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

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 $|\cO|$ malicious outliers contaminate the labels. In that case, under a local Bernstein condition, we show that the $L2$-error rate is bounded by $ rN + AL |\cO|/N$, where $N$ is the total number of observations, $rN$ is the $L2$-error rate in the non-contaminated setting and $A$ is a parameter coming from the local Bernstein condition. When $rN$ is minimax-rate-optimal in a non-contaminated setting, the rate $rN + AL|\cO|/N$ is also minimax-rate-optimal when $|\cO|$ 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. 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 when the noise can be heavy-tailed.