Differentially Private Sparse Linear Regression with Heavy-tailed Responses

Abstract

As a fundamental problem in machine learning and differential privacy (DP), DP linear regression has been extensively studied. However, most existing methods focus primarily on either regular data distributions or low-dimensional cases with irregular data. To address these limitations, this paper provides a comprehensive study of DP sparse linear regression with heavy-tailed responses in high-dimensional settings. In the first part, we introduce the DP-IHT-H method, which leverages the Huber loss and private iterative hard thresholding to achieve an estimation error bound of \( \tilde{O}\biggl( s^{* \frac{1 }{2}} \cdot \biggl(\frac{\log d}{n}\biggr)^{\frac{\zeta}{1 + \zeta}} + s^{* \frac{1 + 2\zeta}{2 + 2\zeta}} \cdot \biggl(\frac{\log^2 d}{n \varepsilon}\biggr)^{\frac{\zeta}{1 + \zeta}} \biggr) \) under the $(\varepsilon, \delta)$-DP model, where $n$ is the sample size, $d$ is the dimensionality, $s^*$ is the sparsity of the parameter, and $\zeta \in (0, 1]$ characterizes the tail heaviness of the data. In the second part, we propose DP-IHT-L, which further improves the error bound under additional assumptions on the response and achieves \( \tilde{O}\Bigl(\frac{(s^*)^{3/2} \log d}{n \varepsilon}\Bigr). \) Compared to the first result, this bound is independent of the tail parameter $\zeta$. Finally, through experiments on synthetic and real-world datasets, we demonstrate that our methods outperform standard DP algorithms designed for ``regular'' data.