Robust Regression with Covariate Filtering: Heavy Tails and Adversarial Contamination
研究协变量和响应变量同时存在重尾分布和对抗性污染时的线性回归问题，提出通过过滤协变量再应用经典稳健估计器的方法，实现计算和统计效率的平衡。
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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.