On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions

TL;DR

This paper establishes a theoretical connection between kernel quadrature rules and random feature expansions, providing bounds on sample complexity based on eigenvalues, and extends results to function approximation and learning guarantees.

Contribution

It demonstrates the equivalence between kernel quadrature and random features, deriving bounds on sample size based on eigenvalues, and extends analysis to broader function approximation and learning contexts.

Findings

Upper bounds on sample size depend on eigenvalues of the integral operator.

Lower bounds are valid for any set of points, regardless of distribution.

Improves the understanding of random features needed for learning with Lipschitz losses.

Abstract

We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a theoretical analysis of the number of required samples for a given approximation error, leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms. In particular, we show that the upper bound may be obtained from independent and identically distributed samples from a specific non-uniform distribution, while the lower bound if valid for any set of points. Applying our results to kernel-based quadrature, while our results are fairly general, we recover known upper and lower bounds for the special cases of Sobolev spaces. Moreover, our results extend to the more general problem of full function approximations (beyond simply computing an integral), with results in L2- and L$\infty$-norm that match known results for special cases. Applying our results to random features, we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with Lipschitz-continuous losses.