The Dynamics of Gradient Descent for Overparametrized Neural Networks

We consider the dynamics of gradient descent (GD) in overparameterized single hidden layer neural networks with a squared loss function. Recently, it has been shown that, under some conditions, the parameter values obtained using GD achieve zero training error and generalize well if the initial conditions are chosen appropriately. Here, through a Lyapunov analysis, we show that the dynamics of neural network weights under GD converge to a point which is close to the minimum norm solution subject to the condition that there is no training error when using the linear approximation to the neural network. To illustrate the application of this result, we show that the GD converges to a prediction function that generalizes well, thereby providing an alternative proof of the generalization results in Arora et al. (2019).