A spectral least-squares-type method for heavy-tailed corrupted regression with unknown covariance \& heterogeneous noise

TL;DR

This paper introduces a computationally efficient spectral least-squares method for robust linear regression in heavy-tailed, corrupted data scenarios with unknown covariance and heterogeneous noise, achieving near-optimal statistical rates.

Contribution

It proposes the first tractable algorithm that attains optimal robustness and statistical guarantees in heavy-tailed corrupted regression with unknown covariance.

Findings

Achieves near-optimal statistical rate with high probability.

Breakdown point is proportional to inverse of data hypercontractivity and condition number.

First algorithm to simultaneously guarantee robustness, efficiency, and optimal statistical performance.

Abstract

We revisit heavy-tailed corrupted least-squares linear regression assuming to have a corrupted $n$-sized label-feature sample of at most $\epsilon n$ arbitrary outliers. We wish to estimate a $p$-dimensional parameter $b^*$ given such sample of a label-feature pair $(y,x)$ satisfying $y=\langle x,b^*\rangle+\xi$ with heavy-tailed $(x,\xi)$. We only assume $x$ is $L^4-L^2$ hypercontractive with constant $L>0$ and has covariance matrix $\Sigma$ with minimum eigenvalue $1/\mu^2>0$ and bounded condition number $\kappa>0$. The noise $\xi$ can be arbitrarily dependent on $x$ and nonsymmetric as long as $\xi x$ has finite covariance matrix $\Xi$. We propose a near-optimal computationally tractable estimator, based on the power method, assuming no knowledge on $(\Sigma,\Xi)$ nor the operator norm of $\Xi$. With probability at least $1-\delta$, our proposed estimator attains the statistical rate $\mu^2\Vert\Xi\Vert^{1/2}(\frac{p}{n}+\frac{\log(1/\delta)}{n}+\epsilon)^{1/2}$ and breakdown-point $\epsilon\lesssim\frac{1}{L^4\kappa^2}$, both optimal in the $\ell_2$-norm, assuming the near-optimal minimum sample size $L^4\kappa^2(p\log p + \log(1/\delta))\lesssim n$, up to a log factor. To the best of our knowledge, this is the first computationally tractable algorithm satisfying simultaneously all the mentioned properties. Our estimator is based on a two-stage Multiplicative Weight Update algorithm. The first stage estimates a descent direction $\hat v$ with respect to the (unknown) pre-conditioned inner product $\langle\Sigma(\cdot),\cdot\rangle$. The second stage estimate the descent direction $\Sigma\hat v$ with respect to the (known) inner product $\langle\cdot,\cdot\rangle$, without knowing nor estimating $\Sigma$.