On the Sampling Problem for Kernel Quadrature

Briol, Francois-Xavier, Oates, Chris J., Cockayne, Jonathan and Girolami, Mark (2017) On the Sampling Problem for Kernel Quadrature. In Proceedings of the 34th International Conference on Machine Learning. vol. 70, pp. 586-595 .

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.

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Contributors

Author: Jonathan Cockayne

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