Adapting to noise tails in private linear regression

TL;DR

This paper introduces differentially private, tail-robust linear regression methods that balance bias, privacy, and robustness, achieving near-optimal rates under various error distributions and high-dimensional settings.

Contribution

It develops novel privacy-preserving robust regression algorithms using Huber loss and iterative hard thresholding, with theoretical analysis under diverse error conditions.

Findings

Achieves near-optimal convergence rates for sub-Gaussian errors.

Explicitly characterizes convergence dependence on error moments and privacy parameters.

Demonstrates effectiveness through simulations and real data applications.

Abstract

While the traditional goal of statistics is to infer population parameters, modern practice increasingly demands protection of individual privacy. One way to address this need is to adapt classical statistical procedures into privacy-preserving algorithms. In this paper, we develop differentially private tail-robust methods for linear regression. The trade-off among bias, privacy, and robustness is controlled by a tunable robustification parameter in the Huber loss. We implement noisy clipped gradient descent for low-dimensional settings and noisy iterative hard thresholding for high-dimensional sparse models. Under sub-Gaussian errors, our method achieves near-optimal convergence rates while relaxing several assumptions required in earlier work. For heavy-tailed errors, we explicitly characterize how the non-asymptotic convergence rate depends on the moment index, privacy parameters, sample size, and intrinsic dimension. Our analysis shows how the moment index influences the choice of robustification parameters and, in turn, the resulting statistical error and privacy cost. By quantifying the interplay among bias, privacy, and robustness, we extend classical perspectives on privacy-preserving robust regression. The proposed methods are evaluated through simulations and two real datasets.