Performance analysis of greedy algorithms for minimising a Maximum Mean Discrepancy

摘要

Abstract We analyse the performance of several iterative algorithms for the quantisation of a probability measure μdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mu $$end{document}, based on the minimisation of a Maximum Mean Discrepancy (MMD). Our analysis includes kernel herding, greedy MMD minimisation and Sequential Bayesian Quadrature (SBQ). We show that the finite-sample-size approximation error, measured by the MMD, decreases as 1/n for SBQ and also for kernel herding and greedy MMD minimisation when using a suitable step-size sequence. The upper bound on the approximation error is slightly better for SBQ, but the other methods are significantly faster, with a computational cost that increases only linearly with the number of points selected. This is illustrated by two numerical examples, with the target measure μdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mu $$end{document} being uniform (a space-filling design application) and with μdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mu $$end{document} a Gaussian mixture. They suggest that the bounds derived in the paper are overly pessimistic, in particular for SBQ. The sources of this pessimism are identified but seem difficult to counter.