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

Peter L. Bartlett, Shahar Mendelson, Petra Philips

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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.

    Original languageEnglish
    Pages (from-to)315-337
    Number of pages23
    JournalESAIM - Probability and Statistics
    Volume14
    Issue number4
    DOIs
    Publication statusPublished - 29 Oct 2010

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