Limit Theorems for Stochastic Gradient Descent in High-Dimensional Single-Layer Networks

Abstract

This paper studies the high-dimensional scaling limits of online stochastic gradient descent (SGD). Building on the recent work of Ben Arous, Gheissari, and Jagannath on the effective dynamics of SGD, we study the critical scaling regime of the step size for single-layer networks. Below this critical regime, the effective dynamics are governed by deterministic (ballistic) limits, whereas at the critical scale, a new correction term emerges that changes the phase diagram. In this regime, near fixed points, the corresponding diffusive (SDE) limits of the effective dynamics reduce to an Ornstein-Uhlenbeck process under certain conditions. These results highlight how the information exponent controls sample complexity and illustrate the limitations of deterministic scaling limits in capturing stochastic fluctuations in high-dimensional learning dynamics.