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The Asymptotic Behaviour of Information Leakage Metrics

Sophie Taylor, Praneeth Kumar Vippathalla, Justin P. Coon

TL;DR

This work provides a unified asymptotic theory for information-theoretic privacy leakage metrics, addressing both global and pointwise leakage. By positing axioms for pointwise leakage and embedding them into a broad global leakage framework, it proves that privacy degrades exponentially under composition of many conditionally independent observations, with a rate governed by the minimum Chernoff information between distinct conditional channels. It shows almost-sure convergence of pointwise leakage to the intrinsic information content and establishes exponential convergence rates for both pointwise and global metrics, applicable to a wide class including mutual information, Sibson and Arimoto mutual information, maximal leakage, min-entropy leakage, and more via f-divergence and g-leakage forms. The results connect privacy guarantees to Bayesian hypothesis testing and the method of types, offering a principled, extensible foundation for evaluating privacy in scenarios with many observations or side-channel data. The framework supports future metric definitions and provides actionable rates for privacy degradation in repeated-query or multi-sample inference contexts.

Abstract

Information theoretic leakage metrics quantify the amount of information about a private random variable $X$ that is leaked through a correlated revealed variable $Y$. They can be used to evaluate the privacy of a system in which an adversary, from whom we want to keep $X$ private, is given access to $Y$. Global information theoretic leakage metrics quantify the overall amount of information leaked upon observing $Y$, whilst their pointwise counterparts define leakage as a function of the particular realisation $Y=y$ that the adversary sees, and thus can be viewed as random variables. We consider an adversary who observes a large number of independent identically distributed realisations of $Y$. We formalise the essential asymptotic behaviour of an information theoretic leakage metric, considering in turn what this means for pointwise and global metrics. With the resulting requirements in mind, we take an axiomatic approach to defining a set of pointwise leakage metrics, as well as a set of global leakage metrics that are constructed from them. The global set encompasses many known measures including mutual information, Sibson mutual information, Arimoto mutual information, maximal leakage, min entropy leakage, $f$-divergence metrics, and g-leakage. We prove that both sets follow the desired asymptotic behaviour. Finally, we derive composition theorems which quantify the rate of privacy degradation as an adversary is given access to a large number of independent observations of $Y$. It is found that, for both pointwise and global metrics, privacy degrades exponentially with increasing observations for the adversary, at a rate governed by the minimum Chernoff information between distinct conditional channel distributions. This extends the work of Wu et al. (2024), who have previously found this to be true for certain known metrics, including some that fall into our more general set.

The Asymptotic Behaviour of Information Leakage Metrics

TL;DR

This work provides a unified asymptotic theory for information-theoretic privacy leakage metrics, addressing both global and pointwise leakage. By positing axioms for pointwise leakage and embedding them into a broad global leakage framework, it proves that privacy degrades exponentially under composition of many conditionally independent observations, with a rate governed by the minimum Chernoff information between distinct conditional channels. It shows almost-sure convergence of pointwise leakage to the intrinsic information content and establishes exponential convergence rates for both pointwise and global metrics, applicable to a wide class including mutual information, Sibson and Arimoto mutual information, maximal leakage, min-entropy leakage, and more via f-divergence and g-leakage forms. The results connect privacy guarantees to Bayesian hypothesis testing and the method of types, offering a principled, extensible foundation for evaluating privacy in scenarios with many observations or side-channel data. The framework supports future metric definitions and provides actionable rates for privacy degradation in repeated-query or multi-sample inference contexts.

Abstract

Information theoretic leakage metrics quantify the amount of information about a private random variable that is leaked through a correlated revealed variable . They can be used to evaluate the privacy of a system in which an adversary, from whom we want to keep private, is given access to . Global information theoretic leakage metrics quantify the overall amount of information leaked upon observing , whilst their pointwise counterparts define leakage as a function of the particular realisation that the adversary sees, and thus can be viewed as random variables. We consider an adversary who observes a large number of independent identically distributed realisations of . We formalise the essential asymptotic behaviour of an information theoretic leakage metric, considering in turn what this means for pointwise and global metrics. With the resulting requirements in mind, we take an axiomatic approach to defining a set of pointwise leakage metrics, as well as a set of global leakage metrics that are constructed from them. The global set encompasses many known measures including mutual information, Sibson mutual information, Arimoto mutual information, maximal leakage, min entropy leakage, -divergence metrics, and g-leakage. We prove that both sets follow the desired asymptotic behaviour. Finally, we derive composition theorems which quantify the rate of privacy degradation as an adversary is given access to a large number of independent observations of . It is found that, for both pointwise and global metrics, privacy degrades exponentially with increasing observations for the adversary, at a rate governed by the minimum Chernoff information between distinct conditional channel distributions. This extends the work of Wu et al. (2024), who have previously found this to be true for certain known metrics, including some that fall into our more general set.
Paper Structure (25 sections, 13 theorems, 135 equations, 6 figures, 2 tables)

This paper contains 25 sections, 13 theorems, 135 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Asymptotic Behaviour of Information Leakage
  4. Global Leakage Example: Mutual Information
  5. Pointwise Leakage Example: Pointwise Maximal Leakage
  6. Pointwise Leakage
  7. Axioms of the Pointwise Leakage Function
  8. Conditions on the Pointwise Leakage Function
  9. Asymptotic Behaviour of Pointwise Leakage and Composition Theorems
  10. Global Leakage
  11. Conditions on the Global Leakage Function
  12. Composition Theorem for Global Leakage
  13. Global Leakage Examples
  14. Privacy and Hypothesis Testing
  15. Discussion and Conclusions
  16. ...and 10 more sections

Key Result

Corollary 1

For a given $Q\textcolor{black}{\in \mathcal{P}^{\circ}_{\mathcal{X}}}$, the global maximum of $f(P,Q)$ is $f(E_{i},Q)$ where $i = \arg \min_{i'}q_{i'}$.

Figures (6)

  • Figure 1: Joint distribution for criminal record survey response and sanitised output
  • Figure 2: Joint distribution for PML simulation with $(p,q,r)=(0.6,0.3,0.1)$, $s=0.4$
  • Figure 3: Empirical CDFs of PML as $n$ increases for the joint distribution specified in Fig. \ref{['fig: setup sim 3x']}
  • Figure 4: An example over the probability simplex $\mathcal{P}_{\mathcal{X}}$ with $|\mathcal{X}|=3$
  • Figure 5: An example over the probability simplex $\mathcal{P}$ with $|\mathcal{Y}|=3$
  • ...and 1 more figures

Theorems & Definitions (26)

  • Remark 1
  • Corollary 1
  • proof
  • Definition 1
  • Lemma 1
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Corollary 2
  • proof
  • ...and 16 more