  In this paper, we show that although the minimizers of cross-entropy and
related classification losses are off at infinity, network weights learned by
gradient flow converge in direction, with an immediate corollary that network
predictions, training errors, and the margin distribution also converge. This
proof holds for deep homogeneous networks -- a broad class of networks allowing
for ReLU, max-pooling, linear, and convolutional layers -- and we additionally
provide empirical support not just close to the theory (e.g., the AlexNet), but
also on non-homogeneous networks (e.g., the DenseNet). If the network further
has locally Lipschitz gradients, we show that these gradients also converge in
direction, and asymptotically align with the gradient flow path, with
consequences on margin maximization, convergence of saliency maps, and a few
other settings. Our analysis complements and is distinct from the well-known
neural tangent and mean-field theories, and in particular makes no requirements
on network width and initialization, instead merely requiring perfect
classification accuracy. The proof proceeds by developing a theory of unbounded
nonsmooth Kurdyka-{\L}ojasiewicz inequalities for functions definable in an
o-minimal structure, and is also applicable outside deep learning.
