High-Dimensional Limit of Stochastic Gradient Flow via Dynamical Mean-Field Theory

• URL: http://arxiv.org/abs/2602.06320v1
• Date: Fri, 06 Feb 2026 02:37:10 GMT
• Title: High-Dimensional Limit of Stochastic Gradient Flow via Dynamical Mean-Field Theory
• Authors: Sota Nishiyama, Masaaki Imaizumi,
• Abstract summary: Modern machine learning models are typically trained via multi-pass gradient descent (SGD) with small batch sizes.<n>We analyze the high-dimensional dynamics of a differential equation called a emphstochastic gradient flow (SGF)<n>We show that the resulting DMFT equations recover several existing high-dimensional descriptions of SGD dynamics as special cases.
• Score: 6.2000582635449994
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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.

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