Spectral Regularization Dynamics]{Spectral Regularization Dynamics: A Continuous-Time Framework for Non-Convex Optimization

Spectral Regularization Dynamics]{Spectral Regularization Dynamics: A Continuous-Time Framework for Non-Convex Optimization | 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 Spectral Regularization Dynamics]{Spectral Regularization Dynamics: A Continuous-Time Framework for Non-Convex Optimization Hasan Dalman, Sara Badur Dalman This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7274493/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract Non-convex optimization is central in machine learning and scientific computing, yet traditional methods often falter at saddle points. This paper introduces Spectral Regularization Dynamics (SRD), a novel continuous-time framework that leverages second-order information to address these challenges. SRD models the optimization trajectory as a system of ordinary differential equations (ODEs) with an autonomous control mechanism that adjusts regularization based on the Hessian?s minimum eigenvalue, ensuring a descent direction. We prove global convergence to critical points using LaSalle?s invariance principle and demonstrate, via the stable manifold theorem, that SRD almost surely avoids saddle points and local maxima. Near strict local minima, it recovers the fast convergence of Newton?s method. Numerical experiments on benchmark problems validate these claims, positioning SRD as a robust theoretical foundation for optimization. Non-Convex Optimization Second-Order Methods Continuous-Time Dynamics Saddle-Point Avoidance Dynamical Systems Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 01 Oct, 2025 Reviews received at journal 19 Sep, 2025 Reviews received at journal 31 Aug, 2025 Reviewers agreed at journal 30 Aug, 2025 Reviews received at journal 22 Aug, 2025 Reviewers agreed at journal 21 Aug, 2025 Reviewers agreed at journal 07 Aug, 2025 Reviewers invited by journal 06 Aug, 2025 Editor assigned by journal 06 Aug, 2025 Submission checks completed at journal 02 Aug, 2025 First submitted to journal 01 Aug, 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. 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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-7274493","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":501392662,"identity":"fad69fa9-5101-4c74-a2fa-aded6aeab164","order_by":0,"name":"Hasan Dalman","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABDklEQVRIiWNgGAWjYBACAwjFzMAHoj42wMTZiNACUsM4s4FBAq6FhxgtzLzEaDFnP/7wcwGDtTyb2OFnj2132NXptp89wPCh7DCDvfQBrFose3KMpWcwpBu2SaeZG+eeSZYwO5OXwDjj3GEGHr4E7A47kMMgzcNwmLFNOsFMOreNWcLsQI4BM28bUAsOlxmcf/74N1CLfZt0+jdpy7Z6CbPzbwyY/+LTcgNoOFBLYpt0jpk0Y9thCbMbQFsY8Wp5Y2bNY5CeDNRSJtl75rjkthtvDA72nEvn4TmDy2Hpj2/zVFjb9kunb5P4uaOa3+x8juGDH2XWcuw92LVANaLxDzDgjpZRMApGwSgYBUQAAOQTVHiM2T/qAAAAAElFTkSuQmCC","orcid":"","institution":"Batman University","correspondingAuthor":true,"prefix":"","firstName":"Hasan","middleName":"","lastName":"Dalman","suffix":""},{"id":501392666,"identity":"8feb4cf0-2049-4958-9985-c2fef3c36965","order_by":1,"name":"Sara Badur Dalman","email":"","orcid":"","institution":"Eskişehir Osmangazi University","correspondingAuthor":false,"prefix":"","firstName":"Sara","middleName":"Badur","lastName":"Dalman","suffix":""}],"badges":[],"createdAt":"2025-08-01 22:23:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7274493/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7274493/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89301275,"identity":"41c72dd2-f37c-4225-8be9-4ad275e76173","added_by":"auto","created_at":"2025-08-18 14:27:01","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1252875,"visible":true,"origin":"","legend":"","description":"","filename":"SRDsp.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7274493/v1_covered_fb1df172-8fe5-4866-9772-7817bbb7c841.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Spectral Regularization Dynamics]{Spectral Regularization Dynamics: A Continuous-Time Framework for Non-Convex Optimization","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"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-machine-learning-and-cybernetics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmlc","sideBox":"Learn more about [International Journal of Machine Learning and Cybernetics](http://actavetscand.biomedcentral.com/)","snPcode":"13042","submissionUrl":"https://submission.nature.com/new-submission/13042/3","title":"International Journal of Machine Learning and Cybernetics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Non-Convex Optimization, Second-Order Methods, Continuous-Time Dynamics, Saddle-Point Avoidance, Dynamical Systems","lastPublishedDoi":"10.21203/rs.3.rs-7274493/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7274493/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Non-convex optimization is central in machine learning and scientific computing, yet traditional methods often falter at saddle points. 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