Universal Approximation with Deep Narrow Networks

TL;DR

This paper proves that deep neural networks with narrow width can approximate any continuous function on compact sets, extending classical results to a dual scenario and including polynomial activation functions, highlighting differences from wide shallow networks.

Contribution

It establishes the universal approximation property for deep narrow networks with arbitrary depth and specific width, covering all practical activation functions including polynomials.

Findings

Deep narrow networks are dense in continuous functions on compact sets.

The result applies to all practical activation functions, including polynomials.

Extensions include non-differentiable activations and reduced width for most functions.

Abstract

The classical Universal Approximation Theorem holds for neural networks of arbitrary width and bounded depth. Here we consider the natural `dual' scenario for networks of bounded width and arbitrary depth. Precisely, let $n$ be the number of inputs neurons, $m$ be the number of output neurons, and let $\rho$ be any nonaffine continuous function, with a continuous nonzero derivative at some point. Then we show that the class of neural networks of arbitrary depth, width $n + m + 2$, and activation function $\rho$, is dense in $C(K; \mathbb{R}^m)$ for $K \subseteq \mathbb{R}^n$ with $K$ compact. This covers every activation function possible to use in practice, and also includes polynomial activation functions, which is unlike the classical version of the theorem, and provides a qualitative difference between deep narrow networks and shallow wide networks. We then consider several extensions of this result. In particular we consider nowhere differentiable activation functions, density in noncompact domains with respect to the $L^p$-norm, and how the width may be reduced to just $n + m + 1$ for `most' activation functions.