[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83079-en":3,"doc-seo-83079-105":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},83079,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Adversarial Robustness for Small Frequency Moments and a Weak Equivalence Theorem for Turnstile Streams","Adversarially robust algorithms are studied for insertion-deletion (turnstile) streams where future updates may depend on past algorithm outputs. After robust (1+ε)-approximation for the second moment F2 in polylogarithmic space, achieving high accuracy for other moments remained open. The work proves (1+ε)-approximate robustness in polylog space for every p∈[0,2], including F0, metric and information-theoretic quantities such as EMD and k-median cost, and Shannon entropy up to ε-additive error, plus Bernstein-function symmetric losses. It also establishes a weak equivalence between oblivious sketching and adversarial robustness, relating efficient sketches to robust turnstile algorithms via L1 embeddability. ","Adversarial Robustness for Small Frequency Moments and a Weak Equivalence Theorem for Turnstile Streams  \nElena Gribelyuk Princeton University  \n[eg5539@princeton. edu](eg5539@princeton. edu)  \nHonghao Lin Carnegie Mellon University [honghaol@andrew. cmu. edu](honghaol@andrew. cmu. edu)  \nDavid P. Woodruff Carnegie Mellon University [dwoodruf@andrew. cmu. edu](dwoodruf@andrew. cmu. edu)  \nHuacheng Yu Princeton University  \n[hy2@cs. princeton. edu](hy2@cs. princeton. edu)  \nSamson Zhou Texas A&M University [samsonzhou@gmail. com](samsonzhou@gmail. com)  \narXiv :2607 .063 12v 1 [ cs .DS] 7 Jul 2026  \nJuly 8, 2026  \nAbstract  \nWe study adversarially robust algorithms for insertion-deletion (turnstile) streams, where future updates may depend on past algorithm outputs. While recent work achieved a robust (1 + ε)-approximation for the second moment F2 in polylogarithmic space, achieving high accuracy for other frequency moments remained a major open question; for p ∈ [0 , 2), includingthe fundamental distinct elements problem (F0 ), only constant-factor approximations were known in sublinear space. We close this gap, showing that (1 + ε)-approximate robustness can be achieved in polylogarithmic space for all p ∈ [0 , 2] . Our approach generalizes the estimatorcorrector-learner framework to non-Hilbert spaces by dynamically maintaining implicit isometric embeddings into L2 and performing regularized kernel ridge regression over adaptively discovered hard queries, yielding the first insertion-deletion algorithms that approximate: (1) the p-th frequency moment Fp up to a (1 + ε)-factor in poly(1/ε, log n) space for all p ∈ [0 , 2], includingthe support size F0 , (2) metric and information-theoretic quantities, including the Earth Mover Distance (EMD) and k-median clustering cost over [∆]d up to an O (d log ∆)-factor, and the Shannon entropy up to an ε-additive error, and (3) non-normed symmetric losses defined by Bernstein functions up to a (1 + ε)-factor. For the Fp moments, our algorithm is optimal up to poly(1/ε, log n) factors. Furthermore, we establish a weak equivalence between classical oblivious sketching and adversarial robustness. We prove that for any sub-multiplicative norm, the existence of an efficient classical linear sketch is equivalent to the existence of an efficient adversarially robust turnstile algorithm, up to polynomial factors, formalizing L 1 embeddability as the fundamental mechanism governing both models.  \n1 Introduction  \nThe streaming model of computation has emerged as a central framework for studying algorithms that process massive amounts of sequential data. In this setting, an underlying dataset is implicitly defined by a stream of updates and an algorithm must maintain a compact summary to approximately answer queries or report relevant statistics about the data. Often, the algorithm is restricted toa single pass over the stream and must use space sublinear in both the stream length m and the universe size n. This framework has proven remarkably effective for reasoning about large-scale  \ndata processing, and has led to a rich body of work spanning algorithm design, lower bounds, and applications.  \nIn the classical streaming setting, a standard assumption in the literature is that the input stream is fixed in advance and independent of the algorithm’s internal randomness. However, this assumption is often violated in modern applications, where the input stream may be generated adaptively based on previous outputs of the algorithm. For instance, queries to a database may depend on previous answers, iterative scientific procedures may update their state using previously computed estimates, and feedback loops may arise in online recommendation platforms and financial systems. In such settings, the input stream may depend on the behavior of the algorithm itself (and on the internal randomness), thus raising the question of whether classical streaming guarantees continue to hold.  \nAdversarially robust ","cbCainKBP4QmACt9","https://ap.wps.com/l/cbCainKBP4QmACt9","pdf",704313,2,1,41,"English","en",105,"# Abstract\n# Introduction\n## Streaming model and limitations\n## Adversarially robust streaming\n### Formal definition of robustness","[{\"question\":\"What problem does the document address in turnstile (insertion-deletion) streams?\",\"answer\":\"It addresses designing algorithms that remain accurate at every time step even when future updates are chosen adaptively based on prior outputs.\"},{\"question\":\"Which frequency moments are approximated with (1+ε) robustness, and under what space constraints?\",\"answer\":\"For all p∈[0,2], the paper achieves (1+ε)-approximate robustness in polylogarithmic space, including the support size F0.\"},{\"question\":\"How is adversarial robustness related to classical oblivious sketching?\",\"answer\":\"It proves a weak equivalence: for any sub-multiplicative norm, the existence of an efficient classical linear sketch is equivalent (up to polynomial factors) to the existence of an efficient adversarially robust turnstile algorithm.\"}]",1784185052,103,{"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},"adversarial-robustness-for-small-frequency-moments-and-a-weak-equivalence-theorem-for-turnstile-streams","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/adversarial-robustness-for-small-frequency-moments-and-a-weak-equivalence-theorem-for-turnstile-streams/83079/",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 document address in turnstile (insertion-deletion) streams?","Question",{"text":75,"@type":76},"It addresses designing algorithms that remain accurate at every time step even when future updates are chosen adaptively based on prior outputs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which frequency moments are approximated with (1+ε) robustness, and under what space constraints?",{"text":80,"@type":76},"For all p∈[0,2], the paper achieves (1+ε)-approximate robustness in polylogarithmic space, including the support size F0.",{"name":82,"@type":73,"acceptedAnswer":83},"How is adversarial robustness related to classical oblivious sketching?",{"text":84,"@type":76},"It proves a weak equivalence: for any sub-multiplicative norm, the existence of an efficient classical linear sketch is equivalent (up to polynomial factors) to the existence of an efficient adversarially robust turnstile algorithm.","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":20,"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"]