On the Information Leakage Envelope of the Gaussian Mechanism

TL;DR

This paper derives a closed-form expression for the maximal leakage envelope of the Gaussian mechanism, providing bounds on information leakage with high probability, especially for Gaussian and certain log-concave priors.

Contribution

It introduces a closed-form PML envelope for the Gaussian mechanism and extends it to general unbounded secrets under specific conditions.

Findings

Closed-form PML envelope for Gaussian secrets.

Extension to unbounded secrets with log-concave priors.

Application of Brascamp-Lieb inequality to establish conditions.

Abstract

We study the pointwise maximal leakage (PML) envelope of the Gaussian mechanism, which characterizes the smallest information leakage bound that holds with high probability under arbitrary post-processing. For the Gaussian mechanism with a Gaussian secret, we derive a closed-form expression for the deterministic PML envelope for sufficiently small failure probabilities. We then extend this result to general unbounded secrets by identifying a sufficient condition under which the envelope coincides with the Gaussian case. In particular, we show that strongly log-concave priors satisfy this condition via an application of the Brascamp-Lieb inequality.