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

arXiv:2607.00876v1 Announce Type: cross 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 Omega(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.

Source: arXiv cs.LG | 2026-07-02