Optimally-Weighted Herding is Bayesian Quadrature

TL;DR

This paper reveals that kernel herding minimizes the Bayesian quadrature posterior variance, and introduces an optimally-weighted herding method that outperforms existing approaches with faster convergence.

Contribution

It establishes a theoretical link between kernel herding and Bayesian quadrature, proposing an optimally-weighted herding method with superior performance.

Findings

Sequential Bayesian quadrature outperforms unweighted herding.

Achieves convergence rate faster than O(1/N).

Provides an upper bound on empirical Bayesian quadrature error.

Abstract

Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves performance superior to any other weighted herding method. We demonstrate empirically a rate of convergence faster than O(1/N). Our results also imply an upper bound on the empirical error of the Bayesian quadrature estimate.