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.
"Empirical risk minimization for heavy-tailed losses." Ann. Statist. 43 (6) 2507 - 2536, December 2015. https://doi.org/10.1214/15-AOS1350