Nonasymptotic convergence analysis for the tamed unadjusted stochastic Langevin algorithm

Nonasymptotic convergence analysis for the tamed unadjusted stochastic Langevin algorithm | 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 Nonasymptotic convergence analysis for the tamed unadjusted stochastic Langevin algorithm Jing Huang, Yin Dai, Yulin Jiao, Lican Kang, Xiliang Lu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7746921/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 In this work, we consider sampling from a target distribution $\pi_{\beta}$ characterized by the density function $ \pi_{\beta}( \theta) = e^{-\beta U(\theta) }/ \int_{\mathbb R^d} e^{-\beta U(\theta)} , \mathrm{d} \theta$ with $\beta>0$. It is well-known that the Euler-Maruyama discretization of overdamped Langevin stochastic differential equations (SDEs) exhibits instability when the potentials exhibit superlinear growth. Building upon the approach proposed in \cite{brosse2019tamed} for mitigating the impact of superlinear drift coefficients in SDEs, we propose a novel Langevin dynamics-based algorithm, termed the Tamed Unadjusted Stochastic Langevin Algorithm (TUSLA), to address the aforementioned sampling problem and establish rigorous performance guarantees. Specifically, we establish a sharp non-asymptotic convergence guarantee in Kullback–Leibler (KL) divergence with the optimal rate of order one, by combining tools from the logarithmic Sobolev inequality (LSI) and the Fokker–Planck equation. As a direct consequence, we further obtain an $O(\lambda^{1/2})$ convergence rate in both Wasserstein-2 and total variation distances, thereby strengthening and generalizing the best-known results in the current literature. Our theoretical findings are supported by comprehensive high-dimensional experiments. Full Text Additional Declarations No competing interests reported. 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-7746921","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":525796967,"identity":"66065f0b-8fce-45b1-a3c9-aa57ad913ea7","order_by":0,"name":"Jing Huang","email":"","orcid":"","institution":"Wuhan University","correspondingAuthor":false,"prefix":"","firstName":"Jing","middleName":"","lastName":"Huang","suffix":""},{"id":525796968,"identity":"fee2a59d-4ee1-42ac-b5dc-8b4ad80f4ae5","order_by":1,"name":"Yin Dai","email":"","orcid":"","institution":"Hefei University","correspondingAuthor":false,"prefix":"","firstName":"Yin","middleName":"","lastName":"Dai","suffix":""},{"id":525796969,"identity":"da1f9345-cfc0-42d5-9f91-d100ec6e8b8d","order_by":2,"name":"Yulin Jiao","email":"","orcid":"","institution":"Wuhan University","correspondingAuthor":false,"prefix":"","firstName":"Yulin","middleName":"","lastName":"Jiao","suffix":""},{"id":525796970,"identity":"6fd8d0f9-4810-4a3f-9169-f0447ecc7884","order_by":3,"name":"Lican Kang","email":"","orcid":"","institution":"Wuhan University","correspondingAuthor":false,"prefix":"","firstName":"Lican","middleName":"","lastName":"Kang","suffix":""},{"id":525796971,"identity":"941c19e1-6a1f-4d04-9e60-84151cd1ca20","order_by":4,"name":"Xiliang Lu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2klEQVRIie3PsQrCMBCA4YRAXCJdr4jvcFCwCNJnaSk4Ka6CoJWCo7voQ3TKHCjYRXfFpSA4ubi4aquDW6KbQ34It9yXEEJstn+swZLXxEbC60kTI2HvHUChfiYQfkkwpwugg2Dqu9fOWZBeO1PsUuqIm9ZExtBdD3xPkL6XKe6jjjiMpkglAzwNZUuQPMqU4KAj/E1mgMd9TR5mUr0yL6nMAQ/Nmigzqf4yLyNZuNludHc3GHurnHe0BItUqZucOFhs+3AdB+1lkV605FX4uaE6zLhvs9lsNlNPqYJBn+xMrN0AAAAASUVORK5CYII=","orcid":"","institution":"Wuhan University","correspondingAuthor":true,"prefix":"","firstName":"Xiliang","middleName":"","lastName":"Lu","suffix":""}],"badges":[],"createdAt":"2025-09-30 04:08:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7746921/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7746921/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93486687,"identity":"46454d9f-f8a7-48d6-82bf-f14995618af4","added_by":"auto","created_at":"2025-10-14 11:12:49","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":6687,"visible":true,"origin":"","legend":"","description":"","filename":"7305c5fec37d4874ae7f3e84b6204129.json","url":"https://assets-eu.researchsquare.com/files/rs-7746921/v1/3f80b94e192f6b1a85c567ec.json"},{"id":97227415,"identity":"3a803570-9d62-4710-99af-0443feaae4bb","added_by":"auto","created_at":"2025-12-02 08:39:27","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":883793,"visible":true,"origin":"","legend":"","description":"","filename":"STATCOMPUTNonasymptoticconvergenceforTUSLA.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7746921/v1_covered_942091ba-93e2-4c83-ad7d-f1e239303b74.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Nonasymptotic convergence analysis for the tamed unadjusted stochastic Langevin algorithm","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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-7746921/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7746921/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this work, we consider sampling from a target distribution $\\pi_{\\beta}$ characterized by the density function $ \\pi_{\\beta}( \\theta) = e^{-\\beta U(\\theta) }/ \\int_{\\mathbb R^d} e^{-\\beta U(\\theta)} \\, \\mathrm{d} \\theta$ with $\\beta\u003e0$. It is well-known that the Euler-Maruyama discretization of overdamped Langevin stochastic differential equations (SDEs) exhibits instability when the potentials exhibit superlinear growth. Building upon the approach proposed in \\cite{brosse2019tamed} for mitigating the impact of superlinear drift coefficients in SDEs, we propose a novel Langevin dynamics-based algorithm, termed the Tamed Unadjusted Stochastic Langevin Algorithm (TUSLA), to address the aforementioned sampling problem and establish rigorous performance guarantees. Specifically, we establish a sharp non-asymptotic convergence guarantee in Kullback–Leibler (KL) divergence with the optimal rate of order one, by combining tools from the logarithmic Sobolev inequality (LSI) and the Fokker–Planck equation. As a direct consequence, we further obtain an $O(\\lambda^{1/2})$ convergence rate in both Wasserstein-2 and total variation distances, thereby strengthening and generalizing the best-known results in the current literature. Our theoretical findings are supported by comprehensive high-dimensional experiments.","manuscriptTitle":"Nonasymptotic convergence analysis for the tamed unadjusted stochastic Langevin algorithm","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-14 11:12:44","doi":"10.21203/rs.3.rs-7746921/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"a75817b2-fd77-417d-a42f-ecd7ae1a1cb9","owner":[],"postedDate":"October 14th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-12-02T08:38:23+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-14 11:12:44","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7746921","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7746921","identity":"rs-7746921","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}