[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83461-en":3,"doc-seo-83461-105":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},83461,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Distributionally Robust Linear Regression With Block Lewis Weights","The paper presents an algorithmic framework for the group distributionally robust (GDR) least squares problem. It formulates the GDR regression task via a carefully constructed least-squares subproblem and leverages accelerated proximal methods. The approach improves on known interior-point methods in moderate-accuracy regimes and recovers state-of-the-art guarantees for the special case of ℓ∞ regression. It also provides schemes that smoothly interpolate between average and distributionally robust losses.","arXiv :2607 .00252v1 [ cs .LG] 30 Jun 2026  \nDistributionally Robust Linear Regression With Block Lewis  \nWeights  \nNaren Sarayu Manoj* Kumar Kshitij Patel†  \nJuly 2, 2026  \nAbstract  \nWe present an algorithm for the group distributionally robust (GDR) least squares problem.  \nGoulinfoilvrelenaarowlg- 􀀼osyfrgritstomrouhmemapo-sresboc,atalvenpainest grsoeaafomme􀀑rriv-mesc cecuoontoltfsinlice fuciv,mnae,noAbitamAkcafLkloeedsor bwlideutlocs wisiokeimsagtsin,atrgonthscepo(Ahnem,ise􀀼capislr}A1/mica3 􀀑etal3sR),  \nproblem to a carefully chosen least squares problem and an application of accelerated proximal methods. Our algorithm improves over known interior point methods for moderate accuracy regimes and matches the state-of-the-art guarantees for the special case of ℓ∞ regression. We also give algorithms that smoothly interpolate between minimizing the average least squares loss and the distributionally robust loss.  \n* TTIC. Email: [nsm@ttic.edu](nsm@ttic.edu. Supported by NSF Award ECCS-2216899)[. Supported by NSF Award ECCS-2216899](nsm@ttic.edu. Supported by NSF Award ECCS-2216899)  \n†Institute for Foundations of Data Science, Yale University. Email: [kumarkshitij.patel@yale.edu](kumarkshitij.patel@yale.edu. This work was)[. This work was](kumarkshitij.patel@yale.edu. This work was)[ ](kumarkshitij.patel@yale.edu. This work was)partly done while the author was a Fellow at the Simons Institute for the Theory of Computing.  \nContents  \n1 Introduction 3  \n1. 1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n1.2 Prior Results, Connections, and Open Problems ..................... 7  \n1.3 Paper Outline ......................................... 8  \n2 Technical Overview 8  \n2.1 Solving Proximal Subproblems ............................... 9  \n2.1.1 The Robust Case (􀀿 = ∞) ............................... 9  \n2.1.2 The Interpolating Case (2 ≤ 􀀿 \u003C ∞) ......................... 11  \n2.2 Iterating Proximal Calls ................................... 11  \n2.3 The Geometry of the Proximal Subproblems and Block Lewis Weights ........ 12  \n2.4 Algorithm for Distributionally Robust Regression .................... 13  \n3 Block Lewis Weights and their Properties 14  \n4 Mirror Descent with Inexact Updates 17  \n5 Optimal MS Acceleration under Custom Euclidean Geometry 20  \n6 Minimizing the Distributionally Robust Loss 25  \n6.1 Smoothly Approximating the Objective .......................... 25  \n6.2 Calculus for LogSumExp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27  \n6.3 Smoothness and Quasi-self-concordance of the Modified Objective .......... 30  \n6.4 Analysis of Algorithm 1 ................................... 32  \n7 Interpolating Between Average and Robust Losses 35  \n7.1 Calculus for the objective ................................... 35  \n7.1.1 Strong Convexity of the Objective ......................... 36  \n7.1.2 Smoothness of the Objective ............................. 41  \n7.2 Facts about the Iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  \n7.3 Proximal Subproblems – Calculus, Algorithms, Proofs .................. 42  \n7.3.1 Hessian Stability ................................... 43  \n7.3.2 Strong Convexity of the Proximal Objective and Friends ............ 47  \n7.3.3 Smoothness of the Proximal Objective ....................... 48  \n7.3.4 Solving the Proximal Subproblems ......................... 50  \n7.4 The Algorithm ......................................... 58  \n8 Empirical Evaluation 59  \n8.1 Synthetic Heterogeneous Regression Construction .................... 59  \n8.1.1 Computing the Robust Optimum via Convex Programming .......... 60  \n8.1.2 Baselines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61  \n8.1.3 Hyperparameter Tuning ............................... 61  \n8.1.4 Empirical Behavior .................................. 62  \n8.2 Real-world Experiment: ACS Income ............................ 6","cbCaimysUYwtGJtU","https://ap.wps.com/l/cbCaimysUYwtGJtU","pdf",1172742,2,1,71,"English","en",105,"# Introduction\n## Our Results\n## Prior Results, Connections, and Open Problems\n## Paper Outline\n# Technical Overview\n## Solving Proximal Subproblems\n## Iterating Proximal Calls\n## Geometry of the Proximal Subproblems and Block Lewis Weights\n## Algorithm for Distributionally Robust Regression\n# Block Lewis Weights and their Properties\n# Mirror Descent with Inexact Updates\n# Optimal MS Acceleration under Custom Euclidean Geometry\n# Minimizing the Distributionally Robust Loss\n## Smoothly Approximating the Objective\n## Calculus for LogSumExp\n## Smoothness and Quasi-self-concordance of the Modified Objective\n## Analysis of Algorithm 1\n# Interpolating Between Average and Robust Losses\n## Calculus for the objective\n## Facts about the Iterates\n## Proximal Subproblems – Calculus, Algorithms, Proofs\n## The Algorithm\n# Empirical Evaluation","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper targets group distributionally robust (GDR) linear regression in a least-squares setting, focusing on robust performance across multiple groups of data.\"},{\"question\":\"How does the proposed method relate GDR regression to a least-squares subproblem?\",\"answer\":\"It converts the GDR problem into a carefully chosen least-squares problem and then applies accelerated proximal methods to solve the resulting structure efficiently.\"},{\"question\":\"What performance claims are made compared with existing approaches?\",\"answer\":\"The algorithm improves over known interior-point methods for moderate accuracy regimes and matches state-of-the-art guarantees for the ℓ∞ regression special case, while also enabling smooth interpolation between average and robust losses.\"}]",1784188123,179,{"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},"distributionally-robust-linear-regression-with-block-lewis-weights","",{"@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/distributionally-robust-linear-regression-with-block-lewis-weights/83461/",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 paper address?","Question",{"text":75,"@type":76},"The paper targets group distributionally robust (GDR) linear regression in a least-squares setting, focusing on robust performance across multiple groups of data.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method relate GDR regression to a least-squares subproblem?",{"text":80,"@type":76},"It converts the GDR problem into a carefully chosen least-squares problem and then applies accelerated proximal methods to solve the resulting structure efficiently.",{"name":82,"@type":73,"acceptedAnswer":83},"What performance claims are made compared with existing approaches?",{"text":84,"@type":76},"The algorithm improves over known interior-point methods for moderate accuracy regimes and matches state-of-the-art guarantees for the ℓ∞ regression special case, while also enabling smooth interpolation between average and robust 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