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

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, [Formula presented].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 R^d 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.