{"id":10337,"date":"2026-02-09T07:02:34","date_gmt":"2026-02-09T07:02:34","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2026\/02\/09\/2602-06320\/"},"modified":"2026-02-09T07:02:34","modified_gmt":"2026-02-09T07:02:34","slug":"2602-06320","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2026\/02\/09\/2602-06320\/","title":{"rendered":"High-Dimensional Limit of Stochastic Gradient Flow via Dynamical Mean-Field Theory"},"content":{"rendered":"

High-Dimensional Limit of Stochastic Gradient Flow via Dynamical Mean-Field Theory
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arXiv:2602.06320v1 Announce Type: new
\nAbstract: Modern machine learning models are typically trained via multi-pass stochastic gradient descent (SGD) with small batch sizes, and understanding their dynamics in high dimensions is of great interest. However, an analytical framework for describing the high-dimensional asymptotic behavior of multi-pass SGD with small batch sizes for nonlinear models is currently missing. In this study, we address this gap by analyzing the high-dimensional dynamics of a stochastic differential equation called a emph{stochastic gradient flow} (SGF), which approximates multi-pass SGD in this regime. In the limit where the number of data samples $n$ and the dimension $d$ grow proportionally, we derive a closed system of low-dimensional and continuous-time equations and prove that it characterizes the asymptotic distribution of the SGF parameters. Our theory is based on the dynamical mean-field theory (DMFT) and is applicable to a wide range of models encompassing generalized linear models and two-layer neural networks. We further show that the resulting DMFT equations recover several existing high-dimensional descriptions of SGD dynamics as special cases, thereby providing a unifying perspective on prior frameworks such as online SGD and high-dimensional linear regression. Our proof builds on the existing DMFT technique for gradient flow and extends it to handle the stochasticity in SGF using tools from stochastic calculus.<\/div>\n

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\n Sota Nishiyama, Masaaki Imaizumi
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High-Dimensional Limit of Stochastic Gradient Flow via Dynamical Mean-Field Theory arXiv:2602.06320v1 Announce Type: new Abstract: Modern machine learning models are typically trained via multi-pass stochastic gradient descent (SGD) with small batch sizes, and understanding their dynamics in high dimensions is of great interest. However, an analytical framework for describing the high-dimensional asymptotic behavior of multi-pass […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,1878,113,112],"tags":[487,332,606],"class_list":["post-10337","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cond-mat-dis-nn","category-cs-lg","category-stat-ml","tag-dimensional","tag-high","tag-stochastic"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/10337"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=10337"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/10337\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=10337"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=10337"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=10337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}