Adapting to Noise Tails in Private Linear Regression

原文摘要

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 article, we develop differentially private tail-robust methods for linear regression. The tradeoff 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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

摘要来源：OpenAlex