Rethinking Breiman's Dilemma in Neural Networks: Phase Transitions of Margin Dynamics - Zhu, Weizhi; Huang, Yifei; Yao, Yuan (Frontiers in Applied Mathematics and Statistics, 2020)
Margin enlargement of training data has been an important strategy for perceptrons in machine learning for the purpose of boosting the confidence of training toward a good generalization ability. Yet Breiman (1999) shows a dilemma: a uniform improvement on margin distribution does not necessarily reduce generalization errors. In this paper, we revisit Breiman's dilemma in deep neural networks with recently proposed spectrally normalized margins from a novel perspective based on phase transitions of normalized margin distributions in training dynamics. Normalized margin distribution of a classifier of the data can be divided into two parts: low/small margins such as some negative margins for misclassified samples vs. high/large margins for high confident correctly classified samples, which often behave differently during the training process. Low margins for training and test datasets are often effectively reduced in training, along with reductions of training and test errors, whereas high margins may exhibit different dynamics, reflecting the trade-off between the expressive power of models and the complexity of data. When data complexity is comparable to the model expressiveness, high margin distributions for both training and test data undergo similar decrease-increase phase transitions during training. In such cases, one can predict the trend of generalization or test error through margin-based generalization bounds with restricted Rademacher complexities, shown in two ways in this paper with early stopping time exploiting such phase transitions. On the other hand, over-expressive models may have both low and high training margins undergoing uniform improvements with a distinct phase transition in test margin dynamics. This reconfirms the Breiman's dilemma associated with over-parameterized neural networks where margins fail to predict overfitting. Experiments are conducted with some basic convolutional networks, AlexNet, VGG-16, and ResNet-18, on several datasets, including Cifar10/100 and mini-ImageNet.