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On the Information Leakage Envelope of the Gaussian Mechanism

Sara Saeidian

TL;DR

The paper investigates the information leakage envelope of the Gaussian mechanism under deterministic post-processing using the pointwise maximal leakage (PML) framework. It derives a closed-form deterministic envelope $\varepsilon_d(\delta)$ for the Gaussian mechanism with a Gaussian secret, showing $\varepsilon_d(\delta)=\log(2/\delta)$ for small $\delta$ when $\sigma_X^2 \le 3\sigma_N^2$, and extends the result to general unbounded secrets under a sufficient variance bound that can be guaranteed for strongly log-concave priors via the Brascamp–Lieb inequality (e.g., when $\beta \le \sqrt{3}\,\sigma_N$). The work thus connects tail-robust privacy guarantees to priors with controlled concentration, providing practical post-processing robust privacy bounds for Gaussian mechanisms. These results inform prior design and privacy analysis in settings where tail leakage, rather than global bounds, governs privacy risk.

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.

On the Information Leakage Envelope of the Gaussian Mechanism

TL;DR

The paper investigates the information leakage envelope of the Gaussian mechanism under deterministic post-processing using the pointwise maximal leakage (PML) framework. It derives a closed-form deterministic envelope for the Gaussian mechanism with a Gaussian secret, showing for small when , and extends the result to general unbounded secrets under a sufficient variance bound that can be guaranteed for strongly log-concave priors via the Brascamp–Lieb inequality (e.g., when ). The work thus connects tail-robust privacy guarantees to priors with controlled concentration, providing practical post-processing robust privacy bounds for Gaussian mechanisms. These results inform prior design and privacy analysis in settings where tail leakage, rather than global bounds, governs privacy risk.

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.
Paper Structure (21 sections, 13 theorems, 98 equations)

This paper contains 21 sections, 13 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Deterministic PML Envelope of the Gaussian Mechanism
  4. Properties of the Gaussian Mechanism
  5. Gaussian Secret
  6. General Unbounded Secret
  7. Discussion and Conclusions
  8. Properties of the Gaussian Mechanism
  9. Proof of Lemma \ref{['lemma:tail_worse']}
  10. Applications of the Bathtub Principle
  11. Finding the Solution of Eq. \ref{['eq:conditonal_dist_optimization']}
  12. Finding the Solution of Eq. \ref{['eq:relaxed_optimization_2']}
  13. Proofs for the Gaussian Mechanism with a Gaussian Secret
  14. Proof of Lemma \ref{['lemma:gaussian_interval_leakage']}
  15. Proof of Lemma \ref{['lemma:interval_leakage_increasing']}
  16. ...and 6 more sections

Key Result

Lemma 1

Consider the Gaussian mechanism $Y = X + N$, where $N \sim \mathcal{N} (0, \sigma_N^2)$ and $X$ and $N$ are independent. Suppose $X$ is unbounded, i.e., $\operatorname{supp}(P_X) = \mathbb{R}$.

Theorems & Definitions (15)

  • Definition 1: saeidian2023pointwise_isit
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Theorem 3
  • Definition 4: Log concave and strongly log concave densities
  • Theorem 5: Brascamp-Lieb inequality
  • Corollary 1
  • ...and 5 more