Mean Field Analysis of Neural Networks: A Central Limit Theorem

TL;DR

This paper establishes a central limit theorem for single-hidden-layer neural networks, characterizing their fluctuations around the mean-field limit as Gaussian processes in the asymptotic regime.

Contribution

It provides a rigorous proof of Gaussian fluctuations in neural networks with large hidden layers and training iterations, using stochastic analysis methods.

Findings

Neural network fluctuations are Gaussian in the large-scale limit.

The fluctuations satisfy a stochastic PDE.

The proof employs weak convergence and Sobolev space techniques.

Abstract

We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural network's fluctuations around its mean-field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. The proof relies upon weak convergence methods from stochastic analysis. In particular, we prove relative compactness for the sequence of processes and uniqueness of the limiting process in a suitable Sobolev space.