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

arXiv eprints
 Pub Date:
 July 2020
 DOI:
 10.48550/arXiv.2007.06007
 arXiv:
 arXiv:2007.06007
 Bibcode:
 2020arXiv200706007T
 Keywords:

 Computer Science  Machine Learning;
 Electrical Engineering and Systems Science  Systems and Control;
 Mathematics  Optimization and Control;
 Statistics  Machine Learning
 EPrint:
 Sejun Park and Geonho Hwang brought to our atention a mistake in the proof of Theorem 5.1. This mistake is corrected in this version with the consequence of increasing the number of neurons per layer from n+1 to 2n+1