Optimality of Staircase Mechanisms for Vector Queries under Differential Privacy

2026-01-21

AI Summary

This work addresses the problem of designing optimal additive noise mechanisms for vector-valued queries under differential privacy, aiming to minimize expected utility loss. Given a query’s sensitivity and an arbitrary norm-monotone cost function, the authors leverage convex rearrangement theory to reduce the infinite-dimensional optimization problem to optimizing over a family of radially symmetric distributions supported on a one-dimensional compact convex set. Building on this reduction, they prove that, in any dimension, under any norm, and for any norm-monotone cost function, there always exists a staircase mechanism that achieves optimality. This result resolves a conjecture posed by Geng et al. and provides a geometric interpretation of the optimality condition. Consequently, the study establishes the universal optimality of staircase mechanisms among all additive differentially private mechanisms, offering a rigorous theoretical foundation and constructive guidance for mechanism design.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We study the optimal design of additive mechanisms for vector-valued queries under $\epsilon$-differential privacy (DP). Given only the sensitivity of a query and a norm-monotone cost function measuring utility loss, we ask which noise distribution minimizes expected cost among all additive $\epsilon$-DP mechanisms. Using convex rearrangement theory, we show that this infinite-dimensional optimization problem admits a reduction to a one-dimensional compact and convex family of radially symmetric distributions whose extreme points are the staircase distributions. As a consequence, we prove that for any dimension, any norm, and any norm-monotone cost function, there exists an $\epsilon$-DP staircase mechanism that is optimal among all additive mechanisms. This result resolves a conjecture of Geng, Kairouz, Oh, and Viswanath, and provides a geometric explanation for the emergence of staircase mechanisms as extremal solutions in differential privacy.

Problem

• differential privacy
• vector queries
• additive mechanisms
• staircase mechanisms
• optimality

Innovation

• staircase mechanism
• differential privacy
• convex rearrangement
• additive noise
• optimality