Improved Analysis of Sparse Linear Regression in Local Differential Privacy Model

Abstract

In this paper, we revisit the problem of sparse linear regression in the local differential privacy (LDP) model. Existing research in the non-interactive and sequentially local models has focused on obtaining the lower bounds for the case where the underlying parameter is $1$-sparse, and extending such bounds to the more general $k$-sparse case has proven to be challenging. Moreover, it is unclear whether efficient non-interactive LDP (NLDP) algorithms exist. To address these issues, we first consider the problem in the $\epsilon$ non-interactive LDP model and provide a lower bound of $\Omega (\frac{\sqrt{dk\log d}}{\sqrt{n}\epsilon })$ on the ${\ell }_2$-norm estimation error for sub-Gaussian data, where $n$ is the sample size and $d$ is the dimension of the space. We propose an innovative NLDP algorithm, the very first of its kind for the problem. As a remarkable outcome, this algorithm also yields a novel and highly efficient estimator as a valuable by-product. Our algorithm achieves an upper bound of $\tilde{O}(\frac{d\sqrt{k}}{\sqrt{n}\epsilon })$ for the estimation error when the data is sub-Gaussian, which can be further improved by a factor of $O(\sqrt{d})$ if the server has additional public but unlabeled data. For the sequentially interactive LDP model, we show a similar lower bound of $\Omega (\frac{\sqrt{dk}}{\sqrt{n}\epsilon })$. As for the upper bound, we rectify a previous method and show that it is possible to achieve a bound of $\tilde{O}(\frac{k\sqrt{d}}{\sqrt{n}\epsilon })$. Our findings reveal fundamental differences between the non-private case, central DP model, and local DP model in the sparse linear regression problem.