Approximation spaces of deep neural networks

TL;DR

This paper investigates the approximation capabilities of deep neural networks, introducing the concept of approximation spaces to quantify their expressivity and examining how network features like skip connections, depth, and activation functions influence these spaces.

Contribution

It constructs approximation spaces for neural networks using classical approximation theory, showing their invariance to skip connections and relating them to Besov spaces for ReLU activations.

Findings

Deep networks can approximate functions with low Besov smoothness.

Skip connections do not alter the fundamental approximation spaces.

Deeper networks expand the class of functions that can be well approximated.

Abstract

We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. Using results from classical approximation theory, we show that this class can be endowed with a (quasi)-norm that makes it a linear function space, called approximation space. We establish that allowing the networks to have certain types of "skip connections" does not change the resulting approximation spaces. We also discuss the role of the network's nonlinearity (also known as activation function) on the resulting spaces, as well as the role of depth. For the popular ReLU nonlinearity and its powers, we relate the newly constructed spaces to classical Besov spaces. The established embeddings highlight that some functions of very low Besov smoothness can nevertheless be well approximated by neural networks, if these networks are sufficiently deep.