[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81942-en":3,"doc-seo-81942-105":31,"detail-sidebar-cat-0-en-105":92},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81942,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Boosting with List-Decodable Codes","Boosting systematically improves the accuracy of learning algorithms using a weak learner. Prior results require O(log(1/ϵ)/γ^2) rounds and are essentially tight for generic concept classes via Freund’s lower bound. This work avoids that barrier for concept classes with a mild closure property: F closed under O(log(1/γ))-Xor. It presents a new boosting algorithm using only O(log(1/ϵ)) weak-learner calls plus one additional batch of O(log(1/ϵ)/γ^2) labeled samples, based on connecting boosting to list-decodable codes.","Boosting with List-Decodable Codes  \nAddison Prairie Stanford  \nLi-Yang Tan Stanford  \narXiv :2607 .05791v1 [ stat .ML] 7 Jul 2026  \nJuly 8, 2026  \nAbstract  \nBoosting is a fundamental technique for generically improving the accuracy of learning algorithms (Schapire 1989) . Existing boosting algorithms construct a strong learner using O(log(~~1~~ϵ)/γ2 ) calls to a γ-advantage weak learner, and this round complexity is known to be optimal for generic boosters that succeed on all concept classes (Freund 1995) .  \nWe show that this lower bound can be circumvented for concept classes that satisfy a mild closure property. Specifically, we present a new boosting algorithm that, for any class F closed under O(log ~~1~~γ )-X˜ OR, strong learns F using O(log ~~1~~ϵ) calls to a γ-advantage weak learner and a single batch of O(log( ~~ 1~~ϵ)/γ2 ) additional samples.  \nOur algorithm arises from a new and simple connection between boosting and list-decodable codes. Viewing the target function as a message, we run the weak learner on its encoding and view the resulting weak hypothesis as a corrupted codeword. Feeding this corrupted codeword to a list decoder, we obtain a small list of candidate hypotheses, at least one of which is a strong hypothesis for the original function. Using additional samples, we identify and output this strong hypothesis.  \n1 Introduction  \nBoosting is a celebrated method for systematically improving the accuracy of learning algorithms. Given black-box access to a weak learner—an algorithm that produces hypotheses with accuracy slightly better than trivial—for a concept class, a boosting algorithm produces a strong learner that achieves arbitrarily high accuracy. Boosting algorithms typically work by sequentially running the weak learner multiple times and aggregating the weak hypotheses. Originally introduced by Schapire [Sch89], boosting is notable as a technique that has proven successful both in theory and in practice.  \nA key measure of a boosting algorithm’s efficiency is the number of calls it makes to the weak learner, also known as its round complexity. To strong learn with accuracy 1 −ϵ given a weak learner with accuracy ~~1~~2 +γ, existing boosting algorithms (e.g., AdaBoost, [FS97]) make O(log( ~~ 1~~ϵ)/γ 2 ) calls to the weak learner. Early work of Freund showed that this is essentially optimal:  \nTheorem ([Fre95]) . For any boosting algorithm B, there exists a concept class F = {Fn}n∈N anda ( ~~1~~2 + γ)-accuracy weak learner W such that B must call W at least Ω(log( ~~ 1~~ϵ)/γ 2 ) times to learn F to 1 − ϵ accuracy.  \nHowever, the hard concept class F used in this lower bound is highly artificial: for each n ∈ N, the slice Fn consists of a single function f : {±1}n → {±1} chosen uniformly at random. As a result, it does not rule out the possibility of more round-efficient boosting for the more structured concept classes that arise in theory and practice.  \nIn this work, we evade Freund’s lower bound by restricting the concept class being learned. Our approach applies to any class F = {Fn}n∈N satisfying a mild closure assumption: we say that F is closed under k-Xor if, for every n ∈ N and f ∈ Fn, the function f⊕k(x1 , ... , xk) = f (x1 ) ⊕ ··· ⊕ f(xk) is in Fn · k . Under such an assumption on F, we give a boosting algorithm that makes significantly fewer calls to W:  \nTheorem 1 (see Theorem 6 for formal statement) . There exists an efficient boosting algorithm that, for any γ,ϵ > 0, given a γ-weak learner W for a concept cla˜ss F closed under O(log ~~1~~γ )-Xor, learns F to accuracy 1 − ϵ with O(log ~~1~~ϵ) calls to W and O(log( ~~ 1~~ϵ)/γ 2 ) additional labeled samples.  \nWe make the following remarks about Theorem 1 .  \nAdditional Labeled Samples. Here, by additional labeled samples, we˜ mean samples beyond those implicitly required for O(log ~~1~~ϵ) calls to the weak learner W. Our O(log(~~1~~ϵ)/γ 2 ) additional samples are modest compared to existing boosting algorithms that make O(log( ~~ 1~~ϵ)/","cbCaifQe6xRikiRk","https://ap.wps.com/l/cbCaifQe6xRikiRk","pdf",432134,4,1,21,"English","en",105,"# Abstract\n# Introduction\n## Round Complexity and Freund’s Lower Bound\n## Closure Under Xor and Main Theorem\n## Additional Labeled Samples and Tradeoffs\n## Mildness of Closure Assumption\n## Uniform-Distribution Boosting","[{\"question\":\"What limitation do existing boosting algorithms face in round complexity?\",\"answer\":\"Existing boosting algorithms achieve accuracy 1−ϵ using O(log(1/ϵ)/γ^2) calls to the weak learner. Freund’s result shows this is essentially optimal for generic concept classes.\"},{\"question\":\"How does the proposed approach improve boosting efficiency?\",\"answer\":\"The new algorithm targets concept classes F that are closed under O(log(1/γ))-Xor. For such classes, it reduces weak-learner calls to O(log(1/ϵ) ) while using an additional batch of O(log(1/ϵ)/γ^2) labeled samples.\"},{\"question\":\"What is the key idea connecting boosting to list-decodable codes?\",\"answer\":\"The method treats the target function as a message: it encodes using the weak learner, interprets the weak hypothesis as a corrupted codeword, then runs a list decoder to obtain a small candidate set containing a strong hypothesis. Additional samples select the correct strong hypothesis.\"}]","Boosting with List-Decodable Codes | PDF",1784177189,53,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"boosting-with-list-decodable-codes","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":20},"https://docshare.wps.com/document/boosting-with-list-decodable-codes/81942/",{"url":53,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What limitation do existing boosting algorithms face in round complexity?","Question",{"text":76,"@type":77},"Existing boosting algorithms achieve accuracy 1−ϵ using O(log(1/ϵ)/γ^2) calls to the weak learner. Freund’s result shows this is essentially optimal for generic concept classes.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the proposed approach improve boosting efficiency?",{"text":81,"@type":77},"The new algorithm targets concept classes F that are closed under O(log(1/γ))-Xor. For such classes, it reduces weak-learner calls to O(log(1/ϵ) ) while using an additional batch of O(log(1/ϵ)/γ^2) labeled samples.",{"name":83,"@type":74,"acceptedAnswer":84},"What is the key idea connecting boosting to list-decodable codes?",{"text":85,"@type":77},"The method treats the target function as a message: it encodes using the weak learner, interprets the weak hypothesis as a corrupted codeword, then runs a list decoder to obtain a small candidate set containing a strong hypothesis. Additional samples select the correct strong hypothesis.","https://schema.org",{"og:url":53,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]