Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees

TL;DR

This paper analyzes the statistical and approximation properties of random Fourier features in kernel ridge regression, providing bounds, optimal sampling strategies, and demonstrating potential speedups and limitations of the method.

Contribution

It offers tight spectral approximation bounds, links these to statistical guarantees, and proposes an improved sampling scheme for Gaussian kernels.

Findings

Random Fourier features can speed up kernel ridge regression.

Sampling from the leverage function improves performance.

The proposed method outperforms standard random Fourier features in low-dimensional settings.

Abstract

Random Fourier features is one of the most popular techniques for scaling up kernel methods, such as kernel ridge regression. However, despite impressive empirical results, the statistical properties of random Fourier features are still not well understood. In this paper we take steps toward filling this gap. Specifically, we approach random Fourier features from a spectral matrix approximation point of view, give tight bounds on the number of Fourier features required to achieve a spectral approximation, and show how spectral matrix approximation bounds imply statistical guarantees for kernel ridge regression. Qualitatively, our results are twofold: on the one hand, we show that random Fourier feature approximation can provably speed up kernel ridge regression under reasonable assumptions. At the same time, we show that the method is suboptimal, and sampling from a modified distribution in Fourier space, given by the leverage function of the kernel, yields provably better performance. We study this optimal sampling distribution for the Gaussian kernel, achieving a nearly complete characterization for the case of low-dimensional bounded datasets. Based on this characterization, we propose an efficient sampling scheme with guarantees superior to random Fourier features in this regime.