Bounded Space Differentially Private Quantiles

Abstract

Estimating the quantiles of a large dataset is a fundamental problem in both the streaming algorithms literature and the differential privacy literature. However, all existing private mechanisms for distribution-independent quantile computation require space at least linear in the input size 𝑛. In this work, we devise a differentially private algorithm for the quantile estimation problem, with strongly sublinear space complexity, in the one-shot and continual observation settings. Our basic mechanism estimates any 𝛼approximate quantile of a length-𝑛 stream over a data universe X with probability 1 − 𝛽 using 𝑂 log( | X |/𝛽) log(𝛼𝜖𝑛) 𝛼𝜖 space while satisfying 𝜖-differential privacy at a single time point. Our approach builds upon deterministic streaming algorithms for non-private quantile estimation instantiating the exponential mechanism using a utility function defined on sketch items, while (privately) sampling from intervals defined by the sketch. We also present another algorithm based on histograms that is especially suited to the multiple quantiles case. We implement our algorithms and experimentally evaluate them on synthetic and real-world datasets.

Differentially Private Online Bayesian Estimation With Adaptive Truncation

“…One limitation of this work is the requirement of considering a parametric distribution family for the population distribution. Estimation of characteristics regarding non-parametric distributions, such the as mean and quantiles, is also a fundamental problem for data streaming applications (Dwork and Lei, 2009;Alabi et al, 2022). A useful extension of this work would be an online estimation methodology of (characteristics) non-parametric distributions.…”

Additive Noise Mechanisms for Making Randomized Approximation Algorithms Differentially Private

“…A sketch for fractional frequency moments F p for 0 ≤ p ≤ 1 has been given by Wang, Pinelis, and Song [27]. A sketch for differentially private quantiles has been given by Alabi, Ben-Eliezer, and Chaturvedi [1]. A technique for stream sanitization has been given by Kaplan and Stemmer [17]; this work resulted in improved differentially private sketches for approximate quantiles.…”

“…One of the main difficulties is that most sublinear-time and streaming algorithms are randomized and give only probabilistic guarantees on the quality of the output. This makes adding noise based on global sensitivity 1 -which is commonly used to get differentially private algorithms -unsuitable for this situation, as in the worst case the global sensitivity of the approximation algorithm 2 is very large even if the global sensitivity of the function being approximated is small. In this paper, we propose a way to get around this issue by showing additive noise mechanisms that only need that (1) the function being approximated has low global sensitivity and (2) the answer of the algorithm is sufficiently concentrated around the true value.…”

Differentially Private Linear Sketches: Efficient Implementations and Applications

“…Recently, researchers have observed that some data sketches are inherently differentially private [Blocki et al, 2012, Smith et al, 2020, while many other data sketches need modifications to the algorithm to be differentially private. In particular, a substantial amount of literature has focused on differentially private data sketches for tasks such as linear algebra [Upadhyay, 2014, Arora et al, 2018, cardinality estimation [Mir et al, 2011, Pagh and Stausholm, 2020, Dickens et al, 2022 and quantile approximation [Tzamos et al, 2020, Gillenwater et al, 2021, Alabi et al, 2022.…”