Learning from MOM’s principles: Le Cam’s approach

Abstract

New robust estimators are introduced, derived from median-of-means principle and Le Cam’s aggregation of tests. Minimax sparse rates of convergence are obtained with exponential probability, under weak moment’s assumptions and possible contamination of the dataset. These derive from general risk bounds of the following informal structure maxminimax rate in the i.i.d. setup,number of outliersnumber of observations.In this result, the number of outliers may be as large as (number of data)×(minimax rate) without affecting the rates. As an example, minimax rates slog(ed∕s)∕N of recovery of s-sparse vectors in Rd holding with exponentially large probability, are deduced for median-of-means versions of the LASSO when the noise has q0 moments for some q0>2, the entries of the design matrix have C0log(ed) moments and the dataset is corrupted by up to C1slog(ed∕s) outliers.