High-dimensional learning dynamics of multi-pass Stochastic Gradient Descent in multi-index models

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 $\kappa \asymp n^\alpha$ where $\alpha \in [0,1)$, 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.