  Recent research has used margin theory to analyze the generalization
performance for deep neural networks (DNNs). The existed results are almost
based on the spectrally-normalized minimum margin. However, optimizing the
minimum margin ignores a mass of information about the entire margin
distribution, which is crucial to generalization performance. In this paper, we
prove a generalization upper bound dominated by the statistics of the entire
margin distribution. Compared with the minimum margin bounds, our bound
highlights an important measure for controlling the complexity, which is the
ratio of the margin standard deviation to the expected margin. We utilize a
convex margin distribution loss function on the deep neural networks to
validate our theoretical results by optimizing the margin ratio. Experiments
and visualizations confirm the effectiveness of our approach and the
correlation between generalization gap and margin ratio.
