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Improved Accuracy for Private Continual Cardinality Estimation in Fully Dynamic Streams via Matrix Factorization

Joel Daniel Andersson, Palak Jain, Satchit Sivakumar

TL;DR

The paper tackles private cardinality estimation in fully dynamic streams under differential privacy. It introduces a structured sensitivity-vector framework and reduces cardinality problems to private continual counting on $ ext{S}_{D,k}$-streams, enabling the use of Toeplitz and tree-based factorizations. The authors derive improved error bounds for CountDistinct, DegreeCount, and TriangleCount, including low-space tree-based variants and a unifying analytical framework. These advances provide more accurate private streaming analytics for realistic update patterns, broadening the practical impact of differential privacy in dynamic data environments.

Abstract

We study differentially-private statistics in the fully dynamic continual observation model, where many updates can arrive at each time step and updates to a stream can involve both insertions and deletions of an item. Earlier work (e.g., Jain et al., NeurIPS 2023 for counting distinct elements; Raskhodnikova & Steiner, PODS 2025 for triangle counting with edge updates) reduced the respective cardinality estimation problem to continual counting on the difference stream associated with the true function values on the input stream. In such reductions, a change in the original stream can cause many changes in the difference stream, this poses a challenge for applying private continual counting algorithms to obtain optimal error bounds. We improve the accuracy of several such reductions by studying the associated $\ell_p$-sensitivity vectors of the resulting difference streams and isolating their properties. We demonstrate that our framework gives improved bounds for counting distinct elements, estimating degree histograms, and estimating triangle counts (under a slightly relaxed privacy model), thus offering a general approach to private continual cardinality estimation in streaming settings. Our improved accuracy stems from tight analysis of known factorization mechanisms for the counting matrix in this setting; the key technical challenge is arguing that one can use state-of-the-art factorizations for sensitivity vector sets with the properties we isolate. Empirically and analytically, we demonstrate that our improved error bounds offer a substantial improvement in accuracy for cardinality estimation problems over a large range of parameters.

Improved Accuracy for Private Continual Cardinality Estimation in Fully Dynamic Streams via Matrix Factorization

TL;DR

The paper tackles private cardinality estimation in fully dynamic streams under differential privacy. It introduces a structured sensitivity-vector framework and reduces cardinality problems to private continual counting on -streams, enabling the use of Toeplitz and tree-based factorizations. The authors derive improved error bounds for CountDistinct, DegreeCount, and TriangleCount, including low-space tree-based variants and a unifying analytical framework. These advances provide more accurate private streaming analytics for realistic update patterns, broadening the practical impact of differential privacy in dynamic data environments.

Abstract

We study differentially-private statistics in the fully dynamic continual observation model, where many updates can arrive at each time step and updates to a stream can involve both insertions and deletions of an item. Earlier work (e.g., Jain et al., NeurIPS 2023 for counting distinct elements; Raskhodnikova & Steiner, PODS 2025 for triangle counting with edge updates) reduced the respective cardinality estimation problem to continual counting on the difference stream associated with the true function values on the input stream. In such reductions, a change in the original stream can cause many changes in the difference stream, this poses a challenge for applying private continual counting algorithms to obtain optimal error bounds. We improve the accuracy of several such reductions by studying the associated -sensitivity vectors of the resulting difference streams and isolating their properties. We demonstrate that our framework gives improved bounds for counting distinct elements, estimating degree histograms, and estimating triangle counts (under a slightly relaxed privacy model), thus offering a general approach to private continual cardinality estimation in streaming settings. Our improved accuracy stems from tight analysis of known factorization mechanisms for the counting matrix in this setting; the key technical challenge is arguing that one can use state-of-the-art factorizations for sensitivity vector sets with the properties we isolate. Empirically and analytically, we demonstrate that our improved error bounds offer a substantial improvement in accuracy for cardinality estimation problems over a large range of parameters.
Paper Structure (48 sections, 41 theorems, 163 equations, 4 figures, 4 tables, 2 algorithms)

This paper contains 48 sections, 41 theorems, 163 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Improved Mechanisms for Continual Counting
  3. Inability to Directly Use Improved Mechanisms
  4. Our Results
  5. Counting Distinct Elements.
  6. Degree Histograms.
  7. Triangle Counting.
  8. Low-Space Error Bounds
  9. Technical Overview
  10. Overview of Framework
  11. Techniques Used in Proofs
  12. Challenges in Analyzing Factorization Mechanisms
  13. Related Work
  14. Differential Privacy under Continual Observation.
  15. Continual Counting and the Binary Tree Mechanism.
  16. ...and 33 more sections

Key Result

Theorem 3.1

Let $T, k, D \in \mathbb{N}$. Let $R$ be a lower-triangular $T \times T$ Toeplitz matrix with non-increasing and non-negative lower-diagonal values. Let $\vec{\Delta} \in \mathcal{S}_{D,k}$. Then,We note that if $k \leq D$, then the interval sum bound on $\vec{\Delta}$ is vacuous. In this case, for where $\|R\|_{1 \to 2}$ is the maximum $\ell_2$ norm of any column in matrix $R$.

Figures (4)

  • Figure 1: Root maximum and root mean expected squared error comparison between different mechanisms for $\mathsf{CountDistinct}$ over $T=2^{50}$ time steps for a fixed flippancy bound $k$. Errors are plotted relative to the exact error bound achieved by JainKRSS23. The upper bounds for $b$-ary trees with subtraction are plotted using \ref{['thm:leading-const-tree-approx']} (including lower order terms), each for the asymptotically optimal choice of $b$. The upper bound for the square-root factorization is using \ref{['cor:toep-error']} (including lower order terms), and the error for the naive factorization is exact (see \ref{['thm:naive']}).
  • Figure 2: Heatmap showing the alternating $k$-sparse sensitivity vectors $\vec{\Delta}$ which maximize $\| \sqrt{A} \vec{\Delta}\|_2$ for each $k\in[1,10]$ and $n=30$. The yellow squares represent $1$, the purple squares represent $-1$ and the teal squares represent $0$.
  • Figure 3: Comparison of the odd node count and the corresponding $\ell_2$ sensitivities for $b=2$ and $h=10$. $F_b(h, k)$ and $\hat{F}_b(h, k)$ are computed as dynamic programs using the formulations in \ref{['lem:full-tree-dp']} and \ref{['lem:partial-tree-dp']}. \ref{['fig:odd-node-comp']} directly plots the value of $F_b(h, k)$ and $\hat{F}_b(h, k)$, and \ref{['fig:tree-sens-comp']} relates them to compute the exact sensitivity of $R_b$ and $\hat{R}_b$ respectively. The top green line in \ref{['fig:tree-sens-comp']} is using the upper bound from JainKRSS23, translated to our setting.
  • Figure 4: Leading constants for the $\rho$-zCDP (\ref{['thm:leading-const-tree-approx']}) and $\varepsilon$-DP (\ref{['thm:leading-const-tree-pure']}) bounds on $\mathsf{MaxSE}(\hat{L}_b, \hat{R}_b, \mathcal{S}_{1, k})$ and $\mathsf{MeanSE}(\hat{L}, \hat{R}_b, \mathcal{S}_{1, k})$ plotted versus odd $b$. In \ref{['fig:zcdp-tree-constant-opt']}, they are minimized for $b=5$ and $b=7$ respectively, attaining minima of $0.609$ and $0.466$. In \ref{['fig:puredp-tree-constant-opt']}, they are minimized for $b=17$ and $b=19$ respectively, attaining minima of $0.342$ and $0.249$.

Theorems & Definitions (92)

  • Theorem 3.1
  • Theorem 3.2: Informal Error Bounds for Continual Counting on $\mathcal{S}_{D,k}$-streams
  • Theorem 3.3: Informal error bounds for tree-based mechanisms
  • Theorem 3.4
  • Definition 5.1: $(\varepsilon, \delta)$-Differential Privacy dp_2006dwork_algorithmic_2013
  • Definition 5.2: $\rho$-Zero-Concentrated Differential Privacy bun_concentrated_2016
  • Definition 5.3: $\ell_p$ Sensitivity
  • Definition 5.4: Gaussian Mechanism bun_concentrated_2016
  • Definition 5.5: Laplace Mechanism dp_2006
  • Definition 5.6: Item Neighbors for $\mathsf{CountDistinct}$
  • ...and 82 more