Differentially Private Continual Release of Histograms and Related Queries

TL;DR

This paper introduces differentially private mechanisms for releasing histograms and related queries in continual data streams, achieving improved error bounds and handling both insertions-only and turnstile models with negative updates.

Contribution

It presents novel output-sensitive mechanisms for continual histogram release with better error bounds, applicable to both insertions-only and turnstile models, without requiring prior knowledge of maximum output values.

Findings

Achieves $O(d ext{log}^2(dq^*)+ ext{log} T)$ error in insertions-only model.

Provides $O(d ext{log}^2(dK)+ ext{log} T)$ error mechanism for turnstile model with negative updates.

First private mechanism with $O( ext{log}^2 K + ext{log} T)$ error for continual counting in turnstile model.

Abstract

We study privately releasing column sums of a $d$-dimensional table with entries from a universe $\chi$ undergoing $T$ row updates, called histogram under continual release. Our mechanisms give better additive $\ell_\infty$-error than existing mechanisms for a large class of queries and input streams. Our first contribution is an output-sensitive mechanism in the insertions-only model ($\chi = \{0,1\}$) for maintaining (i) the histogram or (ii) queries that do not require maintaining the entire histogram, such as the maximum or minimum column sum, the median, or any quantiles. The mechanism has an additive error of $O(d\log^2 (dq^*)+\log T)$ whp, where $q^*$ is the maximum output value over all time steps on this dataset. The mechanism does not require $q^*$ as input. This breaks the $\Omega(d \log T)$ bound of prior work when $q^* \ll T$. Our second contribution is a mechanism for the turnstile model that admits negative entry updates ($\chi = \{-1, 0,1\}$). This mechanism has an additive error of $O(d \log^2 (dK) + \log T)$ whp, where $K$ is the number of times two consecutive data rows differ, and the mechanism does not require $K$ as input. This is useful when monitoring inputs that only vary under unusual circumstances. For $d=1$ this gives the first private mechanism with error $O(\log^2 K + \log T)$ for continual counting in the turnstile model, improving on the $O(\log^2 n + \log T)$ error bound by Dwork et al. [ASIACRYPT 2015], where $n$ is the number of ones in the stream, as well as allowing negative entries, while Dwork et al. [ASIACRYPT 2015] can only handle nonnegative entries ($\chi=\{0,1\}$).