[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81567-en":3,"doc-seo-81567-105":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},81567,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Near-optimal Delta-convex Estimation of Lipschitz Functions","This paper develops a tractable method for estimating an unknown Lipschitz regression function from noisy observations under random design. It converts max-affine, shape-restricted regression ideas into a Lipschitz-preserving nonlinear feature expansion that yields a subclass of delta-convex functions as universal approximators. The estimator achieves a near-minimax convergence rate (up to logarithmic factors) depending on intrinsic data dimension, using adaptive partitioning for unknown complexity, penalty regularization to avoid knowing the Lipschitz constant, and a two-stage convex-to-local optimization scheme. Experiments show competitive results versus nearest-neighbor and kernel methods.","arXiv :2511 . 15615v2 [ stat .ML] 10 Jul 2026  \nPublished in JMLR 27(134):1-41, 2026  \nNear-optimal Delta-convex Estimation of Lipschitz Functions  \nG´abor Bal´azs [gabalz@gandg.ai](gabalz@gandg.ai)  \n[G&G](G&G) , [Cartagena](Cartagena) , [Spain](Spain)  \nAbstract  \nThis paper presents a tractable algorithm for estimating an unknown Lipschitz function from noisy observations and establishes an upper bound on its convergence rate. The approach extends max-affine methods from convex shape-restricted regression to the more general Lipschitz setting. A key component is a nonlinear feature expansion that maps max-affine functions into a subclass of delta-convex functions, which act as universal approximators of Lipschitz functions while preserving their Lipschitz constants. Leveraging this property, the estimator attains the minimax convergence rate (up to logarithmic factors) with respect to the intrinsic dimension of the data under squared loss and subgaussian distributions in the random design setting. The algorithm integrates adaptive partitioning to capture intrinsic dimension, a penalty-based regularization mechanism that removes the need to know the true Lipschitz constant, and a two-stage optimization procedure combining a convex initialization with local refinement. The framework is also straightforward to adapt to convex shape-restricted regression. Experiments demonstrate competitive performance relative to other theoretically justified methods, including nearest-neighbor and kernel-based regressors.  \nKeywords: nonparametric regression, Lipschitz function, squared loss, minimax rate, function approximation, delta-convex function, empirical risk minimization  \n1 Introduction  \nThis paper considers the fundamental problem of estimating an unknown regression function from noisy observations in the random design setting. Suppose we observe n independent and identically distributed (i.i.d.) samples, Dn =. ⟨ (Xi, Yi) : i ∈ [n]⟩, for an unknown real-valued regression function f∗ : X∗ → R on some unknown domain X∗ ⊆ Rd , such that  \nXi ∈ X∗ almost surely (a.s.) , Yi =. f∗ (Xi) + ξi. (1)  \nThe noise ξi is centered, satisfying E[ξi|Xi] = 0 a.s. for all i ∈ [n], where [m] =. {1,..., m} for any positive integer m. We assume that the regression function f∗ is λ∗-Lipschitz on X∗ with respect to (w.r.t.) the Euclidean norm ∥·∥2 for some unknown Lipschitz constant λ∗ ∈ (0 , ∞ ) . We evaluate the estimators using the excess risk under squared loss, for which the minimax (convergence) rate is known to be Θ 􀀀n−2/(2+d∗ ) 􀀁 in terms of the sample size n and the intrinsic data dimension d∗ (Stone, 1982) . Throughout the paper, we use the standard asymptotic order of growth notations: Ω(·), Θ(·), and O (·) .  \nIn convex (shape-restricted) regression (e.g. , Lim, 2014; Han and Wellner, 2016; Bal´azs, 2016; Kur et al. , 2024), the regression function f∗ is known to be convex on a convex domain X∗ , and the goal is to estimate f∗ by a convex function. In this setting, it is common to choose the estimator from the class of max-affine functions (functions defined as the maximum of finitely many affine functions) because they approximate any convex function at the worst-case optimal rate (Bal´azs et al. , 2015) . Moreover, empirical risk minimization  \nBal´azs  \nover max-affine functions using n hyperplanes can be reformulated as a tractable convex optimization problem (solvable in polynomial time w.r.t. d and n; Boyd and Vandenberghe, 2004, Section 6.5.5) . Although several extensions of max-affine functions were proposed (Bagirov et al. , 2010; Sun and Yu, 2019; Siahkamari et al. , 2020), none have been shown to achieve the minimax rate up to logarithmic factors (i.e., near-minimax rate) in the general Lipschitz setting of (1) . In this paper, we fill this gap by using the following extension of max-affine functions:  \nFp(ˆX) =. nf : Rd → R 􀀌 f (x) = a[kx0] bf,k + u⊤f,k (x − ˆxk) + vf,k∥x − ˆxk∥p ,  \n(2)  \nx ∈ Rd , bf,k ∈ R, uf,k ∈ Rd , vf,k ∈ R,","cbCaitcJLsVZNjtl","https://ap.wps.com/l/cbCaitcJLsVZNjtl","pdf",1224001,4,1,41,"English","en",105,"# Introduction\n## Problem setting and assumptions\n## Extension of max-affine methods to Lipschitz functions\n## Feature expansion and approximation argument","[{\"question\":\"What regression setting does the paper address?\",\"answer\":\"The paper studies estimation of an unknown real-valued regression function from noisy i.i.d. observations in a random design setting, evaluated via excess risk under squared loss.\"},{\"question\":\"How does the method extend max-affine approaches to Lipschitz functions?\",\"answer\":\"It introduces a nonlinear feature expansion that maps max-affine functions into a subclass of delta-convex functions, designed to approximate Lipschitz regression functions while preserving Lipschitz constants.\"},{\"question\":\"What enables the near-minimax convergence guarantee?\",\"answer\":\"The construction of an adaptive cover based on intrinsic dimension, combined with approximation bounds and an optimization strategy that uses adaptive partitioning plus penalty regularization, yields a near-minimax rate for subgaussian distributions up to logarithmic factors.\"}]",1784174365,103,{"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},"near-optimal-delta-convex-estimation-of-lipschitz-functions","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/near-optimal-delta-convex-estimation-of-lipschitz-functions/81567/",{"url":52,"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","2026-07-25","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 regression setting does the paper address?","Question",{"text":75,"@type":76},"The paper studies estimation of an unknown real-valued regression function from noisy i.i.d. observations in a random design setting, evaluated via excess risk under squared loss.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method extend max-affine approaches to Lipschitz functions?",{"text":80,"@type":76},"It introduces a nonlinear feature expansion that maps max-affine functions into a subclass of delta-convex functions, designed to approximate Lipschitz regression functions while preserving Lipschitz constants.",{"name":82,"@type":73,"acceptedAnswer":83},"What enables the near-minimax convergence guarantee?",{"text":84,"@type":76},"The construction of an adaptive cover based on intrinsic dimension, combined with approximation bounds and an optimization strategy that uses adaptive partitioning plus penalty regularization, yields a near-minimax rate for subgaussian distributions up to 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