[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117735-en":3,"doc-seo-117735-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},117735,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Machine Learning Architectures for Price Formation Models","Machine learning architectures are developed to numerically solve a mean-field games (MFGs) system arising in commodity price formation. The training process is built on a min–max characterization of the optimal control and price variables, and the study provides a posteriori estimates to assess convergence of the learning-based procedure. A linear–quadratic setting is used to illustrate the approach via numerical experiments, with results tied to an Euler–Lagrange variational formulation.","arXiv :2204 .03968v1 [math .OC] 8 Apr 2022  \nMACHINE LEARNING ARCHITECTURES FOR PRICE FORMATION  \nMODELS  \n􀀒  \nDIOGO GOMES, JULIAN GUTIERREZ, AND MATHIEU LAURIERE  \nAbstract. Here, we study machine learning (ML) architectures to solve a mean-􀀌eld games (MFGs) system arising in price formation models. We formulate a training process that relies on a min-max characterization of the optimal control and price variables. Our main theoretical contribution is the development of a posteriori estimates as a tool to evaluate the convergence of the training process. We illustrate our results with numerical  \nexperiments for a linear-quadratic model.  \n1. introduction  \nHere, we consider machine learning (ML) methods to numerically solve the mean-􀀌eld games (MFGs) price formation model introduced in [27] . This model describes the price $ of a commodity with deterministic supply Q. This commodity is traded in a population of rational agents under a market-clearing condition on a 􀀌nite time horizon T > 0. The price formation problem is reduced to the following MFGs system.  \nProblem 1 . Given m0 2 P (R), H : R2 ! R di􀀋erentiable in the second argument, uT : R ! R, and Q 2 C 1 ([0; T ]), 􀀌nd u; m : [0; T ] 􀀂 R ! R and $ : [0; T ] ! R satisfying m > 0 and  \n>8 􀀀ut + H(x; $ + ux ) = 0 [0; T ] 􀀂 R ;  \n>>  \n> u (T; x) = uT (x) x 2 R ;  \n\u003Cmt 􀀀 (Hp (x; $ + ux)m)x = 0 [0; T ] 􀀂 R ; (1.1)  \n> m(0; x) = m0 (x) x 2 R ;  \n>  \n>: 􀀀RR Hp (x; $ + ux)mdx = Q (t) t 2 [0; T ]:  \nIn the previous problem, the Hamiltonian H is the Legendre transform of a Lagrangian L; that is,  \nH (x; p) = sup f􀀀pv 􀀀 L(x; v)g ; (x; p) 2 R2 ; (1 .2)  \nv2R  \nwhere v 7! L (x; v) is convex for all x 2 R. The existence and uniqueness of solutions for (1.1) was obtained in [27] under convexity assumptions for uT and L, and some further technical assumptions on H. The 􀀌rst equation is solved in the viscosity sense, and the second equation is solved in the distributional sense. Moreover, u is Lipschitz continuous in x, and $ is continuous. Furthermore, the linear-quadratic model admits semi-explicit solutions, as presented in [25] for the game with a 􀀌nite population and random supply.  \nHere, we present a method to approximate solutions to this price formation problem using ML tools. For that, we formulate (1.1) as a constrained minimization problem. Then, we follow an update rule analogous to the dual-ascent method in constrained optimization to optimize the ML parameters. We can verify the convergence of our method using a  \nDate: April 11, 2022 .  \nKey words and phrases. Mean Field Games; Price formation; Common noise; Neural networks.  \nKing Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900 . Saudi Arabia. e-mail: [diogo.gomes@kaust.edu.sa](diogo.gomes@kaust.edu.sa).  \nKing Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900 . [Saudi Arabia. e-mail: julian.gutierrezpineda@kaust.edu.sa](Saudi Arabia. e-mail: julian.gutierrezpineda@kaust.edu.sa).  \nNew York University Shanghai (NYU Shanghai), Shanghai [200122. China. e-mail: ml5197@nyu.edu](200122. China. e-mail: ml5197@nyu.edu).  \nThe authors were partially supported by King Abdullah University of Science and Technology (KAUST) baseline funds and KAUST OSR-CRG2021-4674 .  \n􀀒  \n2 DIOGO GOMES, JULIAN GUTIERREZ, AND MATHIEU LAURIERE  \nposteriori estimates, which are based on the Euler-Lagrange equation that characterizes our minimization problem. The optimal control problem described by (1.1) is brie􀀍y recalled in Section 2, where we present a variational problem that motivates our numerical method. In Section 3, we state the main assumptions for the MFGs price formation model. In addition to developing ML frameworks for price formation models, our main theoretical contribution is an a posteriori estimate that controls the di􀀋erence between the solution of Problem 1 and an approximate solution to an Euler-Lagrange equation. These a posteriori estimates do not","cbCaiohZR68IALLv","https://ap.wps.com/l/cbCaiohZR68IALLv","pdf",9656306,1,21,"English","en",105,"# Introduction\n## Problem formulation and MFGs system\n## Min–max training and constrained minimization\n# Main theoretical contribution\n## A posteriori estimates and convergence criterion\n# Numerical experiments\n## Linear–quadratic model results","[{\"question\":\"What problem does the paper address in price formation models?\",\"answer\":\"It addresses numerically solving a mean-field games (MFGs) system that models commodity price formation under market-clearing conditions over a finite time horizon.\"},{\"question\":\"How is the machine learning training process constructed?\",\"answer\":\"The method formulates the MFGs system as a constrained minimization problem and updates ML parameters using an update rule analogous to dual-ascent in constrained optimization.\"},{\"question\":\"What guarantees convergence of the learned approximations?\",\"answer\":\"The paper derives a posteriori estimates based on the Euler–Lagrange equation. These estimates provide a criterion ensuring that if the residual terms converge to zero, the ML approximations converge to the true solution of the price formation problem.\"}]","Machine Learning Architectures for Price Formation Models | PDF",1785679278,53,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"machine-learning-architectures-for-price-formation-models","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":53},"https://docshare.wps.com/document/machine-learning-architectures-for-price-formation-models/117735/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in price formation models?","Question",{"text":75,"@type":76},"It addresses numerically solving a mean-field games (MFGs) system that models commodity price formation under market-clearing conditions over a finite time horizon.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the machine learning training process constructed?",{"text":80,"@type":76},"The method formulates the MFGs system as a constrained minimization problem and updates ML parameters using an update rule analogous to dual-ascent in constrained optimization.",{"name":82,"@type":73,"acceptedAnswer":83},"What guarantees convergence of the learned approximations?",{"text":84,"@type":76},"The paper derives a posteriori estimates based on the Euler–Lagrange equation. These estimates provide a criterion ensuring that if the residual terms converge to zero, the ML approximations converge to the true solution of the price formation problem.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"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":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"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"]