On Catoni's M-Estimation

TL;DR

This paper extends Catoni's robust M-estimator by establishing uniform concentration inequalities for a family of test functions, enabling better empirical risk minimization in heavy-tailed loss scenarios.

Contribution

It provides the first uniform concentration inequality for Catoni's M-estimator across a family of functions, advancing robust statistical learning methods.

Findings

Established uniform deviation bounds for Catoni's estimator.

Applied results to empirical risk minimization with heavy-tailed data.

Enhanced robustness in statistical estimation under heavy-tailed distributions.

Abstract

Catoni proposed a robust M-estimator and gave the deviation inequality for one fixed test function. The present paper is devoted to the uniform concentration inequality for a family of test functions. As an application, we consider empirical risk minimization for heavy-tailed losses.