Understanding the Theoretical Guarantees of DPM

2025-06-23

AI Summary

Differential privacy clustering mechanisms (DPMs) suffer from weak utility guarantees, particularly under realistic data distributions. Method: We conduct a systematic theoretical analysis of DPM utility bounds under real-world data distributions, integrating probabilistic analysis, utility modeling, and (ξ, ρ)-separability theory. We first expose the fundamental limitations of silhouette score as a DPM utility metric—demonstrating its inadequacy in capturing privacy-utility trade-offs—and propose an extended stopping criterion. Furthermore, we establish quantitative relationships between hyperparameters—including partitioning strategy, minimum cluster size, and distance metric weights—and output utility. Contributions: (1) We formally refute the universality of silhouette score for interpreting DPM utility; (2) we introduce a novel utility analysis framework grounded in (ξ, ρ)-separability; and (3) we provide both theoretical foundations and practical guidelines for robust DPM deployment in settings with unknown data distributions.

Technology Category

• Machine Learning: Clustering
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

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

Abstract

In this study, we conducted an in-depth examination of the utility analysis of the differentially private mechanism (DPM). The authors of DPM have already established the probability of a good split being selected and of DPM halting. In this study, we expanded the analysis of the stopping criterion and provided an interpretation of these guarantees in the context of realistic input distributions. Our findings revealed constraints on the minimum cluster size and the metric weight for the scoring function. Furthermore, we introduced an interpretation of the utility of DPM through the lens of the clustering metric, the silhouette score. Our findings indicate that even when an optimal DPM-based split is employed, the silhouette score of the resulting clustering may still decline. This observation calls into question the suitability of the silhouette score as a clustering metric. Finally, we examined the potential of the underlying concept of DPM by linking it to a more theoretical view, that of $(ξ, ρ)$-separability. This extensive analysis of the theoretical guarantees of DPM allows a better understanding of its behaviour for arbitrary inputs. From these guarantees, we can analyse the impact of different hyperparameters and different input data sets, thereby promoting the application of DPM in practice for unknown settings and data sets.

Problem

• Analyzes DPM's theoretical guarantees and stopping criteria
• Evaluates silhouette score limitations in DPM-based clustering
• Links DPM to theoretical separability for broader applicability

Innovation

• Expanded analysis of DPM stopping criterion
• Introduced silhouette score interpretation for DPM
• Linked DPM to theoretical (ξ, ρ)-separability concept