Thermodynamic Natural Gradient Descent (NGD-T): Regulating Natural-Gradient Steps by a Geometric Speed–Cost Bound

Thermodynamic Natural Gradient Descent (NGD-T): Regulating Natural-Gradient Steps by a Geometric Speed–Cost Bound | 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 Article Thermodynamic Natural Gradient Descent (NGD-T): Regulating Natural-Gradient Steps by a Geometric Speed–Cost Bound Barco You This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8626621/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 We introduce Thermodynamic Natural Gradient Descent (NGD T), an optimizer that enforces a physical speed–cost constraint by combining Fisher preconditioned updates with a dissipation aware step size regulator. Starting from an Entropic Action, we show that Natural Gradient Flow (NGF) uniquely minimizes instantaneous irreversible dissipation for a fixed loss decrease. NGD T implements this principle in discrete updates by (i) preconditioning gradients with an approximate inverse Fisher, (ii) computing the geometric norm Δ_F=∇L^⊤ F^(-1) ∇L, and (iii) mapping a user specified dissipation budget Q_budget to a step size η_T that saturates the speed–cost bound. We present numerically stable constructions for rank deficient Fisher estimates using eigendecomposition or Tikhonov damping, a hybrid nullspace fallback to preserve progress in truncated modes, and a scalable K FAC integration with eigendecomposition caching. On CIFAR 10 experiments NGD T matches Adam in convergence while substantially reducing the predicted irreversible dissipation. NGD T provides a principled, tunable trade off between learning speed and thermodynamic cost and is compatible with standard large scale Fisher approximations. Physical sciences/Mathematics and computing/Computer science Physical sciences/Engineering/Electrical and electronic engineering Physical sciences/Mathematics and computing/Software Physical sciences/Mathematics and computing/Computational science 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. 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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-8626621","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":577734031,"identity":"f4eb4967-ae34-4da6-a564-9a32e22b3f47","order_by":0,"name":"Barco You","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzElEQVRIiWNgGAWjYBACAzDJI8HAD6ITCkjRItkA0mJAtBYQ4wAKFw8wZz/78MMHGQs54/OrEz88MGCQ5xc7gF+LZU+6seQMHgljsxtvN0sAHWY4c3YCAYcdSGNj5uGRSNx24+wGkJYEg9uEtJx/BtZSv3nG2c0/iNNyA2JLggF/7zbibLGc8YwZ5BfDGTd4t1kkGEgQ9os5fxrjh489dfL8/Wc33/xRYSPPL01ACxgw9gAJCbBKCSKUg8EPIOY/QKzqUTAKRsEoGGkAAKhjPnfuxVpPAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-3736-0136","institution":"Heidelberg University","correspondingAuthor":true,"prefix":"","firstName":"Barco","middleName":"","lastName":"You","suffix":""}],"badges":[],"createdAt":"2026-01-17 14:30:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8626621/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8626621/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101204044,"identity":"23a321ec-3b48-4961-91e5-eacb4151e110","added_by":"auto","created_at":"2026-01-27 09:41:22","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":887580,"visible":true,"origin":"","legend":"","description":"","filename":"NGDTNatureCSr1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8626621/v1_covered_93ed1c9c-bd59-4180-b085-655343f6c62e.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"\u003cp\u003eThermodynamic Natural Gradient Descent (NGD-T): Regulating Natural-Gradient Steps by a Geometric Speed–Cost Bound\u003c/p\u003e","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-8626621/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8626621/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We introduce Thermodynamic Natural Gradient Descent (NGD T), an optimizer that enforces a physical speed–cost constraint by combining Fisher preconditioned updates with a dissipation aware step size regulator. 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