# Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation

arXiv: https://arxiv.org/abs/2605.01492 · HTML: https://mlsanity.com/paper/2605.01492/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 2 May 2026.

## Builds on (most cited first, 5 of 5)

- [Membership Inference Attacks against Machine Learning Models](https://mlsanity.com/paper/1610.05820/) (2016)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [On-Average KL-Privacy and its equivalence to Generalization for Max-Entropy Mechanisms](https://mlsanity.com/paper/1605.02277/) (2016)
- [Prediction Errors for Penalized Regressions based on Generalized Approximate Message Passing](https://mlsanity.com/paper/2206.12832/) (2022)
- [Privacy-Accuracy Trade-offs in High-Dimensional LASSO under Perturbation Mechanisms](https://mlsanity.com/paper/2603.26227/) (2026)

## Abstract (arXiv)

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.