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Dobrushin Coefficients of Private Mechanisms Beyond Local Differential Privacy

Leonhard Grosse, Sara Saeidian, Tobias J. Oechtering, Mikael Skoglund

TL;DR

The paper studies contraction of the Dobrushin total-variation coefficient for discrete privacy mechanisms under bounded pointwise maximal leakage (PML), generalizing local differential privacy (LDP) via the input-distribution constraint $\min_x P_X(x)\ge c$. It introduces $(\varepsilon,c)$-PML, derives a tight upper bound on the Dobrushin contraction $\eta_{TV}$, and shows the optimum is achieved by binary-output mechanisms with a clear phase transition at $\varepsilon=\log(2/(Nc))$, recovering LDP limits as $c\to0$ and matching uniform-input results at $c=1/N$. The results extend to general $f$-divergences using Binette's inequality, yielding explicit bounds for relative entropy and Hellinger divergences and a private minimax-risk interpretation where effective sample size scales with a factor $\Xi(\varepsilon,c)$. These findings provide tighter privacy-utility tradeoffs beyond LDP for discrete data and offer practical mechanism designs and risk bounds for statistical tasks under privacy constraints.

Abstract

We investigate Dobrushin coefficients of discrete Markov kernels that have bounded pointwise maximal leakage (PML) with respect to all distributions with a minimum probability mass bounded away from zero by a constant $c>0$. This definition recovers local differential privacy (LDP) for $c\to 0$. We derive achievable bounds on contraction in terms of a kernels PML guarantees, and provide mechanism constructions that achieve the presented bounds. Further, we extend the results to general $f$-divergences by an application of Binette's inequality. Our analysis yields tighter bounds for mechanisms satisfying LDP and extends beyond the LDP regime to any discrete kernel.

Dobrushin Coefficients of Private Mechanisms Beyond Local Differential Privacy

TL;DR

The paper studies contraction of the Dobrushin total-variation coefficient for discrete privacy mechanisms under bounded pointwise maximal leakage (PML), generalizing local differential privacy (LDP) via the input-distribution constraint . It introduces -PML, derives a tight upper bound on the Dobrushin contraction , and shows the optimum is achieved by binary-output mechanisms with a clear phase transition at , recovering LDP limits as and matching uniform-input results at . The results extend to general -divergences using Binette's inequality, yielding explicit bounds for relative entropy and Hellinger divergences and a private minimax-risk interpretation where effective sample size scales with a factor . These findings provide tighter privacy-utility tradeoffs beyond LDP for discrete data and offer practical mechanism designs and risk bounds for statistical tasks under privacy constraints.

Abstract

We investigate Dobrushin coefficients of discrete Markov kernels that have bounded pointwise maximal leakage (PML) with respect to all distributions with a minimum probability mass bounded away from zero by a constant . This definition recovers local differential privacy (LDP) for . We derive achievable bounds on contraction in terms of a kernels PML guarantees, and provide mechanism constructions that achieve the presented bounds. Further, we extend the results to general -divergences by an application of Binette's inequality. Our analysis yields tighter bounds for mechanisms satisfying LDP and extends beyond the LDP regime to any discrete kernel.
Paper Structure (16 sections, 15 theorems, 84 equations, 1 figure)

This paper contains 16 sections, 15 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. $f$-Divergences and Contraction Coefficients
  5. Pointwise Maximal Leakage
  6. Related Work
  7. Dobrushin Coefficients with Bounded PML
  8. Extension to General $f$-Divergences
  9. Disclosure Limits of Subset Membership
  10. Proof of Lemma \ref{['lem:QcdontchangeTVcontr']}
  11. Proof of Lemma \ref{['lem:cardinality']}
  12. Proof of Theorem \ref{['thm:M1bound']}
  13. Mechanisms in $\mathcal{M}_1(\varepsilon,c)$
  14. Mechanisms in $\mathcal{M}_2(\varepsilon,c)$
  15. Mechanisms in $\mathcal{M}_0(\varepsilon,c)$
  16. ...and 1 more sections

Key Result

Theorem 1

Let $c\in(0,1/|\mathcal{X}|]$ and $\varepsilon\geq0$. If a Markov kernel $K$ has PML bounded by $\varepsilon$ for all distributions $P_X$ with a minimum probability mass $\min_{x\in\mathcal{X}} P_X(x)\geq c$, we have,

Figures (1)

  • Figure 1: Numerical evaluation of the bounds presented in Corollaries \ref{['corr:relativeentropy']} and \ref{['corr:hellinger']} for randomly generated distributions in the set $\mathcal{Q}_\mathcal{X}(c)$. $K_i$ for $i=1,2$ are given in Example \ref{['ex:RR']}.

Theorems & Definitions (26)

  • Theorem 1
  • Proposition 1: IssaMaxL
  • Definition 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Lemma 3
  • Theorem 3
  • Remark 1
  • Remark 2
  • ...and 16 more