Implicit regularization of deep residual networks towards neural ODEs

TL;DR

This paper demonstrates that deep residual networks implicitly regularize towards neural ODEs during training with gradient flow, providing a mathematical link between discrete networks and their continuous counterparts.

Contribution

It establishes that residual networks initialized as discretizations of neural ODEs remain close to these ODEs throughout training, with convergence results under certain conditions.

Findings

Residual networks initialized as neural ODE discretizations stay close during training.

Gradient flow converges to a global minimum under Polyak-Lojasiewicz condition.

Numerical experiments support the theoretical results.

Abstract

Residual neural networks are state-of-the-art deep learning models. Their continuous-depth analog, neural ordinary differential equations (ODEs), are also widely used. Despite their success, the link between the discrete and continuous models still lacks a solid mathematical foundation. In this article, we take a step in this direction by establishing an implicit regularization of deep residual networks towards neural ODEs, for nonlinear networks trained with gradient flow. We prove that if the network is initialized as a discretization of a neural ODE, then such a discretization holds throughout training. Our results are valid for a finite training time, and also as the training time tends to infinity provided that the network satisfies a Polyak-Lojasiewicz condition. Importantly, this condition holds for a family of residual networks where the residuals are two-layer perceptrons with an overparameterization in width that is only linear, and implies the convergence of gradient flow to a global minimum. Numerical experiments illustrate our results.