[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121715-en":3,"doc-seo-121715-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},121715,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Trading-Off Payments and Accuracy in Online Classification with Paid Stochastic Experts","Investigates online classification where experts must be paid before issuing predictions. Each payment determines expert accuracy through an unknown Lipschitz productivity function, creating a cost trade-off between classification errors and upfront payments. Over T rounds, an online learning algorithm is proposed whose total cost nearly matches a Bayes-optimal predictor that would know all productivity functions in advance, differing by at most O(K^2(ln T)√T). The method combines Lipschitz bandits with online classification using surrogate losses and is evaluated on synthetic data.","Trading-Off Payments and Accuracy in Online Classification with Paid Stochastic Experts  \nDirk van der Hoeven * 1 Ciara Pike-Burke * 2 Hao Qiu 3 Nicol Cesa-Bianchi 3 4  \nAbstract  \nWe investigate online classification with paid stochastic experts. Here, before making their prediction, each expert must be paid. The amount that we pay each expert directly influences the accuracy of their prediction through some unknown Lipschitz “productivity” function. In each round, the learner must decide how much to pay each expert and then make a prediction. They incura cost equal to a weighted sum of the prediction error and upfront payments for all experts. We introduce an online learning algorithm whose total cost after T rounds exceeds that of a predictor which knows the productivity of all experts in advance by at most O 􀀀 K 2 (ln T)√T􀀁 where K is the number of experts. In order to achieve this result, we combine Lipschitz bandits and online classification with surrogate losses. These tools allow us to improve upon the bound of order T2/3 one would obtain in the standard Lipschitz bandit setting. Our algorithm is empirically evaluated on synthetic data.  \n1. Introduction  \nWe investigate online classification in the framework of prediction with expert advice where, in each round, the learning agent predicts an unknown binary label by aggregating the stochastic predictions of a number of experts. At the end of each round, the learner observes the true label and updates the function used to aggregate experts. In the variant considered in this work, we assume that at the beginning of around the learner allocates a payment to each expert which  \n*Equal contribution 1 Korteweg-de Vries Institute for Mathematics University of Amsterdam, Amsterdam, The Netherlands 2Department of Mathematics, Imperial College London, London, UK 3Universit degli Studi di Milano, Milan, Italy 4Politecnico di Milano, Milan, Italy. Correspondence to: Dirk van der Hoeven \u003C[dirk@dirkvanderhoeven.com](dirk@dirkvanderhoeven.com) > .  \nProceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023 . Copyright 2023 by the author(s) .  \naffects the expert’s performance in that round. This payment model of expert advice is realistic in many scenarios since human annotators will often only give useful advice if they are adequately compensated, and machine annotators may require more computation to return accurate predictions. Moreover, monetary incentives have been studied in crowdsourcing (Ho et al., 2015 ; 2016) . Although this is a different setting to that considered here, it is natural to study the effect of these payments in online binary classification with stochastic expert advice.  \nMotivated by results in crowdsourcing—e.g., (Ho et al., 2016)—we assume that each expert has a productivity function which determines the probability that they predict the label correctly given the payment they received. The productivity function can be different for each expert and is initially unknown to the learner. In each round, the learner pays each expert j = 1 ,..., K some amount cj ∈ [0 , 1] before observing their advice. The accuracy of the advice that expert j returns depends on the amount they are paid through the unknown productivity function, pj : [0 , 1] → [0 , 1], where pj (c) is the probability that expert j is correct when the payment is c. The learner can use the expert advice to improve their prediction, but they also want to minimize the payments to the experts. Therefore, they must trade-off between the price of the expert advice, and any improvements to prediction accuracy it may bring.  \nWe define the learner’s cost over a sequence of T rounds asthe sum of classification mistakes and payments to experts. If the probabilities pj (cj) are known for some c 1 , . . . , cK and for all experts j = 1 ,..., K, then we can write down the Bayes-optimal cost. In particular, if the events that each expert makes a mistake are ind","cbCaicTNkURpfkvA","https://ap.wps.com/l/cbCaicTNkURpfkvA","pdf",2256058,1,22,"English","en",105,"# Introduction\n## Problem setup: paid stochastic expert advice\n## Objective: cost and regret\n## Key challenges and approach","[{\"question\":\"How does paying an expert affect online classification accuracy?\",\"answer\":\"Each expert’s probability of being correct depends on the payment via an unknown Lipschitz productivity function, so the learner’s chosen payments directly influence prediction accuracy.\"},{\"question\":\"What cost and regret does the learner aim to minimize?\",\"answer\":\"The learner’s total cost over T rounds sums classification mistakes and payments. Regret compares the algorithm’s cost to the optimal cost that would be achieved if the productivity functions were known.\"},{\"question\":\"Why is this problem harder than standard Lipschitz bandit settings?\",\"answer\":\"The learner observes expert advice only under the payments it chooses, so learning the relationship between payments and regret introduces an additional and more complex exploration–exploitation trade-off, potentially involving non-smooth effects.\"}]","Trading-Off Payments and Accuracy in Online Classification with Paid Stochastic Experts | PDF",1785806445,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"trading-off-payments-and-accuracy-in-online-classification-with-paid-stochastic-experts","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/trading-off-payments-and-accuracy-in-online-classification-with-paid-stochastic-experts/121715/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",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},"How does paying an expert affect online classification accuracy?","Question",{"text":75,"@type":76},"Each expert’s probability of being correct depends on the payment via an unknown Lipschitz productivity function, so the learner’s chosen payments directly influence prediction accuracy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What cost and regret does the learner aim to minimize?",{"text":80,"@type":76},"The learner’s total cost over T rounds sums classification mistakes and payments. Regret compares the algorithm’s cost to the optimal cost that would be achieved if the productivity functions were known.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is this problem harder than standard Lipschitz bandit settings?",{"text":84,"@type":76},"The learner observes expert advice only under the payments it chooses, so learning the relationship between payments and regret introduces an additional and more complex exploration–exploitation trade-off, potentially involving non-smooth effects.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"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":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":53,"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"]