On Kernel Derivative Approximation with Random Fourier Features

TL;DR

This paper investigates the approximation quality of Random Fourier Features (RFF) for kernel derivatives, demonstrating that existing guarantees can be exponentially improved, thus broadening RFF's applicability to derivative-based kernel tasks.

Contribution

It provides the first theoretical analysis of RFF approximation quality for kernel derivatives, showing that guarantees for kernel values extend to derivatives with exponential improvements.

Findings

Finite-sample guarantees for kernel derivatives are exponentially improved.

RFF approximation guarantees for derivatives match those for kernel values.

Theoretical tools from unbounded empirical process theory are employed.

Abstract

Random Fourier features (RFF) represent one of the most popular and wide-spread techniques in machine learning to scale up kernel algorithms. Despite the numerous successful applications of RFFs, unfortunately, quite little is understood theoretically on their optimality and limitations of their performance. Only recently, precise statistical-computational trade-offs have been established for RFFs in the approximation of kernel values, kernel ridge regression, kernel PCA and SVM classification. Our goal is to spark the investigation of optimality of RFF-based approximations in tasks involving not only function values but derivatives, which naturally lead to optimization problems with kernel derivatives. Particularly, in this paper, we focus on the approximation quality of RFFs for kernel derivatives and prove that the existing finite-sample guarantees can be improved exponentially in terms of the domain where they hold, using recent tools from unbounded empirical process theory. Our result implies that the same approximation guarantee is attainable for kernel derivatives using RFF as achieved for kernel values.