[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84725-en":3,"doc-seo-84725-105":28,"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":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":13,"seo_description":14,"update_tm":26,"read_time":27},84725,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","Optimal Online Discrepancy Minimization in Linear Time","An online algorithm assigns random signs to each arriving vector in a worst-case setting with only the norm constraint ∥vi∥2 ≤ 1. For every prefix, the resulting discrepancy admits a representation as the sum of three coupled standard Gaussian vectors, enabling a tight prefix discrepancy bound with high probability. The algorithm runs in O(dT) time and matches the optimal order for prefix discrepancy previously achieved by slower exponential-time methods.","arXiv :2607 .04388v 1 [ cs .DS] 5 Jul 2026  \nOptimal Online Discrepancy Minimization in Linear Time  \nIshaq Aden-Ali ∗  \nJune 27, 2026  \nAbstract  \nWe provide an online algorithm with the following guarantee: for any fixed sequence of vectorsv1 ,   , vT ∈ Rd with ∥vi ∥2 ≤ 1, the algorithm assigns each arriving vector vt a random sign εt such that every prefix sum Pti=1 εivi can be written as the sum of three coupled standard Gaussian vectors. Our algorithm runs in O (dT) time and achieves the optimal prefix discrepancy bound  \n􀀍 t 􀀍  \n1tT 􀀍 X εivi 􀀍 = O 􀀐 plog T􀀑 ,  \n􀀍 i=1 􀀍 ∞  \nwith high probability. This recovers the optimal bound of Kulkarni, Reis, and Rothvoss, whose algorithm runs in time exponential in T and d. The algorithm and main proof were discovered in a GPT-5.5 Pro Extended conversation prompted by the author.  \nStatement on AI use. The algorithm and main proof were discovered by OpenAI’s GPT-5.5 Pro Extended reasoning model. The proof appeared in a discussion that began with the author prompting the model to try to identify proof barriers to a natural family of algorithms related to the self-balancing walk of Alweiss, Liu, and Sawhney [ALS21] . After a few rounds of prompting, a number of barriers emerged. The author then prompted the model to search for an improved algorithm that could overcome these barriers. After a few more rounds of prompting, the model proposed the algorithm below, which is based on coupling three Gaussian fixed point walks, and provided a self-contained proof of the main theorem. The author subsequently verified and rewrote the main proof, derived the straightforward implications to discrepancy minimization, and wrote the paper. GPT-5.5 Pro was also used to help find typographical errors and search for related work.  \n1 Introduction  \nWe study online discrepancy minimization problems against an oblivious adversary: The adversary generatesa sequence of vectors v 1 , v2 ,   , vT ∈ Rd upfront and reveals them to the player one at a time. Upon receiving vt at time step t ∈ [T] := {1,..., T}, the player must commit to a sign εt ∈ {±1} before receiving the next vector vt+1 . The player’s (resp. adversary’s) goal is to minimize (resp. maximize) the prefix discrepancy  \n􀀍 t 􀀍  \nmax 􀀍 X εivi 􀀍 . (1)  \nt∈[T] 􀀍 i=1 􀀍 ∞  \nSpencer [Spe77] was the first to study a more general version of this problem. A number of papers study this online discrepancy problem with various distributional assumptions on the input [BS20, BJSS20 , BJM + 21 , AT25] . We focus on the worst-case setting where we are only assume ∥vt ∥2 ≤ 1. We refer to this as the online Koml´os problem, after the (offline) Koml´os problem.1  \n∗ Unemployed. Email: [adenali@berkeley.edu](adenali@berkeley.edu)  \n1 Koml´os conjectured that for any collection of vectors v1 , . . . , vT with maxi∈[T] ∥vi∥2 ≤ 1, there are signs {εi} such that ∥ P εivi∥∞ is constant.  \nA sequence of works has studied the online Koml´os problem [ALS21, LSS22 , KRR24] . A common approach in this line of work is to design a randomized online algorithm that generates random signs ε 1 ,ε2 , ... ,εT such that every prefix sum St = Pti=1 εivi is K-subgaussian, i.e. , for any fixed vector θ ∈ Rd ,  \nE[exp(⟨St ,θ⟩)] ≤ exp 􀀒 K~~ ~~2~~ ~~∥2θ∥2~~2~~ 􀀓 .  \nIt then follows from standard concentration arguments that the prefix discrepancy does not exceed O(K√log T) with high probability.2 Alweiss, Liu, and Sawhney [ALS21] introduced the elegant self-balancing walk that keeps every prefix sum Θ( √log T)-subgaussian, yielding a prefix discrepancy bound of O(log T) . Their algorithm runs in linear time. Later, Kulkarni, Reis, and Rothvoss [KRR24] were able to construct a distribution that keeps every prefix sum C-subgaussian for a universal constant C. This allowed them to obtain an O( √log T) bound on the prefix discrepancy, and they showed that this bound is optimal. Unfortunately, their algorithm runs in time exponential in T and d.  \nAn intermediate result of Liu, Sah, and Sawhney [L","cbCaiuKH1X5RXCEJ","https://ap.wps.com/l/cbCaiuKH1X5RXCEJ","pdf",437043,1,"English","en",105,"# Abstract\n# Introduction\n## Online discrepancy and the online Komlós problem\n## Prior randomized online approaches\n## Gaussian fixed point walks and coupling idea\n## Main algorithmic contribution","[{\"question\":\"What online task does the paper address?\",\"answer\":\"It studies online discrepancy minimization where vectors are revealed sequentially, and the player must choose a sign for each vector immediately to control the maximum over all prefix sums.\"},{\"question\":\"What guarantee does the proposed algorithm provide?\",\"answer\":\"For any fixed sequence with ∥vi∥2 ≤ 1, it assigns random signs so that each prefix sum can be expressed via coupling three standard Gaussian vectors, yielding an optimal prefix discrepancy bound with high probability.\"},{\"question\":\"How does the runtime of the new algorithm compare to earlier optimal results?\",\"answer\":\"The algorithm runs in O(dT) time, recovering an optimal bound that previously required exponential time in both T and d.\"}]",1784197870,20,{"code":4,"msg":29,"data":30},"ok",{"site_id":23,"language":22,"slug":31,"title":13,"keywords":32,"description":14,"schema_data":33,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":26},"optimal-online-discrepancy-minimization-in-linear-time","",{"@graph":34,"@context":84},[35,52,67],{"@type":36,"itemListElement":37},"BreadcrumbList",[38,42,46,49],{"item":39,"name":40,"@type":41,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":43,"name":44,"@type":41,"position":45},"https://docshare.wps.com/document/","Document",2,{"item":47,"name":12,"@type":41,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":41,"position":51},"https://docshare.wps.com/document/optimal-online-discrepancy-minimization-in-linear-time/84725/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":22,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":39,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What online task does the paper address?","Question",{"text":74,"@type":75},"It studies online discrepancy minimization where vectors are revealed sequentially, and the player must choose a sign for each vector immediately to control the maximum over all prefix sums.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What guarantee does the proposed algorithm provide?",{"text":79,"@type":75},"For any fixed sequence with ∥vi∥2 ≤ 1, it assigns random signs so that each prefix sum can be expressed via coupling three standard Gaussian vectors, yielding an optimal prefix discrepancy bound with high probability.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the runtime of the new algorithm compare to earlier optimal results?",{"text":83,"@type":75},"The algorithm runs in O(dT) time, recovering an optimal bound that previously required exponential time in both T and 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