It is arguably believed that flatter minima can generalize better. However,
it has been pointed out that the usual definitions of sharpness, which consider
either the maxima or the integral of loss over a $\delta$ ball of parameters
around minima, cannot give consistent measurement for scale invariant neural
networks, e.g., networks with batch normalization layer. In this paper, we
first propose a measure of sharpness, BN-Sharpness, which gives consistent
value for equivalent networks under BN. It achieves the property of scale
invariance by connecting the integral diameter with the scale of parameter.
Then we present a computation-efficient way to calculate the BN-sharpness
approximately i.e., one dimensional integral along the "sharpest" direction.
Furthermore, we use the BN-sharpness to regularize the training and design an
algorithm to minimize the new regularized objective. Our algorithm achieves
considerably better performance than vanilla SGD over various experiment
settings.

本文提出了一种与 BN 层等价的神经网络下具有一致性的、刻画了梯度的度量标准和对应的优化算法，该算法利用该标准对训练进行正则化，获得了与原始 SGD 相比较显著的性能提升。