On the Information Leakage Envelope of the Gaussian Mechanism

2026-01-13

AI Summary

This work investigates high-probability information leakage bounds of the Gaussian mechanism under arbitrary post-processing, formalized through the pointwise maximal leakage (PML) envelope. For Gaussian priors, the authors derive, for the first time, a closed-form expression for the PML envelope under small failure probabilities and extend this result to general unbounded secrets. Leveraging the Brascamp–Lieb inequality, they further establish that strongly log-concave priors satisfy the same leakage envelope as the Gaussian case. By integrating tools from information theory, probability theory, and differential privacy, this study provides tight PML bounds for the Gaussian mechanism under both Gaussian and strongly log-concave priors, offering a precise characterization of high-probability information leakage.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Probabilistic Inference
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Web Mining and Content Analysis: Content-based information diffusion

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.

Problem

• information leakage
• Gaussian mechanism
• pointwise maximal leakage
• privacy
• post-processing

Innovation

• pointwise maximal leakage
• Gaussian mechanism
• information leakage envelope
• strongly log-concave priors
• Brascamp-Lieb inequality