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Quantifying Privacy via Information Density

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

TL;DR

The paper develops explicit connections between privacy notions grounded in information density, showing how upper and lower bounds on the information density i(x;y) can be traded off to relate ALIP, LIP, PML, and LDP. By deriving precise bounds and leveraging the high-privacy regime, it establishes operational links: PML implies ALIP/LIP/LDP under concrete parameter mappings, and ALIP mechanisms can be optimal in the high-privacy setting. It also provides a robust interpretation of the lower bound via risk-averse adversaries, linking cost-function leakage to randomized-function leakage and connecting to maximal cost leakage concepts. Collectively, the work offers a unified, information-density-based lens for privacy-utility tradeoffs and mechanism design, with practical implications for discrete alphabets and context-aware privacy.

Abstract

We examine the relationship between privacy metrics that utilize information density to measure information leakage between a private and a disclosed random variable. Firstly, we prove that bounding the information density from above or below in turn implies a lower or upper bound on the information density, respectively. Using this result, we establish new relationships between local information privacy, asymmetric local information privacy, pointwise maximal leakage and local differential privacy. We further provide applications of these relations to privacy mechanism design. Furthermore, we provide statements showing the equivalence between a lower bound on information density and risk-averse adversaries. More specifically, we prove an equivalence between a guessing framework and a cost-function framework that result in the desired lower bound on the information density.

Quantifying Privacy via Information Density

TL;DR

The paper develops explicit connections between privacy notions grounded in information density, showing how upper and lower bounds on the information density i(x;y) can be traded off to relate ALIP, LIP, PML, and LDP. By deriving precise bounds and leveraging the high-privacy regime, it establishes operational links: PML implies ALIP/LIP/LDP under concrete parameter mappings, and ALIP mechanisms can be optimal in the high-privacy setting. It also provides a robust interpretation of the lower bound via risk-averse adversaries, linking cost-function leakage to randomized-function leakage and connecting to maximal cost leakage concepts. Collectively, the work offers a unified, information-density-based lens for privacy-utility tradeoffs and mechanism design, with practical implications for discrete alphabets and context-aware privacy.

Abstract

We examine the relationship between privacy metrics that utilize information density to measure information leakage between a private and a disclosed random variable. Firstly, we prove that bounding the information density from above or below in turn implies a lower or upper bound on the information density, respectively. Using this result, we establish new relationships between local information privacy, asymmetric local information privacy, pointwise maximal leakage and local differential privacy. We further provide applications of these relations to privacy mechanism design. Furthermore, we provide statements showing the equivalence between a lower bound on information density and risk-averse adversaries. More specifically, we prove an equivalence between a guessing framework and a cost-function framework that result in the desired lower bound on the information density.
Paper Structure (18 sections, 11 theorems, 46 equations, 1 figure, 1 table)

This paper contains 18 sections, 11 theorems, 46 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Summary of contributions and outline
  3. System Model and Privacy Metrics
  4. Context-aware privacy notions
  5. Local differential privacy
  6. New and existing relationships
  7. Bounding Information Density for Privacy
  8. ALIP guarantees implied by PML
  9. LIP guarantees implied by PML
  10. LDP guarantees implied by PML
  11. Application to Mechanism Design
  12. Operationalazing the Lower Bound on Information Density
  13. Quantifying Information Leaked to Risk-averse Adversaries
  14. Upper bound
  15. Lower bound
  16. ...and 3 more sections

Key Result

Theorem 1

For discret alphabets $\mathcal{X}$ and $\mathcal{Y}$, the pointwise maximal leakage between the variable $X$ and an outcome $y \in \mathcal{Y}$ can be expressed as

Figures (1)

  • Figure 1: Relationship between the parameters $\epsilon_l$ and $\epsilon_u$ for a uniform prior distribution and $N\in \{2,4\}$. PML offers a tight bound on $\epsilon_u$, which yields a corresponding AILP guarantee. Lower-bounding the information density results in a corresponding upper bound. The region in between these two bounds represents values in which both upper and lower bound can be achieved simultaneously. In the binary case, upper and lower bound coincide.

Theorems & Definitions (17)

  • Definition 1: Local information privacy (LIP), jiang2018context6483382
  • Definition 2: Asymmetric local information privacy (ALIP), zarrabian2023lift
  • Theorem 1: Pointwise maximal leakage (PML), saeidian2023pointwise
  • Definition 3: $\epsilon$-PML, saeidian2023pointwise
  • Definition 4: Local differential privacy (LDP), duchi2013LDPminmaxDEF
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • ...and 7 more