Dynamical mean-field theory for stochastic gradient descent in Gaussian mixture classification
This work provides theoretical insights into SGD dynamics for a specific non-convex problem, which is incremental as it builds on existing methods like dynamical mean-field theory.
The authors tackled the problem of understanding the learning dynamics of stochastic gradient descent (SGD) in a non-convex setting by analyzing a single-layer neural network classifying a high-dimensional Gaussian mixture, resulting in a closed-form analysis that reveals how the algorithm navigates the loss landscape.
We analyze in a closed form the learning dynamics of stochastic gradient descent (SGD) for a single-layer neural network classifying a high-dimensional Gaussian mixture where each cluster is assigned one of two labels. This problem provides a prototype of a non-convex loss landscape with interpolating regimes and a large generalization gap. We define a particular stochastic process for which SGD can be extended to a continuous-time limit that we call stochastic gradient flow. In the full-batch limit, we recover the standard gradient flow. We apply dynamical mean-field theory from statistical physics to track the dynamics of the algorithm in the high-dimensional limit via a self-consistent stochastic process. We explore the performance of the algorithm as a function of the control parameters shedding light on how it navigates the loss landscape.