ICML 2025

Lightweight Protocols for Distributed Private Quantile Estimation
Anders Aamand 1 Fabrizio Boninsegna 2 Abigail Gentle 3 Jacob Imola 1 Rasmus Pagh 1
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
adaptive algorithms for estimating one quantile
(e.g. the median) when each user holds a sin-
gle 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 ε-
LDP algorithm which can estimate any quan-
tile within error α only requiring O( log B
ε2α2 ) users,
and an (ε, δ)-shuffle DP algorithm requiring only
eO(( 1
ε2 +
1
α2 ) log B) users. Prior (nonadaptive)
algorithms require more users by several logarith-
mic factors in B. We further provide a match-
ing lower bound for adaptive protocols, showing
that our LDP algorithm is optimal in the low-ε
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.
1. Introduction
A strong trend in recent years has been towards federated
computations (Kairouz et al., 2021) in which algorithms are
run on a distributed dataset rather than by collecting data and
performing the computation in a centralized manner. This
trend is motivated by the wish to protect individuals’ data,
as well as organizations’ wish steer clear of liability issues
stemming from collecting and handling private data. The
1BARC and Department of Computer Science, Univer-
sity of Copenhagen, Denmark 2Department of Information En-
gineering, University of Padova, Italy 3School of Computer
Science, University of Sydney, Australia.
Correspondence
to:
Anders Aamand <aa@di.ku.dk>, Abigail Gentle <abi-
gail.gentle@sydney.ed...