A Riemannian Mean Field Formulation for Two-layer Neural Networks with Batch Normalization

TL;DR

This paper models the training dynamics of two-layer neural networks with batch normalization as a Wasserstein gradient flow on a Riemannian manifold, providing new theoretical insights into their behavior.

Contribution

It introduces a Riemannian mean-field formulation for networks with batch normalization and analyzes their training dynamics in the infinite-width limit.

Findings

BN changes the metric in parameter space.

Training dynamics follow Wasserstein gradient flow.

Theoretical results on well-posedness and convergence.

Abstract

The training dynamics of two-layer neural networks with batch normalization (BN) is studied. It is written as the training dynamics of a neural network without BN on a Riemannian manifold. Therefore, we identify BN's effect of changing the metric in the parameter space. Later, the infinite-width limit of the two-layer neural networks with BN is considered, and a mean-field formulation is derived for the training dynamics. The training dynamics of the mean-field formulation is shown to be the Wasserstein gradient flow on the manifold. Theoretical analysis are provided on the well-posedness and convergence of the Wasserstein gradient flow.