  Understanding the training dynamics of deep learning models is perhaps a
necessary step toward demystifying the effectiveness of these models. In
particular, how do data from different classes gradually become separable in
their feature spaces when training neural networks using stochastic gradient
descent? In this study, we model the evolution of features during deep learning
training using a set of stochastic differential equations (SDEs) that each
corresponds to a training sample. As a crucial ingredient in our modeling
strategy, each SDE contains a drift term that reflects the impact of
backpropagation at an input on the features of all samples. Our main finding
uncovers a sharp phase transition phenomenon regarding the {intra-class impact:
if the SDEs are locally elastic in the sense that the impact is more
significant on samples from the same class as the input, the features of the
training data become linearly separable, meaning vanishing training loss;
otherwise, the features are not separable, regardless of how long the training
time is. Moreover, in the presence of local elasticity, an analysis of our SDEs
shows that the emergence of a simple geometric structure called the neural
collapse of the features. Taken together, our results shed light on the
decisive role of local elasticity in the training dynamics of neural networks.
We corroborate our theoretical analysis with experiments on a synthesized
dataset of geometric shapes and CIFAR-10.
