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arXiv · 2502.02990

Lightweight Protocols for Distributed Private Quantile Estimation

Anders Aamand·Fabrizio Boninsegna·Abigail Gentle·Jacob Imola·Rasmus Pagh
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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 $O(\frac{\log B}{\varepsilon^2\alpha^2})$ 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.

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Anders Aamand, Fabrizio Boninsegna, Abigail Gentle, Jacob Imola, Rasmus Pagh. 2025-02-05. Lightweight Protocols for Distributed Private Quantile Estimation. https://arxiv.org/abs/2502.02990

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