[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85249-en":3,"doc-seo-85249-105":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},85249,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Lower Bound on the Cumulative Constraint Violation for the OGD+Projection algorithm for Constrained Online Convex Optimization","Constrained online convex optimization is studied under a round-wise protocol where an action x_t is chosen from a convex set and then a convex loss f_t and convex constraint g_t are revealed, with the requirement g_t(x)≤0 defining feasibility. The work focuses on minimizing static regret while also controlling cumulative constraint violation against a benchmark static feasible action. Prior best algorithms achieve O(√T) regret with dimension-dependent CCV. A new lower bound proves the OGD+Projection CCV is Ω(T^{2/d−1}), establishing the first such result.","arXiv :2607 . 10808v 1 [ cs .LG] 12 Jul 2026  \nLower Bound on the Cumulative Constraint Violation for the OGD+Projection algorithm for Constrained Online Convex  \nOptimization (COCO)  \nHaricharan Balasundaram, Karthick Krishna Mahendran, Rahul Vaze  \nAbstract  \nThe problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action xt ∈ X ⊂ Rd , a convex loss function ft and a convex constraint function gt that drives the constraint gt (x) ≤ 0 are revealed. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV) compared to the benchmark that knows the loss functions and constraint functions ft and gt for all t ahead of time, and chooses a static optimal action that is feasible with respect to all gt (x) ≤ 0. Currently, the best known algorithm is OGD+Projection algorithm of [Vaze and Sinha, 2025] that has simultaneous regret of O( √T) and CCV of O(T1/3) for d = 2 [Balasundaramet al., 2026], and simultaneous regret of O ( √T) and CCV of O( √ T) for any d [Sarkar and Sinha, 2026] .  \nd − 1  \nIn this paper, we show that the CCV of the OGD+Projection algorithm is Ω(T 2d ) . This is the first such lower bound result.  \n1 Introduction  \nWe consider constrained online convex optimization (COCO), where on every round t, an online algorithm first chooses an admissible action xt ∈ X ⊂ Rd , and then the adversary chooses a convex loss/cost function ft : X → R and a constraint function of the form gt (x) ≤ 0 , where gt : X → R is a convex function. Let X ⋆ be the feasible set consisting of all admissible actions that satisfy all constraints gt (x) ≤ 0, t ∈ [T] . We work under the standard assumption that X ⋆ is not empty (called the feasibility assumption) .  \nSince gt ’s are revealed after the action xt is chosen, an online algorithm need not necessarily take feasible actions on each round, and in addition to the static regret  \nRegret [1:T] ≡ sup sup RegretT (x⋆ ), (1)  \n{ft }Tt=1 x⋆ ∈X ⋆  \nwhere RegretT (x⋆ ) ≡ P ft (xt) − P ft (x⋆ ), an additional metric of interest is the cumulative constraint violation (CCV) defined as  \nT T  \nCCV [1:T] ≡ X max(gt (xt), 0) =X(gt (xt ))+ . (2)  \nt=1 t=1  \nThe goal is to design an online algorithm to simultaneously achieve a small regret (1) with respect to any admissible benchmark x⋆ ∈ X ⋆ and a small CCV (2) .  \nWith constraint sets Gt = {x ∈ X : gt (x) ≤ 0} being convex for all t, and the assumption X ⋆ = ∩t Gt ∅, note that the sets St = ∩tτ=1 Gτ are convex and are nested, i.e. St ⊆ St−1 and x⋆ ∈ St for all t. Essentially, set St ’s are sufficient to quantify the CCV.  \n1.1 Prior Work  \nConstrained OCO (COCO): (A) Time-invariant constraints: COCO with time-invariant constraints, i.e., gt = g,∀ t Yuan and Lamperski [2018], Jenatton et al. [2016], Mahdavi et al. [2012], Yi et al. [2021] has been considered extensively, where the function g is assumed to be known to the algorithm a priori. The algorithm is allowed to take actions that are infeasible at any time to avoid the costly projection step of the vanilla projected OGD algorithm and the main objective was to design an efficient algorithm with a small regret and CCV while avoiding the explicit projection step.  \n(B) Time-varying constraints: The more difficult question is solving the COCO problem when the constraint functions, [i.e.](i.e. gt)[ g](i.e. gt)[t](i.e. gt)’s, change arbitrarily with time t. In this setting, all prior work on COCO made the feasibility assumption. One popular algorithm for solving COCO considered a Lagrangian function optimization that is updated using the primal and dual variables Yu et al. [2017], Sun et al. [2017], Yi et al. [2023] . Alternatively, Neely and Yu [2017] and Liakopoulos et al. [2019] used the drift-plus-penalty (DPP) framework Neely [2010] to solve the COCO, but which needed additional assumptions, e.g. the Slater’s condition in Neely and Yu [2017] and with weaker form of the feasibility assumption Neel","cbCaigFlMNXYpvIR","https://ap.wps.com/l/cbCaigFlMNXYpvIR","pdf",358971,1,19,"English","en",105,"# Abstract\n# Introduction\n## Prior Work","[{\"question\":\"What problem setting does the paper study in constrained online convex optimization?\",\"answer\":\"Each round, the learner chooses an action x_t from a convex set, then an adversary reveals a convex loss f_t and a convex constraint g_t with feasibility defined by g_t(x)≤0.\"},{\"question\":\"How are static regret and cumulative constraint violation (CCV) defined?\",\"answer\":\"Static regret compares cumulative loss to that of a static feasible action x* over all rounds. CCV sums the positive parts of constraint violations max(g_t(x_t),0) across rounds.\"},{\"question\":\"What is the main lower bound contribution for the OGD+Projection algorithm?\",\"answer\":\"The paper shows that the CCV of OGD+Projection is lower bounded by Ω(T^{2/d−1}), providing the first result of this type.\"}]",1784202069,48,{"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},"lower-bound-on-the-cumulative-constraint-violation-for-the-ogdprojection-algorithm-for-constrained-online-convex-optimization","",{"@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/lower-bound-on-the-cumulative-constraint-violation-for-the-ogdprojection-algorithm-for-constrained-online-convex-optimization/85249/",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","2026-07-22","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 setting does the paper study in constrained online convex optimization?","Question",{"text":75,"@type":76},"Each round, the learner chooses an action x_t from a convex set, then an adversary reveals a convex loss f_t and a convex constraint g_t with feasibility defined by g_t(x)≤0.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are static regret and cumulative constraint violation (CCV) defined?",{"text":80,"@type":76},"Static regret compares cumulative loss to that of a static feasible action x* over all rounds. CCV sums the positive parts of constraint violations max(g_t(x_t),0) across rounds.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main lower bound contribution for the OGD+Projection algorithm?",{"text":84,"@type":76},"The paper shows that the CCV of OGD+Projection is lower bounded by Ω(T^{2/d−1}), providing the first result of this type.","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":21,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},"General","general"]