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SGD in the Large: Average-case Analysis, Asymptotics, and Stepsize Criticality

Courtney Paquette
Elliot Paquette
Kiwon Lee
Proceedings of Machine Learning Research (2021)


We propose a new framework, inspired by random matrix theory, for analyzing the dynamics of stochastic gradient descent (SGD) when both number of samples and dimensions are large. This framework applies to any fixed stepsize and the finite sum setting. Using this new framework, we show that the dynamics of SGD on a least squares problem with random data become deterministic in the large sample and dimensional limit. Furthermore, the limiting dynamics are governed by a Volterra integral equation. This model predicts that SGD undergoes a phase transition at an explicitly given critical stepsize that ultimately affects its convergence rate, which we also verify experimentally.