[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82563-en":3,"doc-seo-82563-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82563,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","The Binary Tree Mechanism is Optimal for Approximate Differentially Private Continual Counting","Private continual counting is a core problem in differential privacy, releasing all running counts over a binary stream while protecting each individual whose contribution is a single 1. The classic binary tree mechanism with Gaussian noise achieves expected ℓ∞ error scaling as log3/2 n under approximate differential privacy. This work proves a matching lower bound, showing any differentially private mechanism must incur expected ℓ∞ error Ω(log3/2 n), hence asymptotically optimality. It also yields an optimal separation for linear queries relating hereditary discrepancy to private ℓ∞ error.","arXiv :2607 .00876v2 [ cs .DS] 2 Jul 2026  \nThe Binary Tree Mechanism is Optimal for Approximate Differentially Private Continual Counting  \nKonstantina Bairaktari∗ Kasper Green Larsen∗  \nAbstract  \nPrivate 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 ℓ ∞ error proportional to log3/2 n for approximate differential privacy. Whether this dependence on the stream length is necessary has remained a central open problem.  \nIn this work, we resolve the dependence on n by proving that every differentially private mechanism for continual counting must incur expected ℓ∞ error Ω(log3/2 n) . This shows that the binary tree mechanism is asymptotically optimal in the approximate-DP setting.  \nAs a consequence, 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.  \n1 Introduction  \nHow can an organization track the occurrence of sensitive events over time without compromising the privacy of individuals? Such scenarios arise naturally in many settings, including monitoring daily disease cases, counting user interactions on a platform, or measuring responses to a public poll. This motivates the study of private continual counting, the focus of this paper. The standard formulation of the problem models time as discrete steps 1 ,..., n, where at each step a bit indicates whether an event occurred. Each 1 corresponds to the contribution of one individual and each individual participates in at most one event. The goal is to release an accurate running count at every time step without revealing any individual’s contribution.  \nThe study of private continual counting was initiated by [DNPR10, CSS10], who established the foundations for differentially private computations under continual observation. Differential privacy (DP) [DMNS06] is the standard framework for formal privacy guarantees, ensuring that the output of a mechanism reveals little about any individual’s data. Private continual counting has found applications in a broad range of tasks including private online learning [JKT12, TS13], convex optimization [AFKT21] and federated learning [KMS+ 21] . In many of these applications, it appears as a subroutine, and improving its accuracy directly translates to improvements in the downstream task. The appropriate notion of error, however, depends on the application: ℓ∞ error (the expected worst-case error over all time steps) is a natural measure for monitoring tasks, while ℓ2 error (the expected root mean squared error over all time steps) is more relevant for learning applications [HUU23] . In this work, we focus on the ℓ ∞ error.  \nThe most well-known method for private continual counting is the binary tree mechanism [DNPR10, CSS10], which computes carefully structured subset counts in a differentially private manner and combines them to recover the running count at any given time step.  \n∗ Department of Computer Science, Aarhus University. Supported by the European Union (ERC, TUCLA, 101125203) . Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.  \nFor pure differential privacy, the binary tree mechanism uses Laplace noise to make the subset counts ε-differentially private. Here, and throughout the paper, we focus on the regime where ε ∈ (0 , 1) . While the original analysis achieves ℓ∞ error O(log5/2(n)/ε) [DNPR10, CSS10], a more careful analysis","cbCaieVgaTbs9SCG","https://ap.wps.com/l/cbCaieVgaTbs9SCG","pdf",604326,1,33,"English","en",105,"# Introduction\n## Private Continual Counting and Error Notions\n## Binary Tree Mechanism and Known Bounds\n## Main Result and Lower Bound Implications","[{\"question\":\"What is private continual counting in differential privacy?\",\"answer\":\"It is the task of releasing an accurate running count at every time step for a binary stream, while ensuring that the output reveals little about any individual’s contribution.\"},{\"question\":\"Why is the binary tree mechanism important here?\",\"answer\":\"The binary tree mechanism is the best-known standard approach for continually releasing counts under differential privacy by combining structured noisy subset counts.\"},{\"question\":\"What does the paper prove about the dependence on stream length n?\",\"answer\":\"It proves that every differentially private mechanism must have expected ℓ∞ error at least Ω(log3/2 n) in the approximate-DP setting, establishing asymptotic optimality of the binary tree 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