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Differential privacy for symmetric log-concave mechanisms

Staal A. Vinterbo

TL;DR

This work extends the differential privacy analysis of noise-adding mechanisms from Gaussian to the broader class of symmetric log-concave densities, providing a necessary and sufficient condition for $(\epsilon,\delta)$-DP in one dimension and extending it to multidimensional, norm-based sensitivities. Central to the approach is the Subbotin family, which unifies Laplace, Gaussian, and other symmetric noises under a common framework, enabling joint optimization over the noise family and the scale to minimize mean squared error for a given privacy budget. The authors derive closed-form scale bounds for the Laplace and Logistic mechanisms and a Gaussian DP criterion, and they show that, under many settings, Laplace and Logistic mechanisms can outperform Gaussian in terms of utility, especially as the query dimension grows. They further demonstrate that multidimensional noise mechanisms based on iid Subbotin components and $\ell_{p}$-norm sensitivities offer a practical path to dimension-aware privacy-utility tradeoffs, with a formal DP criterion and explicit guidance for selecting the norm and the noise parameter to achieve lower utility loss than fixed Gaussian mechanisms.

Abstract

Adding random noise to database query results is an important tool for achieving privacy. A challenge is to minimize this noise while still meeting privacy requirements. Recently, a sufficient and necessary condition for $(ε, δ)$-differential privacy for Gaussian noise was published. This condition allows the computation of the minimum privacy-preserving scale for this distribution. We extend this work and provide a sufficient and necessary condition for $(ε, δ)$-differential privacy for all symmetric and log-concave noise densities. Our results allow fine-grained tailoring of the noise distribution to the dimensionality of the query result. We demonstrate that this can yield significantly lower mean squared errors than those incurred by the currently used Laplace and Gaussian mechanisms for the same $ε$ and $δ$.

Differential privacy for symmetric log-concave mechanisms

TL;DR

This work extends the differential privacy analysis of noise-adding mechanisms from Gaussian to the broader class of symmetric log-concave densities, providing a necessary and sufficient condition for -DP in one dimension and extending it to multidimensional, norm-based sensitivities. Central to the approach is the Subbotin family, which unifies Laplace, Gaussian, and other symmetric noises under a common framework, enabling joint optimization over the noise family and the scale to minimize mean squared error for a given privacy budget. The authors derive closed-form scale bounds for the Laplace and Logistic mechanisms and a Gaussian DP criterion, and they show that, under many settings, Laplace and Logistic mechanisms can outperform Gaussian in terms of utility, especially as the query dimension grows. They further demonstrate that multidimensional noise mechanisms based on iid Subbotin components and -norm sensitivities offer a practical path to dimension-aware privacy-utility tradeoffs, with a formal DP criterion and explicit guidance for selecting the norm and the noise parameter to achieve lower utility loss than fixed Gaussian mechanisms.

Abstract

Adding random noise to database query results is an important tool for achieving privacy. A challenge is to minimize this noise while still meeting privacy requirements. Recently, a sufficient and necessary condition for -differential privacy for Gaussian noise was published. This condition allows the computation of the minimum privacy-preserving scale for this distribution. We extend this work and provide a sufficient and necessary condition for -differential privacy for all symmetric and log-concave noise densities. Our results allow fine-grained tailoring of the noise distribution to the dimensionality of the query result. We demonstrate that this can yield significantly lower mean squared errors than those incurred by the currently used Laplace and Gaussian mechanisms for the same and .
Paper Structure (24 sections, 20 theorems, 79 equations, 2 figures)

This paper contains 24 sections, 20 theorems, 79 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Summary of findings
  3. PRELIMINARIES
  4. Subbotin densities
  5. Differential privacy
  6. THE ONE-DIMENSIONAL CASE
  7. A condition for real-valued mechanisms
  8. Selected analytic conditions
  9. A benefit of achieving \\ epsilon-differential privacy
  10. Mechanism variance ratios
  11. Logistic vs. Laplace mechanisms
  12. Laplace and Logistic vs. Gaussian mechanisms
  13. THE MULTIDIMENSIONAL CASE
  14. Optimizing query responses using subbotin noise
  15. BACKGROUND AND OTHER RELATED WORKS
  16. ...and 9 more sections

Key Result

Lemma 1

Let $X$ be a random variable distributed according to density $f(x) = e^{-\psi(x)}$ where $\psi : \mathbb{R}\to (-\infty,\infty]$ is convex and even, with $f$ having support $(-a, a)$ for $a \in (0,\infty]$. Then for a real-valued function $q$ on databases with global sensitivity $\Delta \geq 0$ and where Furthermore, if the above holds for scale $s > 0$, then it also holds for scales $s' > s$.

Figures (2)

  • Figure 1: For pairs of Logistic($\log$), Laplace (1), and Gaussian (2) distributions: plots of variance ratios $v_{a,b}$ and contours where the resulting variances are equal.
  • Figure 2: Graphical summary of the mean estimation experiments.

Theorems & Definitions (51)

  • Definition 1
  • Definition 2: $(\epsilon,\delta)$-differential privacy Dwork2006aour-data-ourselves-privacy-via-distributed-noise-generation
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Theorem 3
  • Remark 2
  • Theorem 4
  • Remark 3
  • Theorem 5
  • ...and 41 more