Robust machine learning by median-of-means : theory and practice

schedule le mardi 10 avril 2018 de 10h45 à 11h45

Organisé par : Castillo, Fischer, Giulini, Gribkova, Levrard, Roquain, Sangnier

Intervenant : Guillaume Lecué (CNRS, ENSAE)
Lieu : UPMC, salle 15-16.201

Sujet : Robust machine learning by median-of-means : theory and practice

Résumé :

We introduce new estimators of least-squares minimizers based on median-of-means (MOM) estimators of the mean of real valued random variables. These estimators achieve optimal rates of convergence under minimal assumptions on the dataset. The dataset may also have been corrupted by outliers on which no assumption is granted. We also analyze these new estimators with standard tools from robust statistics. In particular, we revisit the concept of breakdown point. We modify the original definition by studying the number of outliers that a dataset can contain without deteriorating the estimation properties of a given estimator. This new notion of breakdown number, that takes into account the statistical performances of the estimators, is non-asymptotic in nature and adapted  for machine learning purposes. We proved that the breakdown number of our estimator is of the order of (number of observations) * (rate of convergence). For instance,  the breakdown number of our estimators for the problem of estimation of a d-dimensional vector with a noise variance sigma^2 is sigma^2d and it becomes sigma^2 s \log(ed/s) when this vector has only s non-zero component. Beyond this breakdown point, we proved that the rate of convergence achieved by our estimator is (number of outliers divided by (number of observations).

Besides these theoretical guarantees, the major improvement brought by these new estimators is that they are easily computable in practice and are therefore well suited for robust machine learning. In fact, basically any algorithm used to approximate the standard Empirical Risk Minimizer (or its regularized versions) has a robust version approximating our estimators. On top of being robust to outliers, the ``MOM version" of the algorithms are even faster than the original ones, less demanding in memory resources in some situations and well adapted for distributed datasets which makes it particularly attractive for large dataset analysis. As a proof of concept, we study many algorithms for the classical LASSO estimator. It turns out that a first version of our algorithm can be improved a lot in practice by randomizing the blocks on which ``local means" are computed at each step of the descent algorithm. A byproduct of this modification is that our algorithms come with a measure of depth of data that can be used to detect outliers, which is another major issue in Machine learning.

Joint work with Matthieu Lerasle