On the Origin of Implicit Regularization in Stochastic Gradient Descent

For infinitesimal learning rates, stochastic gradient descent (SGD) follows the path of gradient flow on the full batch loss function. However moderately large learning rates can achieve higher test accuracies, and this generalization benefit is not explained by convergence bounds, since the learning rate which maximizes test accuracy is often larger than the learning rate which minimizes training loss. To interpret this phenomenon we prove that for SGD with random shuffling, the mean SGD iterate also stays close to the path of gradient flow if the learning rate is small and finite, but on a modified loss. This modified loss is composed of the original loss function and an implicit regularizer, which penalizes the norms of the minibatch gradients. Under mild assumptions, when the batch size is small the scale of the implicit regularization term is proportional to the ratio of the learning rate to the batch size. We verify empirically that explicitly including the implicit regularizer in the loss can enhance the test accuracy when the learning rate is small. INTRODUCTION In the limit of vanishing learning rates, stochastic gradient descent with minibatch gradients (SGD) follows the path of gradient flow on the full batch loss function (Yaida, 2019) . However in deep networks, SGD often achieves higher test accuracies when the learning rate is moderately large (LeCun et al., 2012; Keskar et al., 2017) . This generalization benefit is not explained by convergence rate bounds (Ma et al., 2018; Zhang et al., 2019) , because it arises even for large compute budgets for which smaller learning rates often achieve lower training losses (Smith et al., 2020). Although many authors have studied this phenomenon (