Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws

We consider the extensive-width regime r dβ for β [0,1), and assume a power-law decay on the (non-negative) second-layer coefficients λj j α for α 0. We present a sharp analysis of the SGD dynamics in the feature learning regime, for both the population limit and the finite-sample (online) discretization, and derive scaling laws for the prediction risk that highlight the power-law dependencies on the optimization time, sample size, and model width. Our analysis combines a precise characterization of the associated matrix Riccati differential equation with novel matrix monotonicity arguments to establish convergence guarantees for the infinite-dimensional effective dynamics.