The purpose of this paper is to discuss empirical risk minimization when the
losses are not necessarily bounded and may have a distribution with heavy
tails. In such situations, usual empirical averages may fail to provide
reliable estimates and empirical risk minimization may provide large excess
risk. However, some robust mean estimators proposed in the literature may be
used to replace empirical means. In this paper, we investigate empirical risk
minimization based on a robust estimate proposed by Catoni. We develop
performance bounds based on chaining arguments tailored to Catoni's mean
estimator.Comment: Published at http://dx.doi.org/10.1214/15-AOS1350 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
We study the problem of estimating the mean of a multivariate distribution based on independent samples. The main result is the proof of existence of an estimator with a non-asymptotic sub-Gaussian performance for all distributions satisfying some mild moment assumptions.
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