Robust statistical learning with Lipschitz and convex loss functions

We obtain estimation and excess risk bounds for Empirical Risk Minimizers (ERM) and minmax Median-Of-Means (MOM) estimators based on loss functions that are both Lipschitz and convex. Results for the ERM are derived under weak assumptions on the outputs and subgaussian assumptions on the design as in Alquier et al. (Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions. arXiv:1702.01402, 2017). The difference with Alquier et al. (2017) is that the global Bernstein condition of this paper is relaxed here into a local assumption. We also obtain estimation and excess risk bounds for minmax MOM estimators under similar assumptions on the output and only moment assumptions on the design. Moreover, the dataset may also contains outliers in both inputs and outputs variables without deteriorating the performance of the minmax MOM estimators. Unlike alternatives based on MOM’s principle (Lecué and Lerasle in Ann Stat, 2017; Lugosi and Mendelson in JEMS, 2016), the analysis of minmax MOM estimators is not based on the small ball assumption (SBA) of Koltchinskii and Mendelson (Int Math Res Not IMRN 23:12991–13008, 2015). In particular, the basic example of non parametric statistics where the learning class is the linear span of localized bases, that does not satisfy SBA (Saumard in Bernoulli 24(3):2176–2203, 2018) can now be handled. Finally, minmax MOM estimators are analysed in a setting where the local Bernstein condition is also dropped out. It is shown to achieve excess risk bounds with exponentially large probability under minimal assumptions insuring only the existence of all objects.