Consistent regression when oblivious outliers overwhelm

TL;DR

This paper demonstrates that robust linear regression can be achieved with nearly linear sample size even when an adversary corrupts most observations, extending previous results to broader design matrices and heavy-tailed noise.

Contribution

It proves the consistency of the Huber loss estimator under minimal assumptions and introduces a simple median-based algorithm for Gaussian designs with optimal guarantees.

Findings

Huber loss estimator is consistent with nearly linear sample size

Optimal error bounds are achieved, matching lower bounds

A simple median-based algorithm works efficiently for Gaussian designs

Abstract

We consider a robust linear regression model $y=X\beta^* + \eta$, where an adversary oblivious to the design $X\in \mathbb{R}^{n\times d}$ may choose $\eta$ to corrupt all but an $\alpha$ fraction of the observations $y$ in an arbitrary way. Prior to our work, even for Gaussian $X$, no estimator for $\beta^*$ was known to be consistent in this model except for quadratic sample size $n \gtrsim (d/\alpha)^2$ or for logarithmic inlier fraction $\alpha\ge 1/\log n$. We show that consistent estimation is possible with nearly linear sample size and inverse-polynomial inlier fraction. Concretely, we show that the Huber loss estimator is consistent for every sample size $n= \omega(d/\alpha^2)$ and achieves an error rate of $O(d/\alpha^2n)^{1/2}$. Both bounds are optimal (up to constant factors). Our results extend to designs far beyond the Gaussian case and only require the column span of $X$ to not contain approximately sparse vectors). (similar to the kind of assumption commonly made about the kernel space for compressed sensing). We provide two technically similar proofs. One proof is phrased in terms of strong convexity, extending work of [Tsakonas et al.'14], and particularly short. The other proof highlights a connection between the Huber loss estimator and high-dimensional median computations. In the special case of Gaussian designs, this connection leads us to a strikingly simple algorithm based on computing coordinate-wise medians that achieves optimal guarantees in nearly-linear time, and that can exploit sparsity of $\beta^*$. The model studied here also captures heavy-tailed noise distributions that may not even have a first moment.