Approximate Differential Privacy of the $\ell_2$ Mechanism

TL;DR

This paper analyzes the $ ext{ extl2}$ mechanism for differentially private $d$-dimensional statistics, showing it outperforms Laplace and Gaussian mechanisms in terms of error across various privacy settings.

Contribution

It provides a comparative analysis of the $ ext{ extl2}$ mechanism's error performance against Laplace and Gaussian mechanisms for approximate differential privacy.

Findings

$ ext{ extl2}$ mechanism has lower error than Laplace and Gaussian mechanisms across privacy parameters.

$ ext{ extl2}$ matches Laplace at $d=1$ and approaches Gaussian as $d$ increases.

The error performance varies with dimension and privacy parameters.

Abstract

We study the $\ell_2$ mechanism for computing a $d$-dimensional statistic with bounded $\ell_2$ sensitivity under approximate differential privacy. Across a range of privacy parameters, we find that the $\ell_2$ mechanism obtains lower error than the Laplace and Gaussian mechanisms, matching the former at $d=1$ and approaching the latter as $d \to \infty$.