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We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk Minimizers (RERM) for regression problems with convex and $L$-Lipschitz loss functions. We consider a setting where $|\mathcal O|$ malicious outliers may contaminate the labels. In that case, we show that the $L_2$-error rate is bounded by $r_N + L |\mathcal O|/N$, where $N$ is the total number of observations and $r_N$ is the $L_2$-error rate in the non-contaminated setting. When $r_N$ is minimax-rate-optimal in a non-contaminated setting, the rate $r_N + L|\mathcal O|/N$ is also minimax-rate-optimal when $|\mathcal O|$ outliers contaminate the label. The main results of the paper can be used for many non-regularized and regularized procedures under weak assumptions on the noise. For instance, we present results for Huber's M-estimators (without penalization or regularized by the $\ell_1$-norm) and for general regularized learning problems in reproducible kernel Hilbert spaces.

ERM and RERM are optimal estimators for regression problems when malicious outliers corrupt the labels