[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82782-en":3,"doc-seo-82782-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82782,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Fast, Parallel, Query-Efficient Binary Classification","Fast, parallel, query-efficient randomized algorithms are developed for the maximum-margin separating hyperplane problem in binary classification. For a dataset of size n with normalized d-dimensional features and margin γ, the work and depth are optimized in terms of the sparsity nnz(Φ) of the feature matrix Φ. One algorithm achieves O(γ−2/3)-parallel depth with improved sequential runtime and O(γ−2/3) matvec query complexity, while a second algorithm further reduces sequential work but increases parallel depth. Both algorithms match near-optimal deterministic matvec complexity while trading determinism for randomness.","arXiv :2607 .04062v1 [math .OC] 5 Jul 2026  \nFast, Parallel, Query-Efficient Binary Classification  \nIshani Karmarkar∗ Liam O’Carroll∗ Aaron Sidford∗  \nJuly 7, 2026  \nAbstract  \nWe study the fundamental classification problem of computing a separating hyperplane for a binary-labeled dataset of size n with normalized d-dimensional features. Letting Φ ∈ Rn ×d denote the feature matrix and γ the margin of the maximum-m˜ argin separating hyperplane, we present a randomized algorithm˜ that solves this problem in O (γ−2/3 nnz(Φ) + γ−2(ω+1)/3)-sequential r˜unning time (work), O (γ−2/3)-parallel (computational) depth, and accesses Φ only through O (γ−2/3˜)-matrix-vector queries (matvecs) . We also present a second, ˜faster randomized algorithm with a O (γ−2/3˜ nnz(Φ) + γ−2)-sequential running time that uses O (γ−2/3)-matvecs to Φ, but  \nachieves only O(γ−4/3)-parallel depth. Both algorithms match the near-optimal deterministic matvec complexity recently established by [42, 43] and achieve improved sequential runtime and parallel depth, albeit at the expense of using randomness.  \nContents  \n1 Introduction 1  \n2 Preliminaries 4  \n3 Technical overview 4  \n4 Reducing ℓ2-ℓ1-games to linear system solving 11  \n5 Linear system solving 22  \n6 Main results 33  \n7 Conclusion 34  \nA Reduction from separating hyperplane problem to ℓ2-ℓ1 games 40  \nB Linear algebraic properties 40  \n∗ Stanford University, {ishanik,ocarroll,[sidford}@stanford.edu](sidford}@stanford.edu)  \n1 Introduction  \nIn this paper, we study the foundational binary data classification problem of finding a linear separator for two sets of points. Concretely, we consider the separating hyperplane, or hard-margin support vector machine (SVM) problem, in which we are given a dataset D = { (ϕi, li)} i∈[n] where ϕi ∈ Bd := {x ∈ Rd : ∥x∥2 ≤ 1} is a normalized feature vector and li ∈ {+1, −1} is a binary label for each i ∈ [n] . We let γD denote the maximum margin of a separating hyperplane for D, i.e. ,  \nγD := max min li⟨w,ϕi⟩, and hence, ∃w⋆ ∈ Bd such that li⟨w⋆ ,ϕi⟩ ≥ γD for all i ∈ [n] . (1)  \nw∈Bd i∈[n]  \nWe consider the problem where, given ρ > 0, we must find a hyperplane wˆ ∈ Bd which achieves a margin of γD − ρ .1 We formalize this problem in the following Definition 1.1.  \nDefinition 1.1 (Maximum Margin Separating Hyperplane Problem) . In the ρ-(maximum margin) separating hyperplane problem, we are given a (binary-labeled) dataset D = { (ϕi ∈ Bd , li ∈{−1, +1})} i∈[n] and ρ > 0, and must output wˆ ∈ Bd such that li⟨wˆ,ϕi⟩ ≥ γD − ρ for all i ∈ [n] . We say that such a wˆ induces a ρ-separating hyperplane for D.  \nThis is an incredibly well-studied problem in machine learning, dating back to [47, 59], with numerous applications (especially as it can be extended to non-linear settings [48]) . Our focus in this work is to introduce new, faster randomized algorithms for this foundational learning problem.  \nOur results. For notational convenience, for a dataset D = { (ϕi ∈ Bd , li ∈ {+1, −1})} i∈[n] we use ΦD ∈ Rn ×d to denote the covariate matrix whose i-th row is given by liϕ⊤i . In addition, for A ∈ Rn ×d , nnz(A) := |{ (i, j) : Aij  0}| + n + d denotes an augmented nonzero count.  \nOur main result, given in Theorems 1.2 and 1.3 below, is two parallel, query-efficient algorithms for the maximum-margin separating hyperplane problem (Definition 1.1) which offer different tradeoffs between complexity metrics. The complexity metrics we study are depth, work (or sequential runtime), and matvec complexity.  \n• Depth and work: Following [35], we say an algorithm has (computational) depth D if the number of sequential rounds of computation is D. In particular, we assume element-wise vector operations (e.g. , adding/scaling vectors) in Rk incur O(1)-depth and O (k)-work and that dot products and matrix-vector multiplications require O (log k) depth. We let ω \u003C 2.3714 denote the fast matrix multiplication (˜FMM) constant [1]; namely, two k × k matrices can be  \nmultiplied in O (kω )-wo","cbCairaPAstB6LOP","https://ap.wps.com/l/cbCairaPAstB6LOP","pdf",1029797,3,1,42,"English","en",105,"# Introduction\n# Preliminaries\n# Technical overview\n# Main results\n# Conclusion","[{\"question\":\"What problem does the paper study in binary classification?\",\"answer\":\"The paper studies the maximum-margin separating hyperplane problem (hard-margin SVM) for a binary-labeled dataset, aiming to find a linear separator achieving margin γD up to a deficit ρ.\"},{\"question\":\"How do the two proposed randomized algorithms differ?\",\"answer\":\"The first algorithm targets stronger parallelism (O(ρ−2/3) depth) with improved work and O(ρ−2/3) matvec complexity. The second algorithm achieves faster sequential work (better ρ dependence) but incurs larger parallel depth (O(ρ−4/3)).\"},{\"question\":\"Which complexity metrics are optimized and how is query efficiency defined?\",\"answer\":\"The paper optimizes work (sequential running time), computational depth (number of sequential rounds), and matvec complexity. Matvec complexity counts the number of oracle queries needed to implement the algorithm using matrix-vector operations with Φ.\"}]",1784182907,106,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"fast-parallel-query-efficient-binary-classification","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/fast-parallel-query-efficient-binary-classification/82782/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper study in binary classification?","Question",{"text":75,"@type":76},"The paper studies the maximum-margin separating hyperplane problem (hard-margin SVM) for a binary-labeled dataset, aiming to find a linear separator achieving margin γD up to a deficit ρ.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the two proposed randomized algorithms differ?",{"text":80,"@type":76},"The first algorithm targets stronger parallelism (O(ρ−2/3) depth) with improved work and O(ρ−2/3) matvec complexity. The second algorithm achieves faster sequential work (better ρ dependence) but incurs larger parallel depth (O(ρ−4/3)).",{"name":82,"@type":73,"acceptedAnswer":83},"Which complexity metrics are optimized and how is query efficiency defined?",{"text":84,"@type":76},"The paper optimizes work (sequential running time), computational depth (number of sequential rounds), and matvec complexity. Matvec complexity counts the number of oracle queries needed to implement the algorithm using matrix-vector operations with Φ.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]