The Binary Tree Mechanism is Optimal for Differentially Private Continual Counting

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. For fixed privacy parameters, the standard binary tree mechanism achieves expected ℓ_∞ error O(log3/2 n) under approximate differential privacy and O(log2 n) under pure differential privacy. Whether these dependences on the stream length are necessary has remained a central open problem. For fixed varepsilon∈(0,1), we prove a lower bound of Omega(log3/2 n) under approximate DP with sufficiently small fixed δ>0, and a lower bound of Omega(log2 n) under pure DP. These bounds establish the optimality of the binary tree mechanism in both settings. The bounds hold for arbitrary mechanisms, even when the entire stream is available in advance. Both proofs use the same decomposition and accumulation of residual noise along a tree. As a consequence of the approximate-DP bound, we also obtain a largest-possible separation between hereditary discrepancy and private ℓ_∞ 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.