Differentially Private Quantiles with Smaller Error

Jacob Imola; Fabrizio Boninsegna; Hannah Keller; Anders Aamand; Amrita Roy Chowdhury; Rasmus Pagh

Paper

Differentially Private Quantiles with Smaller Error

Abstract

In the approximate quantiles problem, the goal is to output

quantile estimates, the ranks of which are as close as possible to

given quantiles

. We present a mechanism for approximate quantiles that satisfies

-differential privacy for a dataset of

real numbers where the ratio between the distance between the closest pair of points and the size of the domain is bounded by

. As long as the minimum gap between quantiles is sufficiently large,

for all

, the maximum rank error of our mechanism is

with high probability. Previously, the best known algorithm under pure DP was due to Kaplan, Schnapp, and Stemmer~(ICML '22), who achieved a bound of

. Our improvement stems from the use of continual counting techniques which allows the quantiles to be randomized in a correlated manner. We also present an

-differentially private mechanism that relaxes the gap assumption without affecting the error bound, improving on existing methods when

is sufficiently close to zero. We provide experimental evaluation which confirms that our mechanism performs favorably compared to prior work in practice, in particular when the number of quantiles

is large.