Optimal Rates for Random Fourier Features

TL;DR

This paper provides a comprehensive theoretical analysis of Random Fourier Features, establishing optimal approximation guarantees in various norms and extending the analysis to derivatives of kernels, enhancing understanding of their effectiveness.

Contribution

It offers the first detailed finite-sample theoretical bounds on RFF approximation quality, including for derivatives, improving understanding of their performance and limitations.

Findings

Optimal uniform norm approximation guarantees for RFFs.

Extension of guarantees to $L^r$ norms for $1 \,\le r < \infty$.

Theoretical bounds for RFF approximation of kernel derivatives.

Abstract

Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show good versatility, they are computationally intensive and have poor scalability to large data as they require operations on Gram matrices. In order to mitigate this serious computational limitation, recently randomized constructions have been proposed in the literature, which allow the application of fast linear algorithms. Random Fourier features (RFF) are among the most popular and widely applied constructions: they provide an easily computable, low-dimensional feature representation for shift-invariant kernels. Despite the popularity of RFFs, very little is understood theoretically about their approximation quality. In this paper, we provide a detailed finite-sample theoretical analysis about the approximation quality of RFFs by (i) establishing optimal (in terms of the RFF dimension, and growing set size) performance guarantees in uniform norm, and (ii) presenting guarantees in $L^r$ ($1\le r<\infty$) norms. We also propose an RFF approximation to derivatives of a kernel with a theoretical study on its approximation quality.