On the Sampling Problem for Kernel Quadrature

TL;DR

This paper investigates the sampling problem in Kernel Quadrature, revealing the importance of sampling distribution and proposing an adaptive tempering method to significantly improve integration accuracy.

Contribution

It introduces a novel adaptive tempering approach using sequential Monte Carlo to optimize sampling distribution in Kernel Quadrature.

Findings

Significant reduction in integration error, up to 4 orders of magnitude.

Sampling distribution critically affects Kernel Quadrature performance.

Proposed method outperforms standard sampling strategies.

Abstract

The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio $s/d$, where $s$ and $d$ encode the smoothness and dimension of the integrand. However, an empirical investigation reveals that the rate constant $C$ is highly sensitive to the distribution of the random points. In contrast to standard Monte Carlo integration, for which optimal importance sampling is well-understood, the sampling distribution that minimises $C$ for Kernel Quadrature does not admit a closed form. This paper argues that the practical choice of sampling distribution is an important open problem. One solution is considered; a novel automatic approach based on adaptive tempering and sequential Monte Carlo. Empirical results demonstrate a dramatic reduction in integration error of up to 4 orders of magnitude can be achieved with the proposed method.