[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84329-en":3,"doc-seo-84329-105":30,"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":13,"seo_description":14,"update_tm":28,"read_time":29},84329,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","PIT-SUN A Deployable Empirical Marginal Transform Framework with Expectation Consistent Recovery for Regression in Recommender Systems","Estimating original-space conditional expectations is fundamental for value-driven recommender systems such as dwell time, GMV, and LTV forecasting. Standard MSE is expectation-consistent in theory, but its gradients become unstable on heavy-tailed, zero-inflated, and multimodal targets, leading to mean collapse and tail shrinkage. Target transformations reduce scale conflict, yet direct inversion destroys expectation consistency unless the inverse is affine. PIT-SUN provides an empirical, deployable marginal recovery closure that preserves E[Y|X] while improving point accuracy, calibration, and ranking with low overhead.","PIT-SUN: A Deployable Empirical Marginal Transform Framework with Expectation-Consistent Recovery for Regression in Recommender Systems  \nMingyu Zhao 1∗, Zhaohan Li2 , Zhenxiong Miao2 , Xu Zhang2 , Dewei Leng2 , Yanan Niu2 , Kun Gai2  \n1Renmin University of China  \n2 Kuaishou Technology  \narXiv :2607 .08202v 1 [ cs .LG] 9 Jul 2026  \nAbstract  \nEstimating original-space conditional expectations is central to value-driven recommender systems, including dwell time, GMV, and LTV forecasting. Standard MSE is expectationconsistent in principle, but its gradients become unstable on heavy-tailed, zero-inflated, and multimodal targets, causing mean collapse and tail shrinkage. Target transformation alleviates this scale conflict, yet any useful nonlinear marginal transform loses expectation consistency under direct inversion. This is not an implementation oversight: a direct inversetransform estimator is universally expectation-consistent only when the inverse transform is affine, which cannot simultaneously provide bounded tail compression. Existing conditionally linear recovery methods restore expectation consistency, but still leave open which coordinate, inverse lookup, recovery base, and deployment monitor should be selected for sparse complex marginals. We propose Probability-IntegralTranSformed Unbiased recovery (PIT-SUN), a deployable empirical marginal recovery framework. PIT-SUN uses one empirical marginal table to define a bounded normal-score coordinate, its inverse-quantile lookup, a variance-controlled recovery base, and drift monitoring, then applies multiplicative SUN recovery to estimate the original-space expectation instead of directly inverting transformed predictions. Experiments on synthetic distributions, public benchmarks, largescale industrial datasets, and online deployment show robust improvements in point accuracy, calibration, and ranking quality with lightweight deployment overhead.  \nIntroduction  \nMany recommender tasks—including dwell time, GMV, and LTV forecasting—predict continuous business values for ranking, allocation, and multi-objective decisions. Here, dwell time denotes user video watching time (Davidson et al. 2010; Yi et al. 2014; Wu, Rizoiu, and Xie 2018; Wanget al. 2020), while LTV and GMV support customer acquisition and value-driven marketing (Wang, Liu, and Miao 2019; Li et al. 2022; Weng et al. 2024) . As shown in Figure 1, these targets are often sparse, heavy-tailed, multimodal, and context-dependent. Although original-space MSE has m (x) := E[Y | X = x] as its Bayes optimum, finite-sample gradients are exposed to extreme values, causing mean collapse or tail shrinkage.  \n∗Work performed during an internship at Kuaishou Technology.  \nMonotonic transformations such as log, sqrt, or BoxCox (Bartlett 1936, 1947; Sakia 1992) stabilize labels but introduce retransformation bias (Neyman and Scott 1960; Duan 1983): T−1(E[T(Y ) | x])  m (x) . This is structural: if direct inversion were expectation-preserving for all conditional distributions, T −1 would have to be affine, ruling out nonlinear tail compression. Thus recommender regression needs more than a better scalar transform. Existing recovery methods such as TranSUN/GTS (Yu et al. 2025) restore expectation consistency, but still rely on manual choices of transformed target and recovery base. The gap is a deployable closure: the coordinate, inverse lookup, recovery base, and empirical monitor must be specified jointly.  \nWe formalize a deployable empirical marginal recovery closure: one empirical marginal table defines the stable coordinate, inverse lookup, recovery base, and drift monitor while preserving m (x) = E [Y | x] . This explains why standard transformations, PIT-only regression, and manual-transform recovery each remain incomplete. We instantiate this closure with Probability-Integral-TranSformed Unbiased recovery (PIT-SUN), summarized in Figure 1 . Our main contributions are:  \n• Impossibility-guided formulation. We show that","cbCaiftuzL3da5Zv","https://ap.wps.com/l/cbCaiftuzL3da5Zv","pdf",1634384,5,1,20,"English","en",105,"# Abstract\n# Introduction\n# Related Work\n## Regression Targets and Existing Adaptation Strategies","[{\"question\":\"Why does standard MSE become problematic for recommender regression targets?\",\"answer\":\"Although MSE is expectation-consistent in principle, its gradients can become unstable when targets are heavy-tailed, zero-inflated, and multimodal. This instability can cause mean collapse and tail shrinkage.\"},{\"question\":\"What is the core limitation of using nonlinear target transformations with direct inversion?\",\"answer\":\"Any nonlinear marginal transform generally loses expectation consistency under direct inversion. The paper notes that expectation-preserving direct inversion would require the inverse transform to be affine, which conflicts with bounded tail compression.\"},{\"question\":\"How does PIT-SUN achieve expectation-consistent recovery in practice?\",\"answer\":\"PIT-SUN uses one empirical marginal table to define a bounded normal-score coordinate, an inverse-quantile lookup, a variance-controlled recovery base, and drift monitoring. It then applies multiplicative SUN recovery to estimate the original-space expectation instead of directly inverting transformed predictions.\"}]",1784194861,50,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"pit-sun-a-deployable-empirical-marginal-transform-framework-with-expectation-consistent-recovery-for-regression-in-recommender-systems","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"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/pit-sun-a-deployable-empirical-marginal-transform-framework-with-expectation-consistent-recovery-for-regression-in-recommender-systems/84329/",4,{"url":52,"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":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","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},"Why does standard MSE become problematic for recommender regression targets?","Question",{"text":76,"@type":77},"Although MSE is expectation-consistent in principle, its gradients can become unstable when targets are heavy-tailed, zero-inflated, and multimodal. This instability can cause mean collapse and tail shrinkage.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the core limitation of using nonlinear target transformations with direct inversion?",{"text":81,"@type":77},"Any nonlinear marginal transform generally loses expectation consistency under direct inversion. The paper notes that expectation-preserving direct inversion would require the inverse transform to be affine, which conflicts with bounded tail compression.",{"name":83,"@type":74,"acceptedAnswer":84},"How does PIT-SUN achieve expectation-consistent recovery in practice?",{"text":85,"@type":77},"PIT-SUN uses one empirical marginal table to define a bounded normal-score coordinate, an inverse-quantile lookup, a variance-controlled recovery base, and drift monitoring. It then applies multiplicative SUN recovery to estimate the original-space expectation instead of directly inverting transformed predictions.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":29,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":22,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":22,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":20,"slug":136},19,"General","general"]