Abstract
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 n^α where α ∈ [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 ∈ [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 → ∞.