AI-Modulated Pólya Urns (AIM-PU): A Unified Framework for Risk-Sensitive Contextual Bandits, Resource Allocation and Portfolio Design

AI-Modulated Pólya Urns (AIM-PU): A Unified Framework for Risk-Sensitive Contextual Bandits, Resource Allocation and Portfolio Design | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article AI-Modulated Pólya Urns (AIM-PU): A Unified Framework for Risk-Sensitive Contextual Bandits, Resource Allocation and Portfolio Design Saran Ishika Maiti, Debashis Chatterjee This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8390479/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract We propose the AI-modulated Pólya Urn(AIM-PU), a stochastic framework bridging classical path-dependent reinforcement processes with adaptive, covariate-driven decision rules. While standard Pólya--Eggenberger urns assume fixed replacement kernels, AIM-PU introduces a heterogeneous reinforcement mechanism where the replacement tensor is modulated by a learned policy adapted to a covariate filtration. We establish that this coupling preserves analytical tractability via martingale approximations. Under mild assumptions on policy regularity and graph irreducibility, we prove: (i) controlled linear growth of the total mass; (ii) almost-sure convergence of the normalized composition; and (iii) a functional central limit theorem (FCLT) where fluctuations converge to a Gaussian mixture. The framework is first verified through extensive synthetic simulation studies that confirm the theoretical convergence rates and distributional stability in volatile, non-stationary environments. We then demonstrate the model's practical utility in risk-sensitive portfolio allocation using real-world financial data (SPY, MSFT, DUK, TSLA) from 2020--2025. By embedding Conditional Value at Risk (CVaR) constraints directly into the urn dynamics, AIM-PU structurally enforces long-term risk limits. Our results show that AIM-PU outperforms stateless contextual bandits in both Sharpe ratio and drawdown minimization, providing a robust, interpretable, and mathematically grounded approach to automated financial decision-making. Generalized Pólya urn stochastic approximation contextual bandits functional central limit theorem risk-sensitive control martingale limit theory Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 21 Jan, 2026 Reviews received at journal 19 Jan, 2026 Reviewers agreed at journal 14 Jan, 2026 Reviewers agreed at journal 12 Jan, 2026 Reviewers invited by journal 12 Jan, 2026 Editor assigned by journal 03 Jan, 2026 Submission checks completed at journal 18 Dec, 2025 First submitted to journal 17 Dec, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8390479","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":573958572,"identity":"54268c03-236d-4713-b63d-78e4344b6b7b","order_by":0,"name":"Saran Ishika Maiti","email":"","orcid":"","institution":"Visva-Bharati University","correspondingAuthor":false,"prefix":"","firstName":"Saran","middleName":"Ishika","lastName":"Maiti","suffix":""},{"id":573958573,"identity":"d2031096-5869-4a32-828e-33772962d072","order_by":1,"name":"Debashis Chatterjee","email":"data:image/png;base64,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","orcid":"","institution":"Visva-Bharati University","correspondingAuthor":true,"prefix":"","firstName":"Debashis","middleName":"","lastName":"Chatterjee","suffix":""}],"badges":[],"createdAt":"2025-12-18 03:23:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8390479/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8390479/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100203050,"identity":"3bb83cef-d038-4bde-b21f-a50e91b834ce","added_by":"auto","created_at":"2026-01-14 05:45:23","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5292,"visible":true,"origin":"","legend":"","description":"","filename":"0cbd4c171011423ea4841421d2983a81.json","url":"https://assets-eu.researchsquare.com/files/rs-8390479/v1/e2a4e2de97506fd1b8dcc038.json"},{"id":100369631,"identity":"e2579f0b-6028-43ad-b7a4-f129f6957f6c","added_by":"auto","created_at":"2026-01-16 07:59:12","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1794823,"visible":true,"origin":"","legend":"","description":"","filename":"PolyaAIupdated2025.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8390479/v1_covered_83bba0a0-128a-488b-a6b0-dd9f4f940c45.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eAI-Modulated Pólya Urns (AIM-PU): A Unified Framework for Risk-Sensitive Contextual Bandits, Resource Allocation and Portfolio Design\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"international-journal-of-data-science-and-analytics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jdsa","sideBox":"Learn more about [International Journal of Data Science and Analytics](http://link.springer.com/journal/41060)","snPcode":"41060","submissionUrl":"https://submission.nature.com/new-submission/41060/3","title":"International Journal of Data Science and Analytics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Generalized Pólya urn, stochastic approximation, contextual bandits, functional central limit theorem, risk-sensitive control, martingale limit theory","lastPublishedDoi":"10.21203/rs.3.rs-8390479/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8390479/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe propose the AI-modulated Pólya Urn(AIM-PU), a stochastic framework bridging classical path-dependent reinforcement processes with adaptive, covariate-driven decision rules. 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