  The tremendous recent progress in analyzing the training dynamics of
overparameterized neural networks has primarily focused on wide networks and
therefore does not sufficiently address the role of depth in deep learning. In
this work, we present the first trainability guarantee of infinitely deep but
narrow neural networks. We study the infinite-depth limit of a multilayer
perceptron (MLP) with a specific initialization and establish a trainability
guarantee using the NTK theory. We then extend the analysis to an infinitely
deep convolutional neural network (CNN) and perform brief experiments.
