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 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.

In the approximate quantiles problem, the goal is to output $m$ quantile estimates, the ranks of which are as close as possible to $m$ given quantiles $0 \leq q_1 \leq\dots \leq q_m \leq 1$. We present a mechanism for approximate quantiles that satisfies $\varepsilon$-differential privacy for a dataset of $n$ real numbers where the ratio between the distance between the closest pair of points and the size of the domain is bounded by $\psi$. As long as the minimum gap between quantiles is sufficiently large, $|q_i-q_{i-1}|\geq \Omega\left(\frac{m\log(m)\log(\psi)}{n\varepsilon}\right)$ for all $i$, the maximum rank error of our mechanism is $O\left(\frac{\log(\psi) + \log^2(m)}{\varepsilon}\right)$ 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 $O\left(\frac{\log(\psi)\log^2(m) + \log^3(m)}{\varepsilon}\right)$. 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 $(\varepsilon,\delta)$-differentially private mechanism that relaxes the gap assumption without affecting the error bound, improving on existing methods when $\delta$ 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 $m$ is large.

This paper presents a novel method for generating differentially private tabular datasets for hierarchical data, specifically focusing on origin-destination (O/D) trips. The approach builds upon the TopDown algorithm, a constraint-based mechanism developed by the U.S. Census to incorporate invariant queries into tabular data. O/D hierarchical data refers to datasets representing trips between geographical areas organized in a hierarchical structure (e.g., region -> province -> city). The proposed method is designed to improve the accuracy of queries covering broader geographical areas, which are derived through aggregation. This feature provides a “zoom-in” effect on the dataset, ensuring that when zoomed back out, the overall picture is preserved. Furthermore, the approach aims to reduce false positive detection. These characteristics can strengthen practitioners' and decision-makers' confidence in adopting differential privacy datasets. The main technical contribution of this paper includes a novel TopDown algorithm that employs constrained optimization with Chebyshev distance minimization, with theoretical guarantees on the maximum absolute error. Additionally, we propose a new integer optimization algorithm that significantly reduces the incidence of false positives. The effectiveness of the proposed approach is validated using real-world and synthetic O/D datasets, demonstrating its ability to generate private data with high utility and a reduced number of false positives. Our experiments focus on O/D datasets with a single trip as a unit of privacy: nevertheless, the proposed approach supports other units of privacy and also applies to any tabular data with a hierarchical structure.