The Binary Tree Mechanism is Optimal for Approximate Differentially Private Continual Counting

2026-07-01

AI Summary

This work investigates the minimax error for continuously counting binary streams under approximate differential privacy. By integrating differential privacy theory, information-theoretic lower bounds, and the hereditary discrepancy framework for linear queries, it establishes—for the first time—an Ω(log^{3/2} n) lower bound on the expected ℓ_∞ error of any private mechanism. This result not only confirms the asymptotic optimality of the classic binary tree mechanism in this setting but also reveals a potentially maximal separation between hereditary discrepancy and privacy-induced error, thereby deepening our understanding of the fundamental trade-offs between privacy and accuracy in continual release scenarios.

Abstract

Private continual counting is a fundamental problem in differential privacy: given a binary stream of length $n$, where each $1$ corresponds to the contribution of one individual, the goal is to release all running counts while protecting the privacy of each individual. The standard algorithm is the binary tree mechanism, whose Gaussian-noise variant achieves expected $\ell_\infty$ error proportional to $\log^{3/2} n$ for approximate differential privacy. Whether this dependence on the stream length is necessary has remained a central open problem. In this work, we resolve the dependence on $n$ by proving that every differentially private mechanism for continual counting must incur expected $\ell_\infty$ error $Ω(\log^{3/2} n)$. This shows that the binary tree mechanism is asymptotically optimal in the approximate-DP setting. As a consequence, we also obtain a largest-possible separation between hereditary discrepancy and private $\ell_\infty$ error for linear queries, showing that the known general upper bound in terms of hereditary discrepancy has the optimal dependence on the number of queries.

Problem

• differential privacy
• continual counting
• binary stream
• error lower bound
• hereditary discrepancy

Innovation

• binary tree mechanism
• approximate differential privacy
• continual counting
• hereditary discrepancy
• lower bound