# On the Optimality of Sample-Based Estimates of the Expectation of the Empirical Minimizer***

Peter L. Bartlett; Shahar Mendelson; Petra Philips

ESAIM: Probability and Statistics (2010)

- Volume: 14, page 315-337
- ISSN: 1292-8100

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topBartlett, Peter L., Mendelson, Shahar, and Philips, Petra. "On the Optimality of Sample-Based Estimates of the Expectation of the Empirical Minimizer***." ESAIM: Probability and Statistics 14 (2010): 315-337. <http://eudml.org/doc/250853>.

@article{Bartlett2010,

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.
},

author = {Bartlett, Peter L., Mendelson, Shahar, Philips, Petra},

journal = {ESAIM: Probability and Statistics},

keywords = {Error bounds; empirical minimization;
data-dependent complexity; error bounds; empirical minimization; data-dependent complexity},

language = {eng},

month = {10},

pages = {315-337},

publisher = {EDP Sciences},

title = {On the Optimality of Sample-Based Estimates of the Expectation of the Empirical Minimizer***},

url = {http://eudml.org/doc/250853},

volume = {14},

year = {2010},

}

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

DA - 2010/10//

PB - EDP Sciences

VL - 14

SP - 315

EP - 337

AB -
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.

LA - eng

KW - Error bounds; empirical minimization;
data-dependent complexity; error bounds; empirical minimization; data-dependent complexity

UR - http://eudml.org/doc/250853

ER -

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