The early phase of training has been shown to be important in two ways for deep neural networks. First, the degree of regularization in this phase significantly impacts the final generalization. Second, it is accompanied by a rapid change in the local loss curvature influenced by regularization choices. Connecting these two findings, we show that stochastic gradient descent (SGD) implicitly penalizes the trace of the Fisher Information Matrix (FIM) from the beginning of training. We argue it is an implicit regularizer in SGD by showing that explicitly penalizing the trace of the FIM can significantly improve generalization. We further show that the early value of the trace of the FIM correlates strongly with the final generalization. We highlight that in the absence of implicit or explicit regularization, the trace of the FIM can increase to a large value early in training, to which we refer as catastrophic Fisher explosion. Finally, to gain insight into the regularization effect of penalizing the trace of the FIM, we show that 1) it limits memorization by reducing the learning speed of examples with noisy labels more than that of the clean examples, and 2) trajectories with a low initial trace of the FIM end in flat minima, which are commonly associated with good generalization.

深度神经网络训练初期对损失函数的局部曲率有巨大影响。随机梯度下降（SGD）在训练早期隐含地惩罚了Fisher信息矩阵的迹，并且这种隐式的正则化方法会显著提高泛化能力。

灾难性费舍尔爆炸：早期阶段费舍尔矩阵影响泛化