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On the Sampling Problem for Kernel Quadrature
File | Description | Size | Format | |
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![]() | Accepted version | 1.28 MB | Adobe PDF | View/Open |
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 cs.LG math.NA stat.CO |
Publication Status: | Published |
Conference Place: | Sydney, Australia |
Appears in Collections: | Mathematics Faculty of Natural Sciences |