Deep neural networks, generic universal interpolation, and controlled ODEs

TL;DR

This paper links the expressiveness of deep neural networks to controllability of dynamical systems, introducing the universal interpolation property, and shows its generic nature and robustness even with random parameters, providing theoretical insights into training dynamics.

Contribution

It introduces the universal interpolation property for neural networks, connects expressiveness to controllability, and demonstrates robustness with untrained or randomly initialized parameters.

Findings

Universal interpolation is generic among neural networks.

Training with random parameters can still achieve expressiveness.

Minimal trainable parameters can suffice for neural ODEs without losing expressiveness.

Abstract

A recent paradigm views deep neural networks as discretizations of certain controlled ordinary differential equations, sometimes called neural ordinary differential equations. We make use of this perspective to link expressiveness of deep networks to the notion of controllability of dynamical systems. Using this connection, we study an expressiveness property that we call universal interpolation, and show that it is generic in a certain sense. The universal interpolation property is slightly weaker than universal approximation, and disentangles supervised learning on finite training sets from generalization properties. We also show that universal interpolation holds for certain deep neural networks even if large numbers of parameters are left untrained, and are instead chosen randomly. This lends theoretical support to the observation that training with random initialization can be successful even when most parameters are largely unchanged through the training. Our results also explore what a minimal amount of trainable parameters in neural ordinary differential equations could be without giving up on expressiveness.