Random Matrix Theory for Stochastic Gradient Descent
This work provides theoretical insights into SGD dynamics for researchers in machine learning, but it is incremental as it builds on existing physics and statistics frameworks.
The paper tackled the problem of understanding learning dynamics in machine learning by applying random matrix theory to stochastic gradient descent, deriving a linear scaling rule between learning rate and batch size and identifying universal aspects of weight matrix dynamics, with tests conducted on Gaussian Restricted Boltzmann Machines and linear one-hidden-layer neural networks.
Investigating the dynamics of learning in machine learning algorithms is of paramount importance for understanding how and why an approach may be successful. The tools of physics and statistics provide a robust setting for such investigations. Here we apply concepts from random matrix theory to describe stochastic weight matrix dynamics, using the framework of Dyson Brownian motion. We derive the linear scaling rule between the learning rate (step size) and the batch size, and identify universal and non-universal aspects of weight matrix dynamics. We test our findings in the (near-)solvable case of the Gaussian Restricted Boltzmann Machine and in a linear one-hidden-layer neural network.