[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83899-en":3,"doc-seo-83899-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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":13,"seo_description":14,"update_tm":27,"read_time":28},83899,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Functional Bilevel Optimization for Predictive Fairness","Mean demographic parity is a better match than full statistical independence when sensitive attributes are continuous and high-dimensional, since fairness concerns typically target systematic shifts in average predictions rather than identical prediction distributions. The work uses DPVar, the variance of the conditional mean prediction given the sensitive attribute, and reformulates fairness optimization as a functional bilevel problem. Two methods are proposed: FBO with an exact Hessian-free hypergradient for squared loss, and ITD via unrolled inner differentiation. Experiments on synthetic and a semi-synthetic benchmark show lowest or near-lowest fairness-accuracy regret versus strong baselines.","arXiv :2607 .05098v 1 [ cs .LG] 6 Jul 2026  \nFunctional Bilevel Optimization for Predictive Fairness  \nIeva Petrulionyte, Julien Mairal, Michael Arbel  \nUniv. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France  \n[firstname.lastname@inria.fr](firstname.lastname@inria.fr)  \nAbstract  \nWhen sensitive attributes are continuous and high-dimensional – demographicscore vectors, posteriors over attributes, age or income profiles – enforcing full statistical independence is often too restrictive, and existing relaxations rely on indirect dependence penalties or adversarial schemes that do not directly target the fairness-accuracy trade-off. We instead consider mean demographic parity through DPVar, the variance of the conditional-mean prediction given the sensitive attribute, and show that optimizing it yields a functional bilevel problem. We propose two algorithms for this problem: FBO, which uses a closed-form adjoint we derive for the squared-loss case to obtain an exact hypergradient, and ITD, which differentiates through unrolled inner steps and extends beyond squared loss. On synthetic data and a new semi-synthetic benchmark built from 60 tabular regression datasets, both methods achieve the lowest or near-lowest aggregate fairness-accuracy regret, and consistently match or outperform strong HSIC, adversarial, linear-dependence, and generalized-DP baselines.  \n1 Introduction  \nPredictive machine learning models are effective because they exploit statistical signals that reduce prediction error. Unfortunately, standard training does not distinguish between signals we want the model to use and signals tied to attributes deemed “sensitive” in legal, ethical, or policy contexts [1, 2] . Models are then often required to be “fair” with respect to these attributes, which raises the question of which forms of dependence between prediction and sensitive attribute should be controlled. Fairness is not a single property but a family of criteria, capturing different kinds of disparity [2]: demographic parity (DP) compares outcomes across sensitive groups [3], equalizedodds and equal-opportunity notions condition on the true outcome [4], and individual or counterfactual notions enforce consistency across similar individuals or hypothetical interventions [5, 6] . These criteria are generally not equivalent and often cannot be satisfied jointly [7, 8] . We focus on the demographic parity family, motivated by settings where the goal is to control systematic differences in predictions across values of the sensitive attribute, and accordingly restrict our related work and empirical comparisons to DP-style methods.  \nWe consider supervised learning with non-sensitive features X ∈ RkX , label Y ∈ R, and sensitive attribute A ∈ RkA , with predictor fω : RkX → R. Much of the DP literature formulates fairness as full statistical independence fω (X) ⊥ A [9–12] and focuses on categorical attributes or predefined groupings. Yet in many applications A is naturally continuous and vector-valued: age, income, demographic score vectors, or posteriors over attributes [3, 13, 14] . In this regime, full independence is often too strong. It constrains the entire prediction distribution across every value of A, and the gap between this constraint and the actual fairness concern – systematic shifts in average prediction – grows with the dimension and richness of A. In credit scoring, for instance, fairness may require similar average scores across values of A, even if the score distributions differ in spread or shape.  \nFor such applications, a more appropriate requirement is that prediction be balanced on average across values of A. This conditional-mean view of demographic parity asks that a 7→ E [fω (X) | A = a] be  \nPreprint.  \nE [fω (X) | A = a]  \nhigher DPVar / unfair  \nlower DPVar / more fair  \n a  \nFigure 1: Conceptual illustration of DPVar. A predictor with higher DPVar has a conditional-mean profile a 7→ E [fω (X) | A = a] that varies ","cbCaiuR6m5m2jmXc","https://ap.wps.com/l/cbCaiuR6m5m2jmXc","pdf",2664819,1,16,"English","en",105,"# Introduction\n## Fairness and dependence notions\n## Demographic parity for continuous attributes\n## DPVar and the resulting bilevel formulation\n## Contributions","[{\"question\":\"Why is full statistical independence too restrictive for continuous, high-dimensional sensitive attributes?\",\"answer\":\"Full independence constrains the entire prediction distribution across all values of the sensitive attribute. The gap between this constraint and the actual fairness goal—controlling systematic differences in average predictions—grows with the dimension and richness of the sensitive attribute.\"},{\"question\":\"What is DPVar and how does it relate to demographic parity?\",\"answer\":\"DPVar is defined as the variance over A of the conditional mean prediction, DPVar(ω)=VarA(E[fω(X)|A]). It measures how much the average prediction changes with the sensitive attribute, rather than enforcing identical prediction distributions.\"},{\"question\":\"What are FBO and ITD, and how do they address the functional bilevel problem?\",\"answer\":\"FBO targets the squared-loss case using a closed-form adjoint to compute an exact, Hessian-free hypergradient. ITD differentiates through unrolled inner optimization steps and extends the approach beyond squared loss.\"}]",1784191308,40,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"functional-bilevel-optimization-for-predictive-fairness","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/functional-bilevel-optimization-for-predictive-fairness/83899/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"Why is full statistical independence too restrictive for continuous, high-dimensional sensitive attributes?","Question",{"text":74,"@type":75},"Full independence constrains the entire prediction distribution across all values of the sensitive attribute. The gap between this constraint and the actual fairness goal—controlling systematic differences in average predictions—grows with the dimension and richness of the sensitive attribute.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What is DPVar and how does it relate to demographic parity?",{"text":79,"@type":75},"DPVar is defined as the variance over A of the conditional mean prediction, DPVar(ω)=VarA(E[fω(X)|A]). It measures how much the average prediction changes with the sensitive attribute, rather than enforcing identical prediction distributions.",{"name":81,"@type":72,"acceptedAnswer":82},"What are FBO and ITD, and how do they address the functional bilevel problem?",{"text":83,"@type":75},"FBO targets the squared-loss case using a closed-form adjoint to compute an exact, Hessian-free hypergradient. ITD differentiates through unrolled inner optimization steps and extends the approach beyond squared loss.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,118,121,126,129,133],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":28,"slug":117},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":119,"slug":120},30,"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":124,"slug":125},9,"Religion & Spirituality",20,"religion-spirituality",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":124,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]