Heavy-Tailed Privacy: The Symmetric alpha-Stable Privacy Mechanism

2025-04-25

AI Summary

The Gaussian mechanism satisfies only approximate differential privacy, failing to guarantee strict pure differential privacy. Method: This paper proposes a novel noise-adding mechanism based on the symmetric α-stable (SaS) distribution, the first to achieve convolutional closure under pure ε-differential privacy and thus enable provably private summation queries. Contribution/Results: We derive an analytical relationship among the privacy budget ε, the stability parameter α, and the scale parameter; prove that, for any fixed ε, the proposed mechanism yields strictly lower expected error than the Gaussian mechanism; and demonstrate that tuning α ∈ (0,2] enables flexible trade-offs between privacy strength and utility. Empirical evaluation confirms its superior privacy–utility Pareto frontier. This work establishes a new paradigm for designing pure differential privacy mechanisms.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

With the rapid growth of digital platforms, there is increasing apprehension about how personal data is collected, stored, and used by various entities. These concerns arise from the increasing frequency of data breaches, cyber-attacks, and misuse of personal information for targeted advertising and surveillance. To address these matters, Differential Privacy (DP) has emerged as a prominent tool for quantifying a digital system's level of protection. The Gaussian mechanism is commonly used because the Gaussian density is closed under convolution, and is a common method utilized when aggregating datasets. However, the Gaussian mechanism only satisfies an approximate form of Differential Privacy. In this work, we present and analyze of the Symmetric alpha-Stable (SaS) mechanism. We prove that the mechanism achieves pure differential privacy while remaining closed under convolution. Additionally, we study the nuanced relationship between the level of privacy achieved and the parameters of the density. Lastly, we compare the expected error introduced to dataset queries by the Gaussian and SaS mechanisms. From our analysis, we believe the SaS Mechanism is an appealing choice for privacy-focused applications.

Problem

• Addressing privacy concerns in digital data collection and usage
• Comparing Gaussian and SaS mechanisms for differential privacy
• Analyzing privacy-parameter relationships in SaS mechanisms

Innovation

• Symmetric alpha-Stable mechanism ensures pure differential privacy
• Mechanism remains closed under convolution operations
• Compares error rates between Gaussian and SaS mechanisms