[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84354-en":3,"doc-seo-84354-105":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},84354,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Adaptive Row Selection Meets Asynchrony in Randomized Kaczmarz","Randomized Kaczmarz suits large sparse least-squares and tomographic reconstruction, and adaptive row selection can reduce iteration counts. When deployed on shared-memory hardware, adaptive sampling may draw from a residual vector concurrently modified by lock-free workers, often with stale views. The study systematically evaluates residual-weighted and greedy Kaczmarz under asynchronous execution using 339 runs on a 96-core node with realized delays. Results identify an ℓ*(T) stability boundary, show threshold-greedy instability at scale, demonstrate under-relaxation as a practical safeguard, and compare consistent-snapshot versus live reads.","arXiv :2607 .083 13v 1 [ cs .DC] 9 Jul 2026  \nAdaptive Row Selection Meets Asynchrony in  \nRandomized Kaczmarz  \nEvan Coleman  \nDepartment of Computer Science  \nUniversity of Mary Washington  \nFredericksburg, Virginia, USA  \n[ecolema4@umw.edu](ecolema4@umw.edu)  \nAbstract  \nRandomized Kaczmarz is a natural fit for large sparse least-squares and tomographic reconstruction, and adaptive row selection can reduce iteration counts. However, deploying adaptive selection on a shared-memory machine means sampling from a residual that lock-free workers are concurrently modifying, often using stale data. We present the first systematic study of this regime: residual-weighted and greedy Kaczmarz under asynchronous execution, measured across 339 runs on a 96-core node with realized (not injected) delays. Four findings carry directly to practice. (i) Stability is governed by a boundary ℓ∗ (T) between sampling aggressiveness and thread count; below it, more aggressive sampling is strictly better, so one should tune to just inside the cliff. (ii) Threshold-greedy selection (the standard accelerated rule) is unstable at high thread counts, diverging almost immediately. (iii) Under-relaxation buys back the cliff at a predictable cost, giving a usable safety knob. (iv) Consistent-snapshot reads admit a rare, scheduling-dependent divergence that live (inconsistent) reads never exhibited and that is also cheaper, making inconsistent reads the right default. We validate the implementation against published sequential results and outline the distributed two-level sampler these measurements motivate.  \nIndex Terms  \nasynchronous iterative methods, randomized Kaczmarz, greedy selection, residual-weighted sampling, shared-memory parallelism  \nI. INTRODUCTION  \nRandomized Kaczmarz [1] is a row-action solver: each step projects the current iterate onto one equation’s hyperplane, costing O(nnz) of a single row with no factorization. That structure makes it a natural choice for large sparse least-squares and for applications such as tomographic reconstruction, where it is the algebraic backbone of the classical Algebraic Reconstruction Technique (ART) method. Adaptive selection (e.g., greedy [2] or residual-power-weighted [3],[4], [5]) accelerates convergence by preferring high-residual rows; asynchronous execution [6],[7] extracts parallelism by letting lock-free workers update shared state without barriers.  \nEach of these mechanisms is well understood in isolation, but deploying them together creates a situation neither analysis covers. Adaptive selection must rank rows by residual magnitude; buton an asynchronous machine the residual is a shared vector that other workers are concurrently modifying (possibly with stale views) . The selector is therefore choosing from an imperfect distribution, where the discrepancy from ideal grows with the number of in-flight updates. Existing adaptive-Kaczmarz theory assumes the residual is current; existing asynchronous theory assumes the sampling distribution is fixed. What actually happens between those assumptions has not been measured, and existing parallel implementations [8], [9], [10] do not analyze it.  \nUsing the power-weighted family P (i) ∝ |ri | ℓ (a single knob from uniform, ℓ → 0, through greedy, ℓ → ∞ ) we instrument residual-weighted and threshold-greedy Kaczmarz across 339 runson a 96-core node with delays that are realized by the hardware rather than synthetically injected, on: (1) a benign dense problem,(2) a coherent tomographic problem, and (3) the standard sparse suite. We find that the read-consistency effect known from asynchronous Jacobi does not govern here [11] . What governs is a stability boundary between how aggressively selection chases residual peaks and how many workers are running, a cliff that aggressive sampling races toward as concurrency grows. The contributions of this paper are:  \n• A stability boundary ℓ∗ (T): We map a boundary in the (sampling exponent ℓ, thread count T) pla","cbCaifIza52uACVh","https://ap.wps.com/l/cbCaifIza52uACVh","pdf",600074,1,15,"English","en",105,"# Introduction\n## Problem Setting and Motivation\n## Contributions and Key Findings\n# Related Work","[{\"question\":\"What challenge arises when combining adaptive row selection with asynchronous execution in Randomized Kaczmarz?\",\"answer\":\"Adaptive selection ranks rows using a residual vector that other workers update concurrently without barriers, so sampling is effectively drawn from an imperfect distribution with discrepancies that grow with in-flight updates.\"},{\"question\":\"What is the stability boundary ℓ*(T) and how should it be used?\",\"answer\":\"ℓ*(T) separates sampling aggressiveness and thread count; below the boundary, more aggressive sampling is strictly better. The practical guidance is to tune the aggressiveness to just inside the cliff determined by the number of workers.\"},{\"question\":\"Why can threshold-greedy selection become unstable at high thread counts?\",\"answer\":\"At large thread counts, threshold-greedy selection falls outside the stability boundary and diverges almost immediately (within a fraction of a sweep), even under different read semantics.\"}]",1784195028,38,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"adaptive-row-selection-meets-asynchrony-in-randomized-kaczmarz","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/adaptive-row-selection-meets-asynchrony-in-randomized-kaczmarz/84354/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 challenge arises when combining adaptive row selection with asynchronous execution in Randomized Kaczmarz?","Question",{"text":75,"@type":76},"Adaptive selection ranks rows using a residual vector that other workers update concurrently without barriers, so sampling is effectively drawn from an imperfect distribution with discrepancies that grow with in-flight updates.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the stability boundary ℓ*(T) and how should it be used?",{"text":80,"@type":76},"ℓ*(T) separates sampling aggressiveness and thread count; below the boundary, more aggressive sampling is strictly better. The practical guidance is to tune the aggressiveness to just inside the cliff determined by the number of workers.",{"name":82,"@type":73,"acceptedAnswer":83},"Why can threshold-greedy selection become unstable at high thread counts?",{"text":84,"@type":76},"At large thread counts, threshold-greedy selection falls outside the stability boundary and diverges almost immediately (within a fraction of a sweep), even under different read semantics.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]