# Robust regression with covariate filtering: Heavy tails and adversarial contamination

arXiv: https://arxiv.org/abs/2009.12976 · HTML: https://mlsanity.com/paper/2009.12976/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Sep 2020.

## Built on by (most cited first, 30 of 50)

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- [RobustTSF: Towards Theory and Design of Robust Time Series Forecasting with Anomalies](https://mlsanity.com/paper/2402.02032/) (2024)
- [Robust Regression Revisited: Acceleration and Improved Estimation Rates](https://mlsanity.com/paper/2106.11938/) (2021)
- [High-dimensional robust regression under heavy-tailed data: Asymptotics and Universality](https://mlsanity.com/paper/2309.16476/) (2023)
- [Outlier-Robust Sparse Mean Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/2211.16333/) (2022)
- [Heavy-tailed Streaming Statistical Estimation](https://mlsanity.com/paper/2108.11483/) (2021)
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- [SoS Certifiability of Subgaussian Distributions and its Algorithmic Applications](https://mlsanity.com/paper/2410.21194/) (2024)
- [Consistent Estimation for PCA and Sparse Regression with Oblivious Outliers](https://mlsanity.com/paper/2111.02966/) (2021)
- [Near Optimal Private and Robust Linear Regression](https://mlsanity.com/paper/2301.13273/) (2023)
- [Robust Estimation for Random Graphs](https://mlsanity.com/paper/2111.05320/) (2021)
- [Near-Optimal Algorithms for Gaussians with Huber Contamination: Mean Estimation and Linear Regression](https://mlsanity.com/paper/2312.01547/) (2023)
- [Insufficient Statistics Perturbation: Stable Estimators for Private Least Squares](https://mlsanity.com/paper/2404.15409/) (2024)
- [Trimmed Maximum Likelihood Estimation for Robust Learning in Generalized Linear Models](https://mlsanity.com/paper/2206.04777/) (2022)
- [Robust and Sparse Estimation of Linear Regression Coefficients with Heavy-tailed Noises and Covariates](https://mlsanity.com/paper/2206.07594/) (2022)
- [Higher degree sum-of-squares relaxations robust against oblivious outliers](https://mlsanity.com/paper/2211.07327/) (2022)
- [Robust Methods for High-Dimensional Linear Learning](https://mlsanity.com/paper/2208.05447/) (2022)
- [Efficient List-Decodable Regression using Batches](https://mlsanity.com/paper/2211.12743/) (2022)
- [Robust estimation algorithms don't need to know the corruption level](https://mlsanity.com/paper/2202.05453/) (2022)
- [Outlier Robust and Sparse Estimation of Linear Regression Coefficients](https://mlsanity.com/paper/2208.11592/) (2022)
- [Corruption-Robust Offline Two-Player Zero-Sum Markov Games](https://mlsanity.com/paper/2403.07933/) (2024)
- [High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise](https://mlsanity.com/paper/2503.19091/) (2025)
- [Robust supervised learning with coordinate gradient descent](https://mlsanity.com/paper/2201.13372/) (2022)
- [Robust Estimation for Nonparametric Families via Generative Adversarial Networks](https://mlsanity.com/paper/2202.01269/) (2022)
- [Linear Regression using Heterogeneous Data Batches](https://mlsanity.com/paper/2309.01973/) (2023)
- [Robust Stochastic Optimization via Gradient Quantile Clipping](https://mlsanity.com/paper/2309.17316/) (2023)

## Abstract (arXiv)

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.