[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82193-en":3,"doc-seo-82193-105":29,"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":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},82193,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Solving Stochastic Fixed-Point Equations with High Probability","Studies stochastic fixed-point equations T(x)=x on normed spaces where T is nonexpansive or contractive and is available only through unbiased stochastic evaluations with bounded second central moment. For given accuracy ε and confidence 1-δ, the goal is to find x such that the fixed-point residual ||T(x)-x||≤ε with probability at least 1-δ. It presents VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces, using a recursive clipped stochastic estimator to enable anytime near-geometric high-probability residual decay.","arXiv :2607 .09097v1 [math .OC] 10 Jul 2026  \nSolving Stochastic Fixed-Point Equations with High Probability  \nJelena Diakonikolas  \nDepartment of Computer Sciences  \nUniversity of Wisconsin-Madison  \n[jelena@cs.wisc.edu](jelena@cs.wisc.edu)  \nAbstract  \nWe study stochastic fixed-point equations T (x) = x over normed spaces (E , ∥·∥), where the operator T is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given ϵ > 0,δ ∈ (0 , 1), the goal is to output x ∈ E such that ∥T(x) − x∥ ≤ ϵ with probability at least  \n1 − δ . We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping τ(x;ξ) itself, we clip stochastic differences at the Lipschitz scale γ ∥x − y∥ . This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound:  \non a single event of probability at least 1 − δ, the residual decreases nearly geometrically across epochs, up to lowerorder logarithmic factors. Under only bounded variance, displaying only the dependence on the target error ϵ and Lipschitz constant γ ∈ (0 , 1] of T , the resulting oracle complexity is min{ϵ−5 ,(1 − γ)−3ϵ −2} . Under a Lipschitzin-expectation oracle, the dependence improves to the corresponding ϵ−3 nonexpansive rate (i.e., for γ = 1), and under samplewise nonexpansiveness to ϵ−2 .  \n1 Introduction  \nFixed-point operator equations T (x) = x are pervasive across scientific disciplines. For instance, they are central to game theory and mathematical programming [FP07, Yan09, CLY25, Tod13, BC17, PB14], they arise as primitives in equilibrium problems describing solutions to partial differential equations (PDEs) central to scientific computing [Dud, AAH+ 24, LTT21, ELSV26], and appear in both classical (e.g., reinforcement learning [Bel57, Put94, BT96, SB18]) and emerging (e.g., deep equilibrium models [BKK19] and consistency models [SDCS23]) areas of machine learning. In many such settings, access to exact operator T evaluations is not available due to either computational considerations or the statistical nature of the problem; instead, the operator is accessed through noisy simulations, sampling, or data. Concrete examples include (i) Bellman equations in reinforcement learning (RL) and stochastic dynamic programming [Bel57, Put94, BT96, SB18], (ii) deep equilibrium models and related implicit-layer architectures in deep learning [BKK19, EGGT+ 21, WK20], which define their output as the solution to a fixed-point equation solved through a noisy (minibatch) oracle, and (iii) a broad range of applications in extreme-scale scientific computing, such as, for example, self-consistent field calculations used across materials science and chemistry [KL23, KS65] .  \nMotivated by these applications, we consider the problem of computing an approximate fixed point of an operator T : E → E on a real normed vector space (E , ∥·∥):  \nfind x ∈ E such that x = T (x), (1)  \nwhere T is accessed only through an unbiased stochastic oracle τ (x;ξ) with bounded variance (more precisely, bounded second central moment), and where ξ ∼ D models the underlying randomness from an unknown distribution D. In particular, we assume  \nEξ∼D [τ(x;ξ)] = T(x), Eξ∼D [∥τ(x;ξ) − T(x)∥2] ≤ σ 2 , ∀x ∈ E. (2)  \nWe study such problems in standard settings where the unknown operator T is γ-Lipschitz with γ ∈ (0 , 1], meaning that for any x, y ∈ E , we have ∥T(x) − T(y)∥ ≤ γ ∥x − y∥ . In particular, the operator is assumed to be either  \nγ-contractive (γ ∈ (0 , 1)) or nonexpansive (γ = 1) with respect to ∥·∥ . Our goal is to return a point ˆx whose fixedpoint residual ∥T(ˆx) − ˆx∥ 1 is at most a target ϵ > 0 with high","cbCait3RWFYdjWFR","https://ap.wps.com/l/cbCait3RWFYdjWFR","pdf",517962,1,27,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It studies computing an approximate fixed point of an operator T(x)=x on normed spaces when T can only be accessed through noisy unbiased stochastic evaluations with bounded variance.\"},{\"question\":\"How is the algorithm’s success measured?\",\"answer\":\"Success is measured by achieving a fixed-point residual ||T(ẋ)-ẋ||≤ε with probability at least 1-δ, aiming for high-probability guarantees rather than only expectation-based guarantees.\"},{\"question\":\"What is VR-GHAL and what main estimator idea does it use?\",\"answer\":\"VR-GHAL is a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. It builds a recursive stochastic estimator using clipped differences at the Lipschitz scale γ||x-y||, rather than clipping the oracle output itself.\"}]",1784178727,68,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"solving-stochastic-fixed-point-equations-with-high-probability","",{"@graph":35,"@context":85},[36,53,68],{"@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/solving-stochastic-fixed-point-equations-with-high-probability/82193/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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},"It studies computing an approximate fixed point of an operator T(x)=x on normed spaces when T can only be accessed through noisy unbiased stochastic evaluations with bounded variance.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the algorithm’s success measured?",{"text":80,"@type":76},"Success is measured by achieving a fixed-point residual ||T(ẋ)-ẋ||≤ε with probability at least 1-δ, aiming for high-probability guarantees rather than only expectation-based guarantees.",{"name":82,"@type":73,"acceptedAnswer":83},"What is VR-GHAL and what main estimator idea does it use?",{"text":84,"@type":76},"VR-GHAL is a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. It builds a recursive stochastic estimator using clipped differences at the Lipschitz scale γ||x-y||, rather than clipping the oracle output itself.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]