Private Continual Counting of Unbounded Streams

TL;DR

This paper introduces a novel differentially private continual counting algorithm for unbounded data streams that achieves smooth error bounds, efficient space and time complexity, and performs comparably to the state-of-the-art in empirical tests.

Contribution

It develops a new matrix factorization technique based on logarithmic perturbations, enabling private counting without prior knowledge of stream length.

Findings

Achieves $O( ext{log}^{2+2eta}(t))$ variance for estimating sums.

Requires $O(t)$ space and $O( ext{log} t)$ amortized time per update.

Performs comparably to the best existing algorithms in empirical evaluations.

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 $t\leq n$ it is able to privately estimate the sum of the first $t$ data points with $O(\log^{2+2\alpha}(t))$ 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 $1.5\times$ theirs for $t$ as large as $2^{24}$.