On the Approximation Properties of Random ReLU Features

TL;DR

This paper investigates the approximation capabilities of random ReLU features, establishing their universality, limitations in approximating complex functions, and comparing their performance to neural networks and other random features.

Contribution

It proves the universality of the RKHS induced by random ReLU features and demonstrates depth separation, showing the advantage of multi-layer networks over shallow models.

Findings

Random ReLU features induce a universal RKHS.

Composite functions are harder to approximate with shallow models.

3-layer neural networks outperform random ReLU features on complex functions.

Abstract

We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes in neural networks. The universality result implies that the random ReLU features method is a universally consistent learning algorithm. We prove that despite the universality of the RKHS induced by the random ReLU features, composition of functions in it generates substantially more complicated functions that are harder to approximate than those functions simply in the RKHS. We also prove that such composite functions can be efficiently approximated by multi-layer ReLU networks with bounded weights. This depth separation result shows that the random ReLU features models suffer from the same weakness as that of shallow models. We show in experiments that the performance of random ReLU features is comparable to that of random Fourier features and, in general, has a lower computational cost. We also demonstrate that when the target function is the composite function as described in the depth separation theorem, 3-layer neural networks indeed outperform both random ReLU features and 2-layer neural networks.