[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82117-en":3,"doc-seo-82117-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},82117,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Improved Error Bounds for Pure Differentially Private Continual Counting via Matrix Factorization","Continual counting under pure differential privacy remains a cornerstone of the continual observation model, yet a persistent asymptotic gap exists between best known upper and lower error bounds for maximum squared error and mean squared error. The upper bound scales as O(ε−2 log3 n) while the lower bound is Ω(ε−2 log2 n). This work improves leading constants using a general matrix factorization mechanism: it lifts a low-dimensional factorization to high dimensions via an explicit matrix construction, enabling efficient implementation and strengthened pure-DP error guarantees for continual counting.","arXiv :2607 .08963v 1 [ cs .DS] 9 Jul 2026  \nImproved Error Bounds for Pure Differentially Private Continual  \nCounting via Matrix Factorization  \nPavel Arkhipov∗ Nikita P. Kalinin†  \nAbstract  \nContinual counting under pure differential privacy is one of the simplest and most wellstudied 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 log3 n), while the lower bound is Ω(ϵ−2 log2 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 log3 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.  \nContents  \n1 Introduction 2  \n2 Upper bound construction 7  \n3 Memory efficient noise correlation 13  \n4 Lower bounds for 0/1 factorizations 15  \n5 Additional results on factorization costs 18  \nA Proof of the upper bound (Theorem 2.5) 23  \nB Proofs of the additional results 26  \nC Factorization costs for tree-based methods 30  \n∗ Institute of Science and Technology Austria, [pavel.arkhipov@ist.ac.at](pavel.arkhipov@ist.ac.at)[ ](pavel.arkhipov@ist.ac.at)†Institute of Science and Technology Austria, [nikita.kalinin@ist.ac.at](nikita.kalinin@ist.ac.at)  \n1 Introduction  \nDifferential privacy [11] has become a standard framework for reasoning about privacy, both in theory and in practice. In many applications, however, data arrive sequentially, and statistics must be released continually as the data stream evolves. This motivates the continual observation model of Dwork, Naor, Pitassi, and Rothblum [12], in which the privacy guarantee must hold for the entire sequence of outputs rather than for a single final release. One of the most fundamental problems in this model is private continual counting under pure differential privacy [11, 12] . Given a binary stream x 1 , . . . , xn ∈ {0, 1} of sensitive updates, the goal is to release, at each time t, an approximation to the prefix sum Pti=1 xi, while ensuring that the entire output sequence satisfies ϵ-differential privacy. Continual counting and its variants serve as building blocks in a range of private streaming algorithms, including continual mean estimation [18], quantile estimation [24], continual histograms [42, 14, 5, 20], continual clustering [34], graph algorithms [15, 41, 9], and continual distinct-elements counting [25, 1] . Thus, improvements to continual counting can benefit a broad range of higher-level private streaming algorithms.  \nThe classical solution to continual counting is the binary tree mechanism [12, 6] . In this mechanism, the stream is represented by the leaves of a binary tree, and noisy partial sums are stored at internal nodes. Each update affects only logarithmically many nodes, which controls the privacy cost, and each prefix sum can be reconstructed from logarithmically many noisy measurements, which controls the error. ","cbCairi9slM97kpv","https://ap.wps.com/l/cbCairi9slM97kpv","pdf",560125,1,33,"English","en",105,"# Introduction\n# Upper bound construction\n# Memory efficient noise correlation\n# Lower bounds for 0/1 factorizations\n# Additional results on factorization costs\n# Proofs","[{\"question\":\"What problem does the document address?\",\"answer\":\"It studies continual counting in the continual observation model under pure differential privacy, where noisy approximations to prefix sums must protect the entire output sequence.\"},{\"question\":\"How does the proposed method improve error bounds?\",\"answer\":\"It introduces a general matrix factorization mechanism that starts from a good low-dimensional factorization and lifts it to larger dimensions with an explicit matrix construction, avoiding reliance on tree-based methods.\"},{\"question\":\"What do the document’s lower bounds show?\",\"answer\":\"It proves an Ω(ε−2 log3 n) lower bound for factorizations whose matrices have entries in {0,1}, matching the upper-bound asymptotic behavior for that class, while extending this to arbitrary factorizations is left 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problem does the document address?","Question",{"text":74,"@type":75},"It studies continual counting in the continual observation model under pure differential privacy, where noisy approximations to prefix sums must protect the entire output sequence.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed method improve error bounds?",{"text":79,"@type":75},"It introduces a general matrix factorization mechanism that starts from a good low-dimensional factorization and lifts it to larger dimensions with an explicit matrix construction, avoiding reliance on tree-based methods.",{"name":81,"@type":72,"acceptedAnswer":82},"What do the document’s lower bounds show?",{"text":83,"@type":75},"It proves an Ω(ε−2 log3 n) lower bound for factorizations whose matrices have entries in {0,1}, matching the upper-bound asymptotic behavior for that class, while extending this to arbitrary factorizations is left 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