SGD vs GD: Rank Deficiency in Linear Networks

In this article, we study the behaviour of continuous-time gradient methods on a two-layer linear network with square loss. A dichotomy between SGD and GD is revealed: GD preserves the rank at initialization while (label noise) SGD diminishes the rank regardless of the initialization. We demonstrate this rank deficiency by studying the time evolution of the determinant of a matrix of parameters. To further understand this phenomenon, we derive the stochastic differential equation (SDE) governing the eigenvalues of the parameter matrix. This SDE unveils a replusive force between the eigenvalues: a key regularization mechanism which induces rank deficiency. Our results are well supported by experiments illustrating the phenomenon beyond linear networks and regression tasks.