High-dimensional learning dynamics of multi-pass Stochastic Gradient Descent in multi-index models
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912911086059520 |
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| author | Fan, Zhou Wang, Leda |
| author_facet | Fan, Zhou Wang, Leda |
| contents | We study the learning dynamics of a multi-pass, mini-batch Stochastic Gradient Descent (SGD) procedure for empirical risk minimization in high-dimensional multi-index models with isotropic random data. In an asymptotic regime where the sample size $n$ and data dimension $d$ increase proportionally, for any sub-linear batch size $κ\asymp n^α$ where $α\in [0,1)$, and for a commensurate ``critical'' scaling of the learning rate, we provide an asymptotically exact characterization of the coordinate-wise dynamics of SGD. This characterization takes the form of a system of dynamical mean-field equations, driven by a scalar Poisson jump process that represents the asymptotic limit of SGD sampling noise. We develop an analogous characterization of the Stochastic Modified Equation (SME) which provides a Gaussian diffusion approximation to SGD.
Our analyses imply that the limiting dynamics for SGD are the same for any batch size scaling $α\in [0,1)$, and that under a commensurate scaling of the learning rate, dynamics of SGD, SME, and gradient flow are mutually distinct, with those of SGD and SME coinciding in the special case of a linear model. We recover a known dynamical mean-field characterization of gradient flow in a limit of small learning rate, and of one-pass/online SGD in a limit of increasing sample size $n/d \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21093 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High-dimensional learning dynamics of multi-pass Stochastic Gradient Descent in multi-index models Fan, Zhou Wang, Leda Machine Learning Optimization and Control Probability Statistics Theory 60F17, 68T05 (Primary), 60G07, 60H10, 90C15 (Secondary) We study the learning dynamics of a multi-pass, mini-batch Stochastic Gradient Descent (SGD) procedure for empirical risk minimization in high-dimensional multi-index models with isotropic random data. In an asymptotic regime where the sample size $n$ and data dimension $d$ increase proportionally, for any sub-linear batch size $κ\asymp n^α$ where $α\in [0,1)$, and for a commensurate ``critical'' scaling of the learning rate, we provide an asymptotically exact characterization of the coordinate-wise dynamics of SGD. This characterization takes the form of a system of dynamical mean-field equations, driven by a scalar Poisson jump process that represents the asymptotic limit of SGD sampling noise. We develop an analogous characterization of the Stochastic Modified Equation (SME) which provides a Gaussian diffusion approximation to SGD. Our analyses imply that the limiting dynamics for SGD are the same for any batch size scaling $α\in [0,1)$, and that under a commensurate scaling of the learning rate, dynamics of SGD, SME, and gradient flow are mutually distinct, with those of SGD and SME coinciding in the special case of a linear model. We recover a known dynamical mean-field characterization of gradient flow in a limit of small learning rate, and of one-pass/online SGD in a limit of increasing sample size $n/d \to \infty$. |
| title | High-dimensional learning dynamics of multi-pass Stochastic Gradient Descent in multi-index models |
| topic | Machine Learning Optimization and Control Probability Statistics Theory 60F17, 68T05 (Primary), 60G07, 60H10, 90C15 (Secondary) |
| url | https://arxiv.org/abs/2601.21093 |