Median-of-means (MOM) based procedures have been recently introduced in learning theory (Lugosi and Mendelson (2019); Lecué and Lerasle (2017)). These estimators outperform classical least-squares estimators when data are heavy-tailed and/or are corrupted. None of these procedures can be implemented, which is the major issue of current MOM procedures (Ann. Statist. 47 (2019) 783–794).

In this paper, we introduce minmax MOM estimators and show that they achieve the same sub-Gaussian deviation bounds as the alternatives (Lugosi and Mendelson (2019); Lecué and Lerasle (2017)), both in small and high-dimensional statistics. In particular, these estimators are efficient under moments assumptions on data that may have been corrupted by a few outliers.

Besides these theoretical guarantees, the definition of minmax MOM estimators suggests simple and systematic modifications of standard algorithms used to approximate least-squares estimators and their regularized versions. As a proof of concept, we perform an extensive simulation study of these algorithms for robust versions of the LASSO.