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On the Optimality of Sample-Based Estimates of the Expectation of theEmpirical Minimizer***

Published online by Cambridge University Press:  29 October 2010

Peter L. Bartlett ,
Shahar Mendelson  and
Petra Philips
Peter L. Bartlett*
Affiliation:
Computer Science Division and Department of Statistics, 367 Evans Hall #3860, University of California, Berkeley, CA, 94720-3860, USA
Shahar Mendelson
Affiliation:
Centre for Mathematics and its Applications (CMA), The Australian National University Canberra, Canberra, ACT, 0200, Australia Department of Mathematics, Technion I.I.T., Haifa, 32000, Israel
Petra Philips
Affiliation:
Friedrich Miescher Laboratory of the Max Planck Society, Tübingen, 72076, Germany
*
Corresponding author: bartlett@cs.berkeley.edu
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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.

Keywords

Error bounds; empirical minimization; data-dependent complexity
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 315 - 337
Copyright
© EDP Sciences, SMAI, 2010

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Footnotes

*

This work was supported in part by National Science Foundation Grant 0434383.

**

This work was supported in part by the Australian Research Council Discovery Grant DP0559465.

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