A mean-field limit for certain deep neural networks

TL;DR

This paper establishes a mean-field limit for deep neural networks with multiple layers trained by stochastic gradient descent, providing a rigorous mathematical framework to describe their training dynamics as the number of neurons per layer grows large.

Contribution

It extends previous mean-field analyses from shallow to deep networks with multiple layers, rigorously deriving the limiting behavior and proving existence and uniqueness of the associated McKean-Vlasov equations.

Findings

Network weights are approximated by ideal particles described by a mean-field model.

The mean-field limit accurately captures the evolution of deep neural networks during training.

Rigorous proof of existence and uniqueness for the McKean-Vlasov problem in this context.

Abstract

Understanding deep neural networks (DNNs) is a key challenge in the theory of machine learning, with potential applications to the many fields where DNNs have been successfully used. This article presents a scaling limit for a DNN being trained by stochastic gradient descent. Our networks have a fixed (but arbitrary) number $L\geq 2$ of inner layers; $N\gg 1$ neurons per layer; full connections between layers; and fixed weights (or "random features" that are not trained) near the input and output. Our results describe the evolution of the DNN during training in the limit when $N\to +\infty$, which we relate to a mean field model of McKean-Vlasov type. Specifically, we show that network weights are approximated by certain "ideal particles" whose distribution and dependencies are described by the mean-field model. A key part of the proof is to show existence and uniqueness for our McKean-Vlasov problem, which does not seem to be amenable to existing theory. Our paper extends previous work on the $L=1$ case by Mei, Montanari and Nguyen; Rotskoff and Vanden-Eijnden; and Sirignano and Spiliopoulos. We also complement recent independent work on $L>1$ by Sirignano and Spiliopoulos (who consider a less natural scaling limit) and Nguyen (who nonrigorously derives similar results).