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Unifying Privacy Measures via Maximal $(α,β)$-Leakage (M$α$beL)

Atefeh Gilani, Gowtham R. Kurri, Oliver Kosut, Lalitha Sankar

TL;DR

This work introduces maximal $(\alpha,\beta)$-leakage (M$\alpha$beL), a two-parameter family of information-leakage measures defined via a guessing-adversary framework to unify existing privacy notions. It provides a computable form and proves key properties (non-negativity, data processing, additivity, continuity) while showing that MaxL, Max-$\alpha$L, LDP, and LRDP emerge as special cases; it also introduces conditional M$\alpha$beL, a vector form, and a reparameterization to $(\alpha,\tau)$-leakage, along with extensions to continuous alphabets. The framework yields an operational interpretation for LRDP and related measurements, and demonstrates that vector variants of these leakages relax differential privacy under Gaussian and Laplacian mechanisms. Overall, the work offers a coherent, tunable continuum between average- and worst-case leakage, enabling tailored privacy-utility tradeoffs across diverse scenarios. The results advance the theory of privacy leakage by unifying disparate measures and providing practical, computable tools for analysis and mechanism design.

Abstract

We introduce a family of information leakage measures called maximal $(α,β)$-leakage (M$α$beL), parameterized by real numbers $α$ and $β$ greater than or equal to 1. The measure is formalized via an operational definition involving an adversary guessing an unknown (randomized) function of the data given the released data. We obtain a simplified computable expression for the measure and show that it satisfies several basic properties such as monotonicity in $β$ for a fixed $α$, non-negativity, data processing inequalities, and additivity over independent releases. We highlight the relevance of this family by showing that it bridges several known leakage measures, including maximal $α$-leakage $(β=1)$, maximal leakage $(α=\infty,β=1)$, local differential privacy (LDP) $(α=\infty,β=\infty)$, and local Renyi differential privacy (LRDP) $(α=β)$, thereby giving an operational interpretation to local Renyi differential privacy. We also study a conditional version of M$α$beL on leveraging which we recover differential privacy and Renyi differential privacy. A new variant of LRDP, which we call maximal Renyi leakage, appears as a special case of M$α$beL for $α=\infty$ that smoothly tunes between maximal leakage ($β=1$) and LDP ($β=\infty$). Finally, we show that a vector form of the maximal Renyi leakage relaxes differential privacy under Gaussian and Laplacian mechanisms.

Unifying Privacy Measures via Maximal $(α,β)$-Leakage (M$α$beL)

TL;DR

This work introduces maximal -leakage (MbeL), a two-parameter family of information-leakage measures defined via a guessing-adversary framework to unify existing privacy notions. It provides a computable form and proves key properties (non-negativity, data processing, additivity, continuity) while showing that MaxL, Max-L, LDP, and LRDP emerge as special cases; it also introduces conditional MbeL, a vector form, and a reparameterization to -leakage, along with extensions to continuous alphabets. The framework yields an operational interpretation for LRDP and related measurements, and demonstrates that vector variants of these leakages relax differential privacy under Gaussian and Laplacian mechanisms. Overall, the work offers a coherent, tunable continuum between average- and worst-case leakage, enabling tailored privacy-utility tradeoffs across diverse scenarios. The results advance the theory of privacy leakage by unifying disparate measures and providing practical, computable tools for analysis and mechanism design.

Abstract

We introduce a family of information leakage measures called maximal -leakage (MbeL), parameterized by real numbers and greater than or equal to 1. The measure is formalized via an operational definition involving an adversary guessing an unknown (randomized) function of the data given the released data. We obtain a simplified computable expression for the measure and show that it satisfies several basic properties such as monotonicity in for a fixed , non-negativity, data processing inequalities, and additivity over independent releases. We highlight the relevance of this family by showing that it bridges several known leakage measures, including maximal -leakage , maximal leakage , local differential privacy (LDP) , and local Renyi differential privacy (LRDP) , thereby giving an operational interpretation to local Renyi differential privacy. We also study a conditional version of MbeL on leveraging which we recover differential privacy and Renyi differential privacy. A new variant of LRDP, which we call maximal Renyi leakage, appears as a special case of MbeL for that smoothly tunes between maximal leakage () and LDP (). Finally, we show that a vector form of the maximal Renyi leakage relaxes differential privacy under Gaussian and Laplacian mechanisms.
Paper Structure (28 sections, 18 theorems, 181 equations, 3 figures)

This paper contains 28 sections, 18 theorems, 181 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Contributions.
  3. Related Work
  4. Organization of the Paper
  5. Overview of Existing Information Leakage Measures
  6. A Unified Measure of Information Leakage
  7. M$\alpha$beL
  8. Conditional M$\alpha$beL
  9. Relationships of Other Leakage Measures with (Conditional) M$\alpha$beL
  10. Vector M$\alpha$beL
  11. Maximal $(\alpha,\tau)$-leakage: A Reparameterization of M$\alpha$beL
  12. Continuous Alphabets
  13. Results for known mechanisms
  14. Conclusion
  15. Proof of Theorem \ref{['theorem:alpha-beta-leakage']}
  16. ...and 13 more sections

Key Result

Theorem 1

Let $X$ and $Y$ take values from finite supports $\mathcal{X}$ and $\mathcal{Y}$, respectively. For $(\alpha,\beta)\in(1,\infty)\times [1,\infty)$, M$\alpha$beL defined in eqn:alpha,beta-leakage-original-def simplifies to where $P_{\Tilde{X}}$ is a probability distribution on the support of $P_{X}$. For $\alpha\to \infty$, since Definition def:maximal-alpha-beta-leakage simplifies to the definiti

Figures (3)

  • Figure 1: Subplots \ref{['alpha-beta-leakage-relationships']} and \ref{['vector-alpha-beta-leakage-relationships']} show the relationships between existing leakage measures with M$\alpha$beL and vector M$\alpha$beL, respectively.
  • Figure 2: Subplot \ref{['alpha-beta-curve']} shows $\beta$ vs $\alpha$ curves across different $\tau$ values. Subplot \ref{['fig:alpha-tau-leakage']} shows relationship between maximal $(\alpha,\tau)$-leakage and other leakage measures as a function of $\alpha$ and $\tau$.
  • Figure 3: Subplots \ref{['fig:lap-b=1']} and \ref{['fig:lap-b=30']} compare vector maximal Renyi leakage and Renyi differential privacy under Laplacian mechanisms with $b=1$ and $b=30$, respectively. Subplots \ref{['fig:gau-sig=1']} and \ref{['fig:gau-sig=20']} compare vector maximal Renyi leakage and Renyi differential privacy under Gaussian mechanisms with $\sigma=1$ and $\sigma=20$, respectively. In all subplots, we consider a situation in which the upper bounds \ref{['eq:lemma-Laplacian']} and \ref{['eq:lemma-Gaussian']} are achieved with equality, and the sensitivity parameter is 1.

Theorems & Definitions (37)

  • Definition 1: Maximal leakage IssaWK2020
  • Definition 2: Maximal $\alpha$-leakage LiaoKS20
  • Definition 3: Local differential privacy kasiviswanathan2011canduchi2013local
  • Definition 4: Differential privacy Dwork2006
  • Definition 5: Maximal realizable leakage IssaWK2020
  • Definition 6: Rényi differential privacy mironov2017renyi
  • Definition 7: Local Rényi differential privacy
  • Definition 8: Maximal Rényi leakage of order $\beta$
  • Definition 9: Maximal $(\alpha,\beta)$-leakage (M$\alpha$beL)
  • Theorem 1
  • ...and 27 more