Implicit bias of deep linear networks in the large learning rate phase

Most theoretical studies explaining the regularization effect in deep learning have only focused on gradient descent with a sufficient small learning rate or even gradient flow (infinitesimal learning rate). Such researches, however, have neglected a reasonably large learning rate applied in most practical applications. In this work, we characterize the implicit bias effect of deep linear networks for binary classification using the logistic loss in the large learning rate regime, inspired by the seminal work by Lewkowycz et al. [26] in a regression setting with squared loss. They found a learning rate regime with a large stepsize named the catapult phase, where the loss grows at the early stage of training and eventually converges to a minimum that is flatter than those found in the small learning rate regime. We claim that depending on the separation conditions of data, the gradient descent iterates will converge to a flatter minimum in the catapult phase. We rigorously prove this claim under the assumption of degenerate data by overcoming the difficulty of the non-constant Hessian of logistic loss and further characterize the behavior of loss and Hessian for non-separable data. Finally, we demonstrate that flatter minima in the space spanned by non-separable data along with the learning rate in the catapult phase can lead to better generalization empirically.