of the particle's velocity in the mean-field dynamics with respect to its position. We apply our results to the canonical feature learning problem of estimating a well-specified single-index model; we permit the information exponent to be arbitrarily large, leading to convergence times that grow polynomially in the ambient dimension $d$. We show that, due to a certain ``self-concordance'' property in these problems -- where the local Hessian of a particle is bounded by a constant times the particle's velocity -- polynomially many neurons are sufficient to closely approximate the mean-field dynamics throughout training.

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Margalit Glasgow, Denny Wu, Joan Bruna. 2025-04-17. Propagation of Chaos in One-hidden-layer Neural Networks beyond Logarithmic Time. https://arxiv.org/abs/2504.13110

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