Improved Error Bounds for Pure Differentially Private Continual Counting via Matrix Factorization

2026-07-09

AI Summary

This work addresses the problem of continual counting under pure differential privacy, aiming to narrow the gap between known upper and lower bounds on both worst-case and average mean squared error. We propose a novel mechanism based on matrix factorization that departs from conventional tree-based approaches by starting from high-quality low-dimensional factorizations, incorporating gradient-based optimization, and explicitly extending to higher dimensions. This method achieves, for the first time in a non-tree-based mechanism, an improved error constant. Furthermore, we establish a lower bound of Ω(ε⁻² log³ n) for the class of {0,1}-matrix factorizations, which asymptotically matches the best-known upper bound. The proposed algorithm significantly improves the leading constants in both worst-case and average mean squared error, offering strong theoretical guarantees alongside practical efficiency.

Abstract

Continual counting under pure differential privacy is one of the simplest and most well-studied problems in the continual observation model. Nevertheless, an asymptotic gap remains between the best known upper and lower bounds for maximum squared error and mean squared error: the upper bound is $O(ε^{-2}\log^3 n)$, while the lower bound is $Ω(ε^{-2}\log^2 n)$, for both error metrics. The best known constant in the upper bound is achieved by the $k$-ary tree mechanism with the subtraction trick, due to Andersson, Pagh, Steiner, and Torkamani (FORC 2025). In this work, we improve the leading constant in the maximum squared error and the mean squared error. Our approach uses a general matrix factorization mechanism, yielding an improved bound for pure-DP continual counting that does not rely on a tree-based construction. The mechanism starts from a good-quality low-dimensional factorization, obtained via gradient-based optimization, and gives an explicit matrix construction that lifts this factorization to arbitrarily large dimensions, further improving its error guarantees. We offer an efficient algorithmic implementation of our mechanism. On the lower-bound side, we prove an $Ω(ε^{-2}\log^3 n)$ lower bound for the class of factorizations whose matrices have entries in $\{0,1\}$, matching the upper-bound asymptotics for this class. This class includes the binary tree mechanism and $k$-ary tree mechanisms without the subtraction trick. Extending this lower bound to arbitrary matrix factorizations, and beyond the matrix mechanism altogether, remains an open problem.

Problem

• differential privacy
• continual counting
• error bounds
• matrix factorization
• lower bound

Innovation

• matrix factorization
• differential privacy
• continual counting
• error bounds
• gradient-based optimization