# Mean Field Analysis of Deep Neural Networks

## TL;DR

This paper provides a rigorous mathematical analysis of multi-layer neural networks in the large-size and long-training limit, revealing their asymptotic behavior and convergence properties.

## Contribution

It introduces a novel limit analysis for deep neural networks, characterizing their behavior via deterministic equations and proving convergence to global minima.

## Key findings

- Limit neural network behavior described by integro-differential equations
- Convergence to global minimum under certain conditions
- Applicable to networks with any number of hidden layers

## Abstract

We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the multi-layer neural network output. The limit procedure is valid for any number of hidden layers and it naturally also describes the limiting behavior of the training loss. The ideas that we explore are to (a) take the limits of each hidden layer sequentially and (b) characterize the evolution of parameters in terms of their initialization. The limit satisfies a system of deterministic integro-differential equations. The proof uses methods from weak convergence and stochastic analysis. We show that, under suitable assumptions on the activation functions and the behavior for large times, the limit neural network recovers a global minimum (with zero loss for the objective function).