Differentially Private Continual Release of Histograms and Related Queries

TL;DR

The paper addresses private continual release of high-dimensional histograms, introducing two parameterized, differential-private mechanisms that beat the conventional $O(d ext{log }T)$ error in regimes where either the maximum query output $q^*$ or the number of fluctuations $K$ is small. By partitioning the input stream into intervals and coupling with a private histogram subroutine, the authors achieve $Oigl(d ext{log}^2 (d q^*)+ ext{log }Tigr)$ error in insertions-only streams for a broad class of histogram queries, and $Oigl(d ext{log}^2 (dK)+ ext{log }Tigr)$ error in turnstile streams with few fluctuations, extending to negative entries and multiple queries. The work also provides extensions to $( ext{ε}, ext{δ})$-DP, natural-number inputs, and varying neighboring definitions, and discusses implications for lower bounds and practical private continual release in ML and streaming contexts. Overall, it advances the state of private continual histogram release by leveraging sparsity and slope-based updates to reduce additive error while preserving adaptive DP guarantees.

Abstract

We study privately releasing column sums of a $d$-dimensional table with entries from a universe $χ$ 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 ($χ= \{0,1\}$) 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 $Ω(d \log T)$ bound of prior work when $q^* \ll T$. Our second contribution is a mechanism for the turnstile model that admits negative entry updates ($χ= \{-1, 0,1\}$). 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 ($χ=\{0,1\}$).

Theorems & Definitions (63)

• Definition 1.1: Continual Histogram
• Definition 2.1: Monotone histogram query
• Theorem 1
• Theorem 2
• Definition 3.1
• Definition 3.2: Differential privacy Dwork2006
• Definition 3.3: $L_p$-sensitivity
• Definition 3.4: Laplace Distribution
• Definition 3.5: Differential privacy in the adaptive continual release model pricedpco
• Lemma 3.1