Previous studies yielded discouraging results for item-level locally differentially private linear regression with $s^*$-sparsity assumption, where the minimax rate for $nm$ samples is $\mathcal{O}(s^{*}d / nm\varepsilon^2)$. This can be challenging for high-dimensional data, where the dimension $d$ is extremely large. In this work, we investigate user-level locally differentially private sparse linear regression. We show that with $n$ users each contributing $m$ samples, the linear dependency of dimension $d$ can be eliminated, yielding an error upper bound of $\mathcal{O}(s^{*2} / nm\varepsilon^2)$. We propose a framework that first selects candidate variables and then conducts estimation in the narrowed low-dimensional space, which is extendable to general sparse estimation problems with tight error bounds. Experiments on both synthetic and real datasets demonstrate the superiority of the proposed methods. Both the theoretical and empirical results suggest that, with the same number of samples, locally private sparse estimation is better conducted when multiple samples per user are available.

本研究针对高维数据中局部差分隐私线性回归的挑战，提出了一种用户级稀疏线性回归方法，证明在每个用户贡献多个样本的情况下，可以消除维度的线性依赖。研究结果表明，与相同样本数量相比，利用多个样本进行局部隐私稀疏估计可以获得更好的效果，具有重要的理论和实践意义。

给定每个用户多个样本的更好局部隐私稀疏估计