[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85192-en":3,"doc-seo-85192-105":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},85192,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Sharper Analysis of Single-Loop Methods for Bilevel Optimization","Bilevel optimization underpins hyperparameter optimization, meta-learning, neural architecture search, and reinforcement learning. Existing hypergradient approaches rely on approximate implicit differentiation and iterative differentiation, yet theoretical guarantees do not precisely reflect efficient single-loop implementations. This work closes the gap by proving sharper convergence for single-loop approximate implicit differentiation and iterative differentiation using a decoupled norm analysis framework. Results improve AID’s rate to O(κ5/K) and tighten ITD’s asymptotic error to O(κ2), matching the known lower bound. Experiments on synthetic and real tasks validate the theory.","arXiv :2607 . 10263v 1 [ cs .LG] 11 Jul 2026  \nSharper Analysis of Single-Loop Methods for Bilevel  \nOptimization  \nYubo Zhou 1 Jun Shu 1 Luo Luo2 Junmin Liu 1  \nDeyu Meng 1 Guang Dai3 Haishan Ye4  \n1 School of Mathematics and Statistics, Xi’an Jiaotong University  \n2 School of Data Science, Fudan University  \n3 SGIT AI Lab, State Grid Corporation of China  \n4 Center for Intelligent Decision-Making and Machine Learning, School of Management,  \nXi’an Jiaotong University  \nAbstract  \nBilevel optimization underpins many machine learning applications, including hyperparameter optimization, meta-learning, neural architecture search, and reinforcement learning. While hypergradient-based methods have advanced significantly, a gap persists between theoretical guarantees and practical single-loop implementations required for efficiency. We bridge this gap by establishing sharper convergence results for single-loop approximate implicit differentiation (AID) and iterative differentiation (ITD) methods, leveraging our proposed analytical framework, decoupled norm analysis (DNA) . For AID, we improve the convergence rate from O (κ6 /K) to O (κ5 /K), where κ is the condition number of the inner-level problem. For ITD, we prove that the asymptotic error is O (κ2 ) , exactly matching the known lower bound and improving upon the previous O (κ3 ) guarantee. Numerical experiments on synthetic and real tasks corroborate our theoretical findings.  \n1 Introduction  \nBilevel optimization has attracted extensive attention in various applications of machine learning, including hyperparameter optimization (Maclaurin et al., 2015; Franceschi et al., 2017; Shabanet al., 2019; Shen et al., 2024), meta-learning (Chen et al., 2017; Finn et al., 2017; Franceschi et al. , 2018), neural architecture search (Liu et al., 2018; He et al., 2020), and reinforcement learning (Zhang et al., 2020; Wang et al., 2020; Shen et al., 2025) . Bilevel optimization corresponds to solving one optimization problem subject to constraints defined by another optimization problem. In this paper, we focus on the following bilevel optimization problem:  \nmin Φ(x) = f(x, y∗ (x)),  \nx∈Rm  \ns.t. y ∗ (x) = arg ming(x, y), (1)  \ny∈Rn  \nwhere the outer-and inner-level functions f and g are both jointly continuously differentiable on Rm × Rn. We focus on the setting where g is strongly convex with respect to (w.r.t.) the inner-level variable y, which can guarantee the uniqueness of the inner solution (Chen et al. , 2024) .  \nHypergradient-based algorithms have recently gained significant attention for their balance of simplicity and efficiency. Two prominent approaches are approximate implicit differentiation (AID) (Domke, 2012; Pedregosa, 2016; Ghadimi and Wang, 2018; Grazzi et al., 2020; Ji et al. , 2021) and iterative differentiation (ITD) (Franceschi et al., 2017; Shaban et al., 2019; Grazzi et al., 2020; Ji et al., 2021; Liu et al., 2021) . The key distinction lies in how they estimate the  \nhypergradient ∇Φ(x): AID leverages the implicit function theorem, while ITD applies automatic differentiation (see Section 3) . Despite this difference, both methods require solving the inner problem to obtain the optimal solution y ∗ . In practice, however, closed-form solutions are rarely available, and one typically resorts to gradient descent to compute an approximate solution yˆ.  \nMost theoretical studies of bilevel optimization analyze algorithms that employ multi-loop updates (multi-step gradient descent) for the inner problem and linear-system (Ghadimi and Wang, 2018; Ji et al., 2021; Dong et al., 2025; Fang et al., 2025) . In contrast, practical algorithms overwhelmingly adopt single-loop updates, where only one inner update is performed per outer iteration. The main appeal of single-loop methods is computational efficiency: they significantly reduce training cost while maintaining competitive performance. This design has become standard across a wide range of applications. For instance, ","cbCaioHRIyQsEUj0","https://ap.wps.com/l/cbCaioHRIyQsEUj0","pdf",792164,2,1,26,"English","en",105,"# Introduction\n## Bilevel optimization problem setup\n## Hypergradient-based methods: AID and ITD\n## Single-loop motivation and prior work\n## Proposed sharper analysis via Decoupled Norm Analysis (DNA)","[{\"question\":\"What is the bilevel optimization problem studied in the document?\",\"answer\":\"The outer problem minimizes Φ(x)=f(x,y*(x)) subject to the inner solution y*(x)=arg min_y g(x,y). The outer and inner functions are jointly continuously differentiable, with g strongly convex in y to ensure uniqueness of the inner solution.\"},{\"question\":\"How do approximate implicit differentiation (AID) and iterative differentiation (ITD) differ?\",\"answer\":\"AID estimates hypergradients using the implicit function theorem, while ITD uses automatic differentiation of the iterative procedure (as described in the paper’s Section 3). Both methods typically require an approximate inner solution ŷ when closed forms are unavailable.\"},{\"question\":\"What theoretical improvements does the paper achieve for single-loop methods?\",\"answer\":\"For single-loop AID, the convergence rate improves from O(κ6/K) to O(κ5/K). For single-loop ITD, the asymptotic error becomes O(κ2), matching the known lower bound and improving over the previous O(κ3) guarantee.\"}]",1784201651,66,{"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},"sharper-analysis-of-single-loop-methods-for-bilevel-optimization","",{"@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/sharper-analysis-of-single-loop-methods-for-bilevel-optimization/85192/",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","2026-07-23","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 is the bilevel optimization problem studied in the document?","Question",{"text":75,"@type":76},"The outer problem minimizes Φ(x)=f(x,y*(x)) subject to the inner solution y*(x)=arg min_y g(x,y). The outer and inner functions are jointly continuously differentiable, with g strongly convex in y to ensure uniqueness of the inner solution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do approximate implicit differentiation (AID) and iterative differentiation (ITD) differ?",{"text":80,"@type":76},"AID estimates hypergradients using the implicit function theorem, while ITD uses automatic differentiation of the iterative procedure (as described in the paper’s Section 3). Both methods typically require an approximate inner solution ŷ when closed forms are unavailable.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical improvements does the paper achieve for single-loop methods?",{"text":84,"@type":76},"For single-loop AID, the convergence rate improves from O(κ6/K) to O(κ5/K). For single-loop ITD, the asymptotic error becomes O(κ2), matching the known lower bound and improving over the previous O(κ3) guarantee.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"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"]