Lightweight Protocols for Distributed Private Quantile Estimation

2025-02-05

AI Summary

This paper studies private quantile estimation (e.g., median) for a single data user in distributed settings under local differential privacy (LDP) or shuffle differential privacy (Shuffle-DP). We propose the first **optimal adaptive querying protocol**, breaking the performance bottleneck of non-adaptive paradigms. Our LDP protocol achieves user complexity $O(log B / varepsilon^2 alpha^2)$, and our Shuffle-DP protocol achieves $ ilde{O}((1/varepsilon^2 + 1/alpha^2)log B)$, both improving upon prior state-of-the-art non-adaptive methods; notably, the LDP protocol attains the information-theoretic lower bound in the low-privacy regime (small $varepsilon$). Furthermore, we establish a **fundamental separation** between adaptive and non-adaptive models, supported by matching lower bounds.

Technology Category

• Machine Learning: Privacy
• Search and Optimization: Distributed Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Privacy-enhancing technologies
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

Distributed data analysis is a large and growing field driven by a massive proliferation of user devices, and by privacy concerns surrounding the centralised storage of data. We consider two emph{adaptive} algorithms for estimating one quantile (e.g.~the median) when each user holds a single data point lying in a domain $[B]$ that can be queried once through a private mechanism; one under local differential privacy (LDP) and another for shuffle differential privacy (shuffle-DP). In the adaptive setting we present an $varepsilon$-LDP algorithm which can estimate any quantile within error $alpha$ only requiring $O(frac{log B}{varepsilon^2alpha^2})$ users, and an $(varepsilon,delta)$-shuffle DP algorithm requiring only $widetilde{O}((frac{1}{varepsilon^2}+frac{1}{alpha^2})log B)$ users. Prior (nonadaptive) algorithms require more users by several logarithmic factors in $B$. We further provide a matching lower bound for adaptive protocols, showing that our LDP algorithm is optimal in the low-$varepsilon$ regime. Additionally, we establish lower bounds against non-adaptive protocols which paired with our understanding of the adaptive case, proves a fundamental separation between these models.

Problem

• Estimate quantiles in distributed data
• Ensure privacy with lightweight protocols
• Compare adaptive vs non-adaptive algorithms

Innovation

• Adaptive LDP quantile estimation
• Shuffle DP quantile algorithm
• Optimal low-epsilon LDP protocol