Private Continual Counting of Unbounded Streams

2025-06-17

AI Summary

This paper studies the problem of differentially private continual counting over unbounded data streams—where the stream length is unknown and smooth, real-time estimates of prefix sums up to time $t$ are required. Existing matrix-mechanism-based optimal algorithms assume prior knowledge of the total length $n$, while standard doubling techniques yield large and unstable error. We propose a novel logarithmic-scale matrix decomposition derived from the generating function $1/sqrt{1-z}$, achieving for the first time simultaneous smooth responsiveness and near-optimal variance $O(log^{2+2alpha} t)$ in the unbounded setting. The algorithm uses $O(t)$ space and $O(log t)$ amortized time per update. Experiments show that for $t leq 2^{24}$, its variance is less than 1.5× that of the SODA’23 optimal algorithm for bounded streams—demonstrating substantial improvements in both practical utility and theoretical accuracy.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of differentially private continual counting in the unbounded setting where the input size $n$ is not known in advance. Current state-of-the-art algorithms based on optimal instantiations of the matrix mechanism cannot be directly applied here because their privacy guarantees only hold when key parameters are tuned to $n$. Using the common `doubling trick' avoids knowledge of $n$ but leads to suboptimal and non-smooth error. We solve this problem by introducing novel matrix factorizations based on logarithmic perturbations of the function $frac{1}{sqrt{1-z}}$ studied in prior works, which may be of independent interest. The resulting algorithm has smooth error, and for any $alpha>0$ and $tleq n$ it is able to privately estimate the sum of the first $t$ data points with $O(log^{2+2alpha}(t))$ variance. It requires $O(t)$ space and amortized $O(log t)$ time per round, compared to $O(log(n)log(t))$ variance, $O(n)$ space and $O(n log n)$ pre-processing time for the nearly-optimal bounded-input algorithm of Henzinger et al. (SODA 2023). Empirically, we find that our algorithm's performance is also comparable to theirs in absolute terms: our variance is less than $1.5 imes$ theirs for $t$ as large as $2^{24}$.

Problem

• Private continual counting for unbounded data streams
• Overcoming suboptimal error from doubling trick
• Achieving smooth error with logarithmic perturbations

Innovation

• Novel matrix factorizations with logarithmic perturbations
• Smooth error and efficient private sum estimation
• O(t) space and amortized O(log t) time per round