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.

翻译：传统统计学的目标在于推断总体参数，而现代实践日益要求保护个体隐私。满足这一需求的一种途径是将经典统计程序改造为隐私保护算法。本文针对线性回归问题，提出了差分隐私的尾部鲁棒方法。偏差、隐私性与鲁棒性之间的权衡通过Huber损失中的可调鲁棒化参数进行控制。我们在低维场景下实现了带噪声的裁剪梯度下降法，在高维稀疏模型中实现了带噪声的迭代硬阈值法。在亚高斯误差条件下，我们的方法在放宽早期研究所需若干假设的同时，达到了接近最优的收敛速率。对于重尾误差，我们明确刻画了非渐近收敛速率如何依赖于矩指数、隐私参数、样本量和内在维度。我们的分析揭示了矩指数如何影响鲁棒化参数的选择，进而影响最终的统计误差与隐私代价。通过量化偏差、隐私性与鲁棒性之间的相互作用，我们拓展了隐私保护鲁棒回归的经典视角。所提方法通过仿真实验和两个真实数据集进行了验证。