A geometric dissipation bound on the lifespan of gradient-based adaptation

A geometric dissipation bound on the lifespan of gradient-based adaptation | 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 Physical Sciences - Article A geometric dissipation bound on the lifespan of gradient-based adaptation Zhan Peng Jing This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8482405/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The stability of adaptive learning systems is commonly attributed to algorithmic design choices, such as optimization strategies, architectural features, or regularization schemes. However, the physical constraints governing the sustained operation of gradient-based adaptation remain poorly understood. Here we identify a finite geometric dissipation budget that bounds the functional lifespan of gradient-based adaptive systems. By tracking the cumulative geometric displacement of parameters during optimization, we show that system collapse is not primarily determined by specific hyperparameters or model architectures, but instead correlates with the exhaustion of this dissipation budget. Across multiple architectures and optimization algorithms, we observe a consistent dissipation horizon: while learning dynamics may differ in speed, the total geometric path length accumulated at collapse remains bounded. We further demonstrate that exhaustion of this geometric budget coincides with a divergence in the Hessian spectral radius, indicating a loss of adaptive plasticity and a transition into a dynamically constrained regime. Although our analysis employs an operational Euclidean proxy for geometric dissipation, we argue that the existence of a finite adaptive horizon reflects a structural property of the learning landscape rather than a coordinate artifact. These findings establish a quantitative physical limit on continual adaptation and suggest that sustained learning requires mechanisms that actively regulate or reset geometric dissipation, which are absent in current gradient-based artificial systems. Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Statistical physics Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Nonlinear phenomena Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Thermodynamics Full Text Additional Declarations There is NO Competing Interest. Cite Share Download PDF Status: Posted Version 1 posted 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-8482405","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Physical Sciences - Article","associatedPublications":[],"authors":[{"id":574056942,"identity":"b9288d7c-64da-48eb-82f5-c0dfd45b93d7","order_by":0,"name":"Zhan Peng Jing","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA60lEQVRIiWNgGAWjYBACxmYGBuYfFf/l7O8/PgDkS8gQp4XhDLMxw4G0BJAWHqJsYmZsY05sOJBjAOIQ1sLcznv4dWEbG2Njw5nPr27UWPAwsB8+ugG/w/jSrGec42FmZuzdZp1zDOgwnrS0G/i18JgZ8JRJsLEx824zzmEDapHgMSNCC5sBDw8bzzPjnH/EaTF+zNOWICHBw8P8OLeNSFsYZ5w5YGAgwWbGnNsnAbSNgF8M+88Yf/hQcaB+gwTz48853+rk+NkPH8OvpYGBTQLKhjDY8CkHAXlg1HyAsuGMUTAKRsEoGAUoAACw7EOJ7M2GYgAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0009-0007-2823-6647","institution":"Department of Mathematics, Northeastern University","correspondingAuthor":true,"prefix":"","firstName":"Zhan","middleName":"Peng","lastName":"Jing","suffix":""}],"badges":[],"createdAt":"2025-12-30 14:50:35","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8482405/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8482405/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104045796,"identity":"065fdff5-89d2-4501-b44b-fadbda544bd3","added_by":"auto","created_at":"2026-03-06 06:11:58","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1046939,"visible":true,"origin":"","legend":"Article File","description":"","filename":"nature3.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8482405/v1_covered_90cfc79c-c6bf-4a61-a06a-1ad4a9c13aaa.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"A geometric dissipation bound on the lifespan of gradient-based adaptation","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8482405/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8482405/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The stability of adaptive learning systems is commonly attributed to algorithmic design choices, such as optimization strategies, architectural features, or regularization schemes. However, the physical constraints governing the sustained operation of gradient-based adaptation remain poorly understood. Here we identify a finite geometric dissipation budget that bounds the functional lifespan of gradient-based adaptive systems.\r\n\r\nBy tracking the cumulative geometric displacement of parameters during optimization, we show that system collapse is not primarily determined by specific hyperparameters or model architectures, but instead correlates with the exhaustion of this dissipation budget. Across multiple architectures and optimization algorithms, we observe a consistent dissipation horizon: while learning dynamics may differ in speed, the total geometric path length accumulated at collapse remains bounded.\r\n\r\nWe further demonstrate that exhaustion of this geometric budget coincides with a divergence in the Hessian spectral radius, indicating a loss of adaptive plasticity and a transition into a dynamically constrained regime. 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