A Statistical Viewpoint on Differential Privacy: Hypothesis Testing, Representation and Blackwell's Theorem (2409.09558v2)

Abstract: Differential privacy is widely considered the formal privacy for privacy-preserving data analysis due to its robust and rigorous guarantees, with increasingly broad adoption in public services, academia, and industry. Despite originating in the cryptographic context, in this review paper we argue that, fundamentally, differential privacy can be considered a \textit{pure} statistical concept. By leveraging David Blackwell's informativeness theorem, our focus is to demonstrate based on prior work that all definitions of differential privacy can be formally motivated from a hypothesis testing perspective, thereby showing that hypothesis testing is not merely convenient but also the right language for reasoning about differential privacy. This insight leads to the definition of $f$-differential privacy, which extends other differential privacy definitions through a representation theorem. We review techniques that render $f$-differential privacy a unified framework for analyzing privacy bounds in data analysis and machine learning. Applications of this differential privacy definition to private deep learning, private convex optimization, shuffled mechanisms, and U.S.\ Census data are discussed to highlight the benefits of analyzing privacy bounds under this framework compared to existing alternatives.

• The paper recasts differential privacy as a hypothesis testing problem to rigorously evaluate privacy guarantees.
• It introduces f-differential privacy, a unified framework using a trade-off function derived from Blackwell's Theorem.
• The approach yields practical improvements in areas like private deep learning, convex optimization, shuffled mechanisms, and public data release.

A Statistical Viewpoint on Differential Privacy: Hypothesis Testing, Representation, and Blackwell's Theorem

This essay provides an analytical review of the paper titled "A Statistical Viewpoint on Differential Privacy: Hypothesis Testing, Representation and Blackwell's Theorem" by Weijie J. Su. The paper advances the understanding of differential privacy (DP) by framing it as a fundamentally statistical concept through the lens of hypothesis testing, facilitated by Blackwell's theorem.

Introduction and Motivation

Differential privacy has become a cornerstone in privacy-preserving data analysis, predominantly due to its robust, mathematically rigorous guarantees. Despite its roots in cryptography, the interpretation of DP through statistical techniques reveals its fundamental nature as a statistical concept. This paper posits that the conventional definition of DP can be formalized from a hypothesis testing perspective, subsequently leading to the notion of $f$-differential privacy, a unifying framework encompassing various DP definitions.

Core Contributions

Hypothesis Testing Interpretation

The paper asserts that any meaningful differential privacy definition should be construed as a hypothesis testing problem. At its core, differential privacy is designed to make it difficult for an attacker to distinguish between neighboring datasets. By modeling this challenge as a hypothesis test between two alternative realities (whether an individual's data is included or not), the authors leverage a powerful statistical framework to evaluate privacy guarantees.

Representation Theorem and Blackwell's Theorem

A significant theoretical contribution of the paper is the Representation Theorem, which demonstrates that any DP definition should inherently depend on the trade-off function between the type I and type II errors. According to the theorem:

$D(P, Q) = d(T(P, Q)),$

where $D$ is the differential privacy metric, $d$ is a link function, and $T$ represents the trade-off function. This result is rooted in Blackwell's Theorem, which provides the necessary condition for comparing probability distributions through hypothesis testing under certain transformations.

$f$-Differential Privacy

Based on the representation theorem, the paper introduces $f$-differential privacy. An algorithm $M$ is said to be $f$-differentially private if the trade-off function between the distributions $M(S)$ and $M(S')$ for any neighboring datasets $S$ and $S'$ satisfies:

$T(M(S), M(S')) \geq f.$

This framework generalizes existing DP definitions and provides a tighter, information-lossless representation of privacy guarantees. It includes well-known forms like pure DP and approximate DP and extends to divergence-based DPs like concentrated DP and Rényi DP.

Applications and Implications

Private Deep Learning

In the context of deep learning, particularly with DP-SGD (Differentially Private Stochastic Gradient Descent), the paper highlights the improved privacy bounds that can be achieved using $f$-differential privacy. The Edgeworth accountant technique further refines the composition of privacy losses, enhancing the analysis over traditional DP approaches.

Private Convex Optimization

The technique of releasing the last iterate in strongly convex optimization processes demonstrates significant privacy amplification. By employing $f$-differential privacy, researchers achieve tighter privacy bounds even for extended training durations, which are crucial for practical applications in machine learning.

Shuffled Mechanisms

Analyzing shuffled mechanisms through the $f$-differential privacy framework provides enhanced privacy guarantees by effectively handling mixture distributions. This is particularly relevant in federated settings where inputs are shuffled to obscure individual contributions.

U.S. Census Data

For accountable public data release, such as the U.S. Census, $f$-differential privacy offers substantial improvements over Rényi DP. By carefully analyzing the mixture properties of discrete Gaussian mechanisms used in the Census disclosure avoidance system, the approach achieves considerable reductions in privacy budget requirements.

Future Directions

The paper outlines several challenges and future research opportunities:

1. Technical Evaluation of Trade-off Functions: Simplifying the computation of trade-off functions and developing more efficient algorithms for $f$-differential privacy would broaden its applicability.
2. Non-Asymptotic Bounds: Investigating the trade-offs between asymptotic and non-asymptotic bounds could unveil optimal conditions for different settings.
3. Extensions to Non-Euclidean Data: Expanding the framework to handle non-Euclidean data will foster new applications in diverse domains.

Conclusion

By re-contextualizing differential privacy within a statistical hypothesis testing framework, this paper provides a foundational shift in understanding privacy guarantees. The $f$-differential privacy framework not only unifies various DP definitions but also enhances practical applicability across fields like machine learning and public data releases. This work paves the way for more precise and robust privacy-preserving techniques, crucial for the integrity and utility of data analysis in the modern era.