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

By Guillaume Lecué, Matthieu Lerasle. 2017

175 citations recorded by the source corpus.

We introduce new estimators for robust machine learning 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(d/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 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.

Subfields: Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

Citations in the corpus

Evaluation Gaps in Machine Learning Practice (2022, background): “…the training data (e.g., distributional shifts, outliers, perturbations, poisoning attacks; and with connections to robust estimation of statistics [101]), while application-centric evaluations pay attention to desired behaviors such as the (in)sensitivity of the model to certain classes of…”
Mean Estimation and Regression Under Heavy-Tailed Distributions: A Survey (2019, unknown): “Similar methods were used in [58] and [47] to study the regularization framework.”
Robust multivariate mean estimation: The optimality of trimmed mean (2019, background): “Such estimators have been studied under a (somewhat more restrictive) adversarial contamination model by Lecué and Lerasle [17] and by Minsker [22], see also see Rodriguez and Valdora [24].”
Robust estimation via robust gradient estimation (2018, background): “Lecué and Lerasle (2017) proposed a median-of-means approach to solve ERM under the -contaminated setting.”