Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory

TL;DR

This paper demonstrates that deep residual neural networks can universally approximate any continuous function by linking their capabilities to nonlinear control theory and specific properties of activation functions.

Contribution

It provides a general sufficient condition based on control theory for residual networks to achieve universal approximation, applicable to simple architectures with minimal weight constraints.

Findings

Residual networks can approximate any continuous function on compact sets.

Activation functions satisfying certain differential equations are key to universal approximation.

Simple architectures with two-value weights are sufficient for universal approximation.

Abstract

In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a general sufficient condition for a residual network to have the power of universal approximation by asking the activation function, or one of its derivatives, to satisfy a quadratic differential equation. Many activation functions used in practice satisfy this assumption, exactly or approximately, and we show this property to be sufficient for an adequately deep neural network with $n+1$ neurons per layer to approximate arbitrarily well, on a compact set and with respect to the supremum norm, any continuous function from $\mathbb{R}^n$ to $\mathbb{R}^n$. We further show this result to hold for very simple architectures for which the weights only need to assume two values. The first key technical contribution consists of relating the universal approximation problem to controllability of an ensemble of control systems corresponding to a residual network and to leverage classical Lie algebraic techniques to characterize controllability. The second technical contribution is to identify monotonicity as the bridge between controllability of finite ensembles and uniform approximability on compact sets.