{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T12:01:55Z","timestamp":1777982515690,"version":"3.51.4"},"reference-count":26,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2025,8,1]],"date-time":"2025-08-01T00:00:00Z","timestamp":1754006400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2025,8,1]],"date-time":"2025-08-01T00:00:00Z","timestamp":1754006400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T00:00:00Z","timestamp":1745280000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Approximation Theory"],"published-print":{"date-parts":[[2025,8]]},"DOI":"10.1016\/j.jat.2025.106160","type":"journal-article","created":{"date-parts":[[2025,2,27]],"date-time":"2025-02-27T07:54:51Z","timestamp":1740642891000},"page":"106160","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":2,"special_numbering":"C","title":["A note on diffusion limits for stochastic gradient descent"],"prefix":"10.1016","volume":"309","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8248-2151","authenticated-orcid":false,"given":"Alberto","family":"Lanconelli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher S.A.","family":"Lauria","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/j.jat.2025.106160_b1","series-title":"On-line Learning in Neural Networks","first-page":"9","article-title":"On-line learning and stochastic approximations","author":"Bottou","year":"1998"},{"issue":"2","key":"10.1016\/j.jat.2025.106160_b2","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1137\/16M1080173","article-title":"Optimization methods for large-scale machine learning","volume":"60","author":"Bottou","year":"2018","journal-title":"SIAM Rev."},{"issue":"2","key":"10.1016\/j.jat.2025.106160_b3","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/j.jeconom.2020.07.042","article-title":"The continuous-time limit of score-driven volatility models","volume":"221","author":"Buccheri","year":"2021","journal-title":"J. Econometrics"},{"key":"10.1016\/j.jat.2025.106160_b4","doi-asserted-by":"crossref","unstructured":"P. Chaudhari, S. Soatto, Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks, in: International Conference on Learning Representations, 2018.","DOI":"10.1109\/ITA.2018.8503224"},{"key":"10.1016\/j.jat.2025.106160_b5","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/S0304-4076(99)00053-6","article-title":"Reconsidering the continuous time limit of the GARCH(1,1) process","volume":"96","author":"Corradi","year":"2000","journal-title":"J. Econometrics"},{"issue":"7","key":"10.1016\/j.jat.2025.106160_b6","article-title":"Adaptive subgradient methods for online learning and stochastic optimization","volume":"12","author":"Duchi","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"10.1016\/j.jat.2025.106160_b7","unstructured":"R.M. Gower, N. Loizou, X. Qian, A. Sailanbayev, E. Shulgin, P. Richt\u00e1rik, SGD: General Analysis and Improved Rates, in: International Conference on Machine Learning, Los Angeles, United States, 2019."},{"key":"10.1016\/j.jat.2025.106160_b8","unstructured":"D. Kingma, J. Ba, Adam: A Method for Stochastic Optimization, in: International Conference on Learning Representations, 2014."},{"key":"10.1016\/j.jat.2025.106160_b9","article-title":"Numerical solution of stochastic differential equations","author":"Kloeden","year":"2011"},{"key":"10.1016\/j.jat.2025.106160_b10","series-title":"Advances in Neural Information Processing Systems","article-title":"On the validity of modeling SGD with stochastic differential equations (SDEs)","author":"Li","year":"2021"},{"key":"10.1016\/j.jat.2025.106160_b11","series-title":"Proceedings of the 34th International Conference on Machine Learning","first-page":"2101","article-title":"Stochastic modified equations and adaptive stochastic gradient algorithms","volume":"vol. 70","author":"Li","year":"2017"},{"issue":"40","key":"10.1016\/j.jat.2025.106160_b12","first-page":"1","article-title":"Stochastic modified equations and dynamics of stochastic gradient algorithms I: Mathematical foundations","volume":"20","author":"Li","year":"2019","journal-title":"J. Mach. Learn. Res."},{"key":"10.1016\/j.jat.2025.106160_b13","series-title":"Continuous-time limit of stochastic gradient descent revisited","author":"Mandt","year":"2015"},{"key":"10.1016\/j.jat.2025.106160_b14","article-title":"Non-asymptotic analysis of stochastic approximation algorithms for machine learning","volume":"24","author":"Moulines","year":"2011","journal-title":"Adv. Neural Inf. Process. Syst."},{"issue":"1","key":"10.1016\/j.jat.2025.106160_b15","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/0304-4076(90)90092-8","article-title":"ARCH models as diffusion approximations","volume":"45","author":"Nelson","year":"1990","journal-title":"J. Econometrics"},{"key":"10.1016\/j.jat.2025.106160_b16","first-page":"1574","article-title":"Robust stochastic approximation approach to stochastic programming","volume":"19","author":"Nemirovski","year":"2009","journal-title":"Soc. Ind. Appl. Math."},{"key":"10.1016\/j.jat.2025.106160_b17","doi-asserted-by":"crossref","DOI":"10.1007\/s41745-019-0098-4","article-title":"Stochastic gradient descent and its variants in machine learning","volume":"99","author":"Netrapalli","year":"2019","journal-title":"J. Indian Inst. Sci."},{"key":"10.1016\/j.jat.2025.106160_b18","unstructured":"L.M. Nguyen, P.H. Nguyen, M.v. Dijk, P. Richt\u00e1rik, K. Scheinberg, M. Tak\u00e1\u010d, SGD and Hogwild! Convergence Without the Bounded Gradients Assumption, Proc. Mach. Learn. Res. 80."},{"key":"10.1016\/j.jat.2025.106160_b19","unstructured":"T.H. Nguyen, U. Simsekli, M. G\u00fcrb\u00fczbalaban, G. Richard, First Exit Time Analysis of Stochastic Gradient Descent Under Heavy-Tailed Gradient Noise, in: NeurIPS, 2019."},{"issue":"3","key":"10.1016\/j.jat.2025.106160_b20","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1214\/aoms\/1177729586","article-title":"A stochastic approximation method","volume":"22","author":"Robbins","year":"1951","journal-title":"Ann. Math. Stat."},{"issue":"6088","key":"10.1016\/j.jat.2025.106160_b21","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1038\/323533a0","article-title":"Learning representations by back-propagating errors","volume":"323","author":"Rumelhart","year":"1986","journal-title":"Nature"},{"key":"10.1016\/j.jat.2025.106160_b22","series-title":"A tail-index analysis of stochastic gradient noise in deep neural networks","author":"Simsekli","year":"2019"},{"key":"10.1016\/j.jat.2025.106160_b23","doi-asserted-by":"crossref","DOI":"10.1007\/3-540-28999-2","article-title":"Multidimensional diffusion processes","author":"Stroock","year":"1997"},{"issue":"4","key":"10.1016\/j.jat.2025.106160_b24","doi-asserted-by":"crossref","first-page":"1694","DOI":"10.1214\/16-AOS1506","article-title":"Asymptotic and finite-sample properties of estimators based on stochastic gradients","volume":"45","author":"Toulis","year":"2017","journal-title":"Ann. Statist."},{"key":"10.1016\/j.jat.2025.106160_b25","series-title":"Adaptation and Learning in Automatic Systems","author":"Tsypkin","year":"1971"},{"key":"10.1016\/j.jat.2025.106160_b26","article-title":"Asymptotic analysis via stochastic differential equations of gradient descent algorithms in statistical and computational paradigms","volume":"21","author":"Wang","year":"2020","journal-title":"J. Mach. Learn. Res."}],"container-title":["Journal of Approximation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0021904525000188?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0021904525000188?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T07:53:51Z","timestamp":1750319631000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0021904525000188"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8]]},"references-count":26,"alternative-id":["S0021904525000188"],"URL":"https:\/\/doi.org\/10.1016\/j.jat.2025.106160","relation":{},"ISSN":["0021-9045"],"issn-type":[{"value":"0021-9045","type":"print"}],"subject":[],"published":{"date-parts":[[2025,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"A note on diffusion limits for stochastic gradient descent","name":"articletitle","label":"Article Title"},{"value":"Journal of Approximation Theory","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.jat.2025.106160","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2025 The Authors. Published by Elsevier Inc.","name":"copyright","label":"Copyright"}],"article-number":"106160"}}