Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising

TL;DR

This paper improves the Gaussian mechanism for differential privacy by providing a more accurate variance calibration and a denoising step, significantly enhancing privacy-utility trade-offs especially in high-dimensional settings.

Contribution

It introduces an analytically calibrated Gaussian mechanism with optimal variance and a post-processing denoising step, addressing limitations of the original mechanism.

Findings

Analytical calibration reduces noise variance by at least one-third.

Denoising significantly improves accuracy in high-dimensional data.

The new mechanism outperforms classical approaches in privacy-utility balance.

Abstract

The Gaussian mechanism is an essential building block used in multitude of differentially private data analysis algorithms. In this paper we revisit the Gaussian mechanism and show that the original analysis has several important limitations. Our analysis reveals that the variance formula for the original mechanism is far from tight in the high privacy regime ($\varepsilon \to 0$) and it cannot be extended to the low privacy regime ($\varepsilon \to \infty$). We address these limitations by developing an optimal Gaussian mechanism whose variance is calibrated directly using the Gaussian cumulative density function instead of a tail bound approximation. We also propose to equip the Gaussian mechanism with a post-processing step based on adaptive estimation techniques by leveraging that the distribution of the perturbation is known. Our experiments show that analytical calibration removes at least a third of the variance of the noise compared to the classical Gaussian mechanism, and that denoising dramatically improves the accuracy of the Gaussian mechanism in the high-dimensional regime.