Robust regression with covariate filtering: Heavy tails and adversarial contamination
This addresses robust regression for data with heavy tails and adversarial outliers, which is an incremental improvement over existing methods limited to sub-Gaussian or uncontaminated settings.
The paper tackles linear regression with heavy-tailed and adversarially contaminated covariates and responses by modifying classical robust estimators with a covariate filtering step, achieving near-optimal error rates, such as with the Huber regression estimator.
We study the problem of linear regression where both covariates and responses are potentially (i) heavy-tailed and (ii) adversarially contaminated. Several computationally efficient estimators have been proposed for the simpler setting where the covariates are sub-Gaussian and uncontaminated; however, these estimators may fail when the covariates are either heavy-tailed or contain outliers. In this work, we show how to modify the Huber regression, least trimmed squares, and least absolute deviation estimators to obtain estimators which are simultaneously computationally and statistically efficient in the stronger contamination model. Our approach is quite simple, and consists of applying a filtering algorithm to the covariates, and then applying the classical robust regression estimators to the remaining data. We show that the Huber regression estimator achieves near-optimal error rates in this setting, whereas the least trimmed squares and least absolute deviation estimators can be made to achieve near-optimal error after applying a postprocessing step.