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International Journal for Uncertainty Quantification
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ISSN Imprimer: 2152-5080
ISSN En ligne: 2152-5099

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International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.v2.i1.30
pages 21-38

DISTANCES AND DIAMETERS IN CONCENTRATION INEQUALITIES: FROM GEOMETRY TO OPTIMAL ASSIGNMENT OF SAMPLING RESOURCES

Tim Sullivan
Freie Universität Berlin
Houman Owhadi
Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

RÉSUMÉ

This note reviews, compares and contrasts three notions of "distance" or "size" that arise often in concentration-of-measure inequalities. We review Talagrand′s convex distance and McDiarmid′s diameter, and consider in particular the normal distance on a topological vector space 𝒳, which corresponds to the method of Chernoff bounds, and is in some sense "natural" with respect to the duality structure on 𝒳. We show that, notably, with respect to this distance, concentration inequalities on the tails of linear, convex, quasiconvex and measurable functions on 𝒳 are mutually equivalent. We calculate the normal distances that correspond to families of Gaussian and of bounded random variables in ℝN, and to functions of N empirical means. As an application, we consider the problem of estimating the confidence that one can have in a quantity of interest that depends upon many empirical—as opposed to exact—means and show how the normal distance leads to a formula for the optimal assignment of sampling resources.