Robust machine learning by medianofmeans : theory and practice
Abstract
We introduce new estimators for robust machine learning based on medianofmeans (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 nonasymptotic 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 ddimensional vector with a noise variance sigma^2 is sigma^2d and it becomes sigma^2 s log(d/s) when this vector has only s nonzero component. Beyond this breakdown point, we proved that the rate of convergence achieved by our estimator is (number of outliers) divided by (number of observation). Besides these theoretical guarantees, the major improvement brought by these new estimators is that they are easily computable in practice. In fact, basically any algorithm used to approximate the standard Empirical Risk Minimizer (or its regularized versions) has a robust version approximating our estimators. As a proof of concept, we study many algorithms for the classical LASSO estimator. A byproduct of the MOM algorithms is a measure of depth of data that can be used to detect outliers.
 Publication:

arXiv eprints
 Pub Date:
 November 2017
 arXiv:
 arXiv:1711.10306
 Bibcode:
 2017arXiv171110306L
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 48 pages, 6 figures