The Annals of Statistics

Empirical risk minimization for heavy-tailed losses

Christian Brownlees, Emilien Joly, and Gábor Lugosi

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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.

Article information

Ann. Statist. Volume 43, Number 6 (2015), 2507-2536.

Received: June 2014
Revised: May 2015
First available in Project Euclid: 7 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F35: Robustness and adaptive procedures
Secondary: 62F12: Asymptotic properties of estimators

Empirical risk minimization heavy-tailed data robust regression robust $k$-means clustering Catoni’s estimator


Brownlees, Christian; Joly, Emilien; Lugosi, Gábor. Empirical risk minimization for heavy-tailed losses. Ann. Statist. 43 (2015), no. 6, 2507--2536. doi:10.1214/15-AOS1350.

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