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.

本文研究随机梯度下降（SGD）的学习率对准确性的影响，证明当学习率适当大时，SGD的迭代路径离梯度下降路径更近，这种现象可通过引入一个隐式正则化项进行解释，并通过实验证明在适当的学习率下包含隐式正则化项可以提高测试准确性。

随机梯度下降中隐式正则化的起源