Distribution-Free Robust Linear Regression

We study random design linear regression with no assumptions on the distribution of the covariates and with a heavy-tailed response variable. When learning without assumptions on the covariates, we establish boundedness of the conditional second moment of the response variable as a necessary and sufficient condition for achieving deviation-optimal excess risk rate of convergence. In particular, combining the ideas of truncated least squares, median-of-means procedures and aggregation theory, we construct a non-linear estimator achieving excess risk of order $d/n$ with the optimal sub-exponential tail. While the existing approaches to learning linear classes under heavy-tailed distributions focus on proper estimators, we highlight that the improperness of our estimator is necessary for attaining non-trivial guarantees in the distribution-free setting considered in this work. Finally, as a byproduct of our analysis, we prove an optimal version of the classical bound for the truncated least squares estimator due to Gy\"{o}rfi, Kohler, Krzyzak, and Walk.