Differential Privacy as a Mutual Information Constraint (1608.03677v1)

Abstract: Differential privacy is a precise mathematical constraint meant to ensure privacy of individual pieces of information in a database even while queries are being answered about the aggregate. Intuitively, one must come to terms with what differential privacy does and does not guarantee. For example, the definition prevents a strong adversary who knows all but one entry in the database from further inferring about the last one. This strong adversary assumption can be overlooked, resulting in misinterpretation of the privacy guarantee of differential privacy. Herein we give an equivalent definition of privacy using mutual information that makes plain some of the subtleties of differential privacy. The mutual-information differential privacy is in fact sandwiched between $\epsilon$-differential privacy and $(\epsilon,\delta)$-differential privacy in terms of its strength. In contrast to previous works using unconditional mutual information, differential privacy is fundamentally related to conditional mutual information, accompanied by a maximization over the database distribution. The conceptual advantage of using mutual information, aside from yielding a simpler and more intuitive definition of differential privacy, is that its properties are well understood. Several properties of differential privacy are easily verified for the mutual information alternative, such as composition theorems.

• The paper introduces MI-DP, redefining differential privacy as a mutual information constraint that quantifies individual data exposure.
• It positions MI-DP between ε-DP and (ε, δ)-DP, emphasizing trade-offs in privacy strength across existing frameworks.
• Using information-theoretic methods, the paper provides rigorous proofs and composition results for handling complex database queries.

Differential Privacy as a Mutual Information Constraint

The paper under discussion presents an in-depth exploration of differential privacy (DP) through the lens of mutual information. The authors, Paul Cuff and Lanqing Yu from Princeton University, introduce a novel perspective by examining differential privacy as a mutual information constraint. Their approach seeks to provide a more intuitive understanding of differential privacy by relating it directly to information theory principles, particularly conditional mutual information.

Main Contributions

1. Redefinition of Differential Privacy: The authors propose an equivalent definition of differential privacy centered on mutual information. Traditional differential privacy guarantees that the output of a database query does not reveal much about any individual entry. This is quantified through parameters $\epsilon$ and $\delta$. The authors redefine the privacy constraint using mutual information, specifically examining how much information about a single database entry can be gained when conditioned on the rest of the database. This framework is termed $\epsilon$-mutual-information differential privacy (MI-DP).
2. Positioning MI-DP within Existing Privacy Definitions: The research places MI-DP within the spectrum of existing privacy definitions. The authors demonstrate that MI-DP lies between $\epsilon$-differential privacy and $(\epsilon, \delta)$-differential privacy in terms of privacy strength. In practical terms, this means that any mechanism satisfying $\epsilon$-differential privacy also satisfies MI-DP, and mechanisms satisfying MI-DP also meet $(\epsilon', \delta)$-differential privacy for appropriately chosen parameters.
3. Properties and Implications: By leveraging the well-understood properties of mutual information, several attributes of differential privacy become more evident. The paper discusses the advantages of using mutual information, such as the immediate verification of composition theorems and the ability to clearly articulate the strong adversary assumption implicit in differential privacy. The strong adversary scenario assumes that all database entries except one are known when trying to infer information about the single unknown entry.
4. Equivalence Under Bounded Conditions: The authors claim the equivalence between MI-DP and $(\epsilon,\delta)$-DP when the range or domain of the query functions is finite. This ties back to the information-theoretic principles where mutual information offers a measurable depiction of privacy guarantees.

Numerical and Theoretical Insights

The paper provides a rigorous mathematical treatment of the problem space, including theorems and corollaries that offer nuanced understanding of differentials in privacy constraints. For instance, the authors provide a mathematical proof showing how mutual information provides a clearer picture of privacy compromises than traditional definitions. Notably, the equivalence relationship establishes a strong groundwork for assessing privacy without resorting to heuristic arguments.

Practical and Theoretical Implications

• Database Privacy Assurance:

This new viewpoint clarifies what differential privacy can and cannot guarantee in terms of privacy, namely the impact of correlation among database entries, which may affect privacy if not considered accurately within the MI framework.

• Complex Query Handling:

By using mutual information, the paper simplifies understanding the interactions between numerous, potentially dependent queries on a dataset. This aspect is crucial for organizations that handle multiple queries and need to ensure cumulative privacy protection.

• Future Research Directions:

The insights provided could spur further exploration into other information-theoretic measures like R{e}nyi entropy as potential metrics for assessing privacy, opening avenues for privacy research that exploits different mathematical properties.

Conclusion

This paper makes a significant contribution by recasting differential privacy through the lens of mutual information – a well-established concept in information theory. This redefinition not only facilitates an enhanced understanding of privacy guarantees but also provides new analytical tools to evaluate privacy in complex systems. Such insights are invaluable in safeguarding privacy in an increasingly data-reliant world, ensuring that privacy preservation techniques keep pace with advances in data analytics. Going forward, the exploration into mutual information as a differential privacy constraint could inspire novel methodologies for designing privacy-aware systems in diverse domains.