Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation

TL;DR

This paper introduces a Gram-based anisotropic objective perturbation method for high-dimensional private LASSO, effectively handling heterogeneous covariate scales without extra privacy costs, enhancing stability and accuracy.

Contribution

It proposes a novel pre-distortion strategy that counteracts covariate anisotropy, improving private LASSO stability and efficiency without data-dependent preprocessing.

Findings

Significantly stabilizes convergence of private LASSO algorithms.

Improves statistical efficiency and privacy performance over standard methods.

Provides theoretical analysis using AMP framework and state evolution.

Abstract

We study high-dimensional LASSO under differential privacy via objective perturbation with heterogeneous covariate scales. In practical scenarios, covariates often exhibit diverse scales; however, standard preprocessing is problematic under privacy constraints, as it consumes additional privacy budget. This heterogeneity induces effective anisotropy in the objective perturbation via the inverse Gram matrix of covariates, which can degrade the stability and accuracy of algorithms. To address this, we propose a Gram-based anisotropic objective perturbation, a ``pre-distortion" strategy that counteracts the distortion from the covariate structure to restore isotropy in the estimation process. Using an Approximate Message Passing (AMP) framework and state evolution analysis, we demonstrate that our proposed perturbation significantly stabilizes convergence and improves both statistical efficiency and privacy performance compared to standard uniform noise injection. Our results provide theoretical insights into designing stable and efficient private estimators without relying on data-dependent preprocessing.