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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.
@article{PS_2010__14__315_0,
author = {Bartlett, Peter L. and Mendelson, Shahar and Philips, Petra},
title = {On the optimality of sample-based estimates of the expectation of the empirical minimizer},
journal = {ESAIM: Probability and Statistics},
pages = {315--337},
publisher = {EDP-Sciences},
volume = {14},
year = {2010},
doi = {10.1051/ps:2008036},
mrnumber = {2779487},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2008036/}
}
TY - JOUR AU - Bartlett, Peter L. AU - Mendelson, Shahar AU - Philips, Petra TI - On the optimality of sample-based estimates of the expectation of the empirical minimizer JO - ESAIM: Probability and Statistics PY - 2010 SP - 315 EP - 337 VL - 14 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps:2008036/ DO - 10.1051/ps:2008036 LA - en ID - PS_2010__14__315_0 ER -
%0 Journal Article %A Bartlett, Peter L. %A Mendelson, Shahar %A Philips, Petra %T On the optimality of sample-based estimates of the expectation of the empirical minimizer %J ESAIM: Probability and Statistics %D 2010 %P 315-337 %V 14 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps:2008036/ %R 10.1051/ps:2008036 %G en %F PS_2010__14__315_0
Bartlett, Peter L.; Mendelson, Shahar; Philips, Petra. On the optimality of sample-based estimates of the expectation of the empirical minimizer. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 315-337. doi: 10.1051/ps:2008036
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