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Composition for Pufferfish Privacy

Jiamu Bai, Guanlin He, Xin Gu, Daniel Kifer, Kiwan Maeng

TL;DR

It is proved necessary and sufficient conditions that must be added to ensure linear composition for Pufferfish mechanisms, hence avoiding privacy collapse and showing that achieving both the interpretable semantics of Pufferfish for correlated data and composition benefits requires adopting differentially private mechanisms to Pufferfish.

Abstract

When creating public data products out of confidential datasets, inferential/posterior-based privacy definitions, such as Pufferfish, provide compelling privacy semantics for data with correlations. However, such privacy definitions are rarely used in practice because they do not always compose. For example, it is possible to design algorithms for these privacy definitions that have no leakage when run once but reveal the entire dataset when run more than once. We prove necessary and sufficient conditions that must be added to ensure linear composition for Pufferfish mechanisms, hence avoiding such privacy collapse. These extra conditions turn out to be differential privacy-style inequalities, indicating that achieving both the interpretable semantics of Pufferfish for correlated data and composition benefits requires adopting differentially private mechanisms to Pufferfish. We show that such translation is possible through a concept called the $(a,b)$-influence curve, and many existing differentially private algorithms can be translated with our framework into a composable Pufferfish algorithm. We illustrate the benefit of our new framework by designing composable Pufferfish algorithms for Markov chains that significantly outperform prior work.

Composition for Pufferfish Privacy

TL;DR

It is proved necessary and sufficient conditions that must be added to ensure linear composition for Pufferfish mechanisms, hence avoiding privacy collapse and showing that achieving both the interpretable semantics of Pufferfish for correlated data and composition benefits requires adopting differentially private mechanisms to Pufferfish.

Abstract

When creating public data products out of confidential datasets, inferential/posterior-based privacy definitions, such as Pufferfish, provide compelling privacy semantics for data with correlations. However, such privacy definitions are rarely used in practice because they do not always compose. For example, it is possible to design algorithms for these privacy definitions that have no leakage when run once but reveal the entire dataset when run more than once. We prove necessary and sufficient conditions that must be added to ensure linear composition for Pufferfish mechanisms, hence avoiding such privacy collapse. These extra conditions turn out to be differential privacy-style inequalities, indicating that achieving both the interpretable semantics of Pufferfish for correlated data and composition benefits requires adopting differentially private mechanisms to Pufferfish. We show that such translation is possible through a concept called the -influence curve, and many existing differentially private algorithms can be translated with our framework into a composable Pufferfish algorithm. We illustrate the benefit of our new framework by designing composable Pufferfish algorithms for Markov chains that significantly outperform prior work.
Paper Structure (35 sections, 1 theorem, 68 equations, 3 figures, 8 tables, 2 algorithms)

This paper contains 35 sections, 1 theorem, 68 equations, 3 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Differential Privacy
  4. Posterior-based Privacy Definitions
  5. Related Work
  6. Privacy Collapse
  7. Necessary Conditions for Composition and Post-processing Invariance
  8. Necessary Condition for Composition
  9. Enforcing Post-processing Invariance
  10. Constructing General Pufferfish Mechanisms with Composition
  11. The MQM and Max-Influence song2017pufferfish
  12. Sufficient Condition for Composition
  13. Example ${\textcolor{black}{$a(b)$}}$-influence Curves
  14. Example Pufferfish Mechanisms
  15. Pufferfish Laplace Mechanism
  16. ...and 20 more sections

Key Result

Theorem 1

Consider Example ex:gaussian where $\mathbf{x}\sim\mathcal{N}(0,\Sigma)$ is truncated to $[-\gamma,\gamma]^n$ and $\Sigma$ is induced by the Gaussian-process kernel $\Sigma_{jk}=\exp\!\bigl(-(j-k)^2/\ell\bigr)$. Fix a region length $\delta<<\gamma$ and define region secrets as $s_i(r)\equiv\{x_i\in[

Figures (3)

  • Figure 1: Example illustration with ${\textcolor{black}{$\mathcal{W}$}}_\epsilon={\textcolor{black}{$\mathcal{M}$}}_1, {\textcolor{black}{$\mathcal{M}$}}_2, {\textcolor{black}{$\mathcal{M}$}}_3$, with each mechanism having 3, 2, and 4 unique outputs. In this illustration, $k=4$ vectors are picked.
  • Figure 2: An illustration of $X_R, X_Q, X_N$ in Markov Chain.
  • Figure 3: ${\textcolor{black}{$a(b)$}}$-influence curves under different priors: (a) binary Markov chains and (b) multivariate Gaussian priors.

Theorems & Definitions (8)

  • definition 1: Pure Differential Privacy dwork2006calibrating
  • definition 2: Per-Entry DP
  • definition 3: Group DP
  • definition 4: Posterior-based Privacy
  • definition 5: Pufferfish Privacy pufferfish
  • definition 6: Max-Influence song2017pufferfish
  • definition 7: ${\textcolor{black}{$a(b)$}}$-influence
  • Theorem 1