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Tous les séminaires

Lundi 20 mars 2017

Intervenant : Guillaume LECUE (ENSAE)

"Learning from MOM's principals"

De 15h à 16h15, en salle 08 à l'ENSAE : 3 avenue Pierre Larousse à Malakoff (Tram T3 : "Porte de Vanves" ou Métro 13 : "Porte de Vanves" ou "Malakoff Plateau de Vanves")

 

joint work with Matthieu Lerasle

 

We obtain estimation error rates and oracle inequalities for Birgé's T-estimators based on regularized median-of-means tests. The results 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 X. The obtained rates are minimax optimal. Various norms may be used for regularization. When they have some sparsity inducing power we recover sparse rates of convergence and sparse oracle inequalities. Moreover, the procedure allows for robust estimation in the sense that a large part of the data may have nothing to do with the oracle we want to reconstruct. The number of such irrelevant data (which can be seen as outliers) may be as large as (sample size)*(rate of convergence) as long as the quantity of useful data is larger than a proportion of the number of observations. As a proof of concept, we obtain the ``exact'' minimax rate of convergence s \log(ed/s)/N for the problem of recovery of a s-sparse vector in R^d via a median-of-means version of the LASSO under a L_{q_0} assumption on the noise for some q_0>2 and a C_0\log(ed) moment assumption on the design matrix. As mentionned previously this result holds with exponentially large probability as if the noise and the design were independent and standard gaussian random variables..

 

Ce séminaire est organisé par :

 

Alexandre TSYBAKOV         (Laboratoire de Statistique-CREST)

 

Cristina BUTUCEA                (Laboratoire de Statistique-CREST)