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arXiv · 2210.16286

A Functional-Space Mean-Field Theory of Partially-Trained Three-Layer Neural Networks

Zhengdao Chen·Eric Vanden-Eijnden·Joan Bruna
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Abstract

To understand the training dynamics of neural networks, prior studies have considered the mean-field limit of two-layer neural networks as the width tends to infinity, establishing theoretical guarantees for its convergence under gradient flow training as well as approximation and generalization capabilities. In this work, we study the infinite-width limit of a type of three-layer neural network where the first-layer weights are randomly sampled and untrained. To rigorously define the limiting model, we extend the mean-field theory by lifting the representation of neurons from Euclidean to functional spaces. This allows us to establish the mean-field training dynamics as a functional gradient flow with a time-varying kernel that remains positive-definite under suitable assumptions, thus proving a linear-rate convergence of its training loss. Furthermore, we define novel function spaces that contain the solutions obtained through the mean-field training dynamics and prove Rademacher complexity bounds for these spaces. Notably, our analysis applies to a range of scaling choices of the model, resulting in two distinct regimes of the mean-field limit that both exhibit feature learning through training.

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Zhengdao Chen, Eric Vanden-Eijnden, Joan Bruna. 2026-07-07. A Functional-Space Mean-Field Theory of Partially-Trained Three-Layer Neural Networks. https://arxiv.org/abs/2210.16286

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