Outlier-robust sparse/low-rank least-squares regression and robust matrix completion

TL;DR

This paper develops robust high-dimensional regression and matrix completion methods that tolerate adversarial label contamination, providing near-optimal rates and practical estimators with tunable parameters independent of failure probability.

Contribution

It introduces a new theoretical framework and estimators for robust sparse, low-rank, and matrix completion problems with adversarial noise, using novel concentration inequalities.

Findings

Achieves near-optimal estimation rates under adversarial contamination.

Proposes tractable estimators based on a new sorted Huber loss.

Numerical results show the proposed methods outperform classical approaches.

Abstract

We study high-dimensional least-squares regression within a subgaussian statistical learning framework with heterogeneous noise. It includes $s$-sparse and $r$-low-rank least-squares regression when a fraction $\epsilon$ of the labels are adversarially contaminated. We also present a novel theory of trace-regression with matrix decomposition based on a new application of the product process. For these problems, we show novel near-optimal "subgaussian" estimation rates of the form $r(n,d_{e})+\sqrt{\log(1/\delta)/n}+\epsilon\log(1/\epsilon)$, valid with probability at least $1-\delta$. Here, $r(n,d_{e})$ is the optimal uncontaminated rate as a function of the effective dimension $d_{e}$ but independent of the failure probability $\delta$. These rates are valid uniformly on $\delta$, i.e., the estimators' tuning do not depend on $\delta$. Lastly, we consider noisy robust matrix completion with non-uniform sampling. If only the low-rank matrix is of interest, we present a novel near-optimal rate that is independent of the corruption level $a$. Our estimators are tractable and based on a new "sorted" Huber-type loss. No information on $(s,r,\epsilon,a)$ are needed to tune these estimators. Our analysis makes use of novel $\delta$-optimal concentration inequalities for the multiplier and product processes which could be useful elsewhere. For instance, they imply novel sharp oracle inequalities for Lasso and Slope with optimal dependence on $\delta$. Numerical simulations confirm our theoretical predictions. In particular, "sorted" Huber regression can outperform classical Huber regression.