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On the optimality of sample-based estimates of the expectation of the empirical minimizer

Bartlett, Peter L., Mendelson, Shahar, & Philips, Petra (2010) On the optimality of sample-based estimates of the expectation of the empirical minimizer. ESAIM : Probability and Statistics, 14(Jan), pp. 315-337.

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Abstract

We study sample-based estimates of the expectation of the function produced by the empirical minimization algorithm. We investigate the extent to which one can estimate the rate of convergence of the empirical minimizer in a data dependent manner. We establish three main results. First, we provide an algorithm that upper bounds the expectation of the empirical minimizer in a completely data-dependent manner. This bound is based on a structural result due to Bartlett and Mendelson, which relates expectations to sample averages. Second, we show that these structural upper bounds can be loose, compared to previous bounds. In particular, we demonstrate a class for which the expectation of the empirical minimizer decreases as O(1/n) for sample size n, although the upper bound based on structural properties is Ω(1). Third, we show that this looseness of the bound is inevitable: we present an example that shows that a sharp bound cannot be universally recovered from empirical data.

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ID Code: 43980
Item Type: Journal Article
Keywords: error bounds, data-dependent complexity, empirical minimization
DOI: 10.1051/ps:2008036
ISSN: 1292-8100
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200)
Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > STATISTICS (010400)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Past > Schools > Mathematical Sciences
Copyright Owner: Copyright 2010 EDP Sciences
Deposited On: 18 Aug 2011 10:45
Last Modified: 01 Mar 2012 00:34

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