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On the Sampling Problem for Kernel Quadrature

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Title: On the Sampling Problem for Kernel Quadrature
Authors: Briol, F-X
Oates, CJ
Cockayne, J
Chen, WY
Girolami, M
Item Type: Conference Paper
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.
Issue Date: 1-Jan-2017
Date of Acceptance: 14-Apr-2017
URI: http://hdl.handle.net/10044/1/53203
Publisher: PMLR
Start Page: 586
End Page: 595
Journal / Book Title: Proceedings of the 34th International Conference on Machine Learning
Volume: 70
Copyright Statement: Copyright 2017 by the author(s).
Conference Name: International Conference on Machine Learning (ICML)
Keywords: stat.ML
Publication Status: Published
Conference Place: Sydney, Australia
Appears in Collections:Mathematics
Faculty of Natural Sciences