(Joint work with Matthieu Lerasle)
We obtain theoretical and practical performances for median of means estimators.
From a theoretical point of view, estimation and prediction error bounds achieved by the MOM estimators hold with exponentially large probability—as in the gaussian framework with independent noise—under only weak moments assumptions on the data and without assuming independence between the noise and the design. Moreover, MOM procedures are robust since a large part of the data may have nothing to do with the oracle we want to reconstruct. Our general risk bound is of order max(minimax rate of convergence in the i.i.d. setup, (number of outliers)/number of observations)). In particular, the number of outliers may be as large as (number of data)*(minimax rate) without affecting the statistical properties of the MOM estimator.
A regularization norm may also be used together with the MOM criterium. In that case, any norm can be used for regularization. When it has some sparsity inducing power we recover sparse rates of convergence and sparse oracle inequalities. For example, the minimax rate s log(d/s)/N of recovery of a s-sparse vector in R^d is achieved by a median-of-means version of the LASSO when the noise has q_0 moments for some q_0>2, the design matrix C_0log(d) moments and the dataset is corrupted by s log(d/s) outliers. This result holds with exponentially large probability as if the noise and the design were i.i.d. Gaussian random variables.
On the practical side, MOM estimators (and their associated regularized versions) can easily be implemented. Actually, most gradient descent algorithms used to implement (non-robust) estimators like the LASSO can easily be transformed into a robust one by using a MOM approach.