# Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression

arXiv: https://arxiv.org/abs/1903.08192 · HTML: https://mlsanity.com/paper/1903.08192/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2019.

## Built on by (most cited first, 23 of 23)

- [Heavy-tailed Streaming Statistical Estimation](https://mlsanity.com/paper/2108.11483/) (2021)
- [ReLU Regression with Massart Noise](https://mlsanity.com/paper/2109.04623/) (2021)
- [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)
- [Hardness and Algorithms for Robust and Sparse Optimization](https://mlsanity.com/paper/2206.14354/) (2022)
- [Understanding In-Context Learning of Linear Models in Transformers Through an Adversarial Lens](https://mlsanity.com/paper/2411.05189/) (2024)
- [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)
- [Higher degree sum-of-squares relaxations robust against oblivious outliers](https://mlsanity.com/paper/2211.07327/) (2022)
- [Efficient List-Decodable Regression using Batches](https://mlsanity.com/paper/2211.12743/) (2022)
- [On the well-spread property and its relation to linear regression](https://mlsanity.com/paper/2206.08092/) (2022)
- [DecoR: Deconfounding Time Series with Robust Regression](https://mlsanity.com/paper/2406.07005/) (2024)
- [Robust Linear Regression for General Feature Distribution](https://mlsanity.com/paper/2202.02080/) (2022)
- [Asymptotic Characterisation of Robust Empirical Risk Minimisation Performance in the Presence of Outliers](https://mlsanity.com/paper/2305.18974/) (2023)
- [Distribution-Independent Regression for Generalized Linear Models with Oblivious Corruptions](https://mlsanity.com/paper/2309.11657/) (2023)
- [Robust Sparse Regression with Non-Isotropic Designs](https://mlsanity.com/paper/2410.23937/) (2024)
- [Robust Regression with Adaptive Contamination in Response: Optimal Rates and Computational Barriers](https://mlsanity.com/paper/2604.04228/) (2026)
- [Corruption-tolerant Algorithms for Generalized Linear Models](https://mlsanity.com/paper/2212.05430/) (2022)
- [A Bayesian Robust Regression Method for Corrupted Data Reconstruction](https://mlsanity.com/paper/2212.12787/) (2022)
- [A Novel Framework for Improving the Breakdown Point of Robust Regression Algorithms](https://mlsanity.com/paper/2305.12220/) (2023)
- [Purification Of Contaminated Convolutional Neural Networks Via Robust Recovery: An Approach with Theoretical Guarantee in One-Hidden-Layer Case](https://mlsanity.com/paper/2407.11031/) (2024)
- [Information-Computation Tradeoffs for Noiseless Linear Regression with Oblivious Contamination](https://mlsanity.com/paper/2510.10665/) (2025)
- [Quantile-based Loss Filtering for Outlier-Robust Stochastic Gradient Descent](https://mlsanity.com/paper/2609.13040/) (2026)

## Abstract (arXiv)

We study the problem of robust linear regression with response variable corruptions. We consider the oblivious adversary model, where the adversary corrupts a fraction of the responses in complete ignorance of the data. We provide a nearly linear time estimator which consistently estimates the true regression vector, even with 1-o(1) fraction of corruptions. Existing results in this setting either don't guarantee consistent estimates or can only handle a small fraction of corruptions. We also extend our estimator to robust sparse linear regression and show that similar guarantees hold in this setting. Finally, we apply our estimator to the problem of linear regression with heavy-tailed noise and show that our estimator consistently estimates the regression vector even when the noise has unbounded variance (e.g., Cauchy distribution), for which most existing results don't even apply. Our estimator is based on a novel variant of outlier removal via hard thresholding in which the threshold is chosen adaptively and crucially relies on randomness to escape bad fixed points of the non-convex hard thresholding operation.