Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks

Abstract

We study the gradient descent (GD) dynamics of a depth-2 linear neural network with a single input and output. We show that GD converges at an explicit linear rate to a global minimum of the training loss, even with a large stepsize -- about $2/\textrm{sharpness}$. It still converges for even larger stepsizes, but may do so very slowly. We also characterize the solution to which GD converges, which has lower norm and sharpness than the gradient flow solution. Our analysis reveals a trade off between the speed of convergence and the magnitude of implicit regularization. This sheds light on the benefits of training at the ``Edge of Stability'', which induces additional regularization by delaying convergence and may have implications for training more complex models.