Near-Optimal Algorithms for Gaussians with Huber Contamination: Mean Estimation and Linear Regression

We study the fundamental problems of Gaussian mean estimation and linear regression with Gaussian covariates in the presence of Huber contamination. Our main contribution is the design of the first sample near-optimal and almost linear-time algorithms with optimal error guarantees for both these problems. Specifically, for Gaussian robust mean estimation on $\mathbb R^d$ with contamination parameter $\epsilon \in (0, \epsilon_0)$ for a small absolute constant $\epsilon_0$, we give an algorithm with sample complexity $n = \tilde{O}(d/\epsilon^2)$ and almost linear runtime that approximates the target mean within $\ell_2$-error $O(\epsilon)$. This improves on prior work that achieved this error guarantee with polynomially suboptimal sample and time complexity. For robust linear regression, we give the first algorithm with sample complexity $n = \tilde{O}(d/\epsilon^2)$ and almost linear runtime that approximates the target regressor within $\ell_2$-error $O(\epsilon)$. This is the first polynomial sample and time algorithm achieving the optimal error guarantee, answering an open question in the literature. At the technical level, we develop a methodology that yields almost-linear time algorithms for multi-directional filtering that may be of broader interest.