  Deep learning has aroused extensive attention due to its great empirical
success. The efficiency of the block coordinate descent (BCD) methods has been
recently demonstrated in deep neural network (DNN) training. However,
theoretical studies on their convergence properties are limited due to the
highly nonconvex nature of DNN training. In this paper, we aim at providing a
general methodology for provable convergence guarantees for this type of
methods. In particular, for most of the commonly used DNN training models
involving both two- and three-splitting schemes, we establish the global
convergence to a critical point at a rate of ${\cal O}(1/k)$, where $k$ is the
number of iterations. The results extend to general loss functions which have
Lipschitz continuous gradients and deep residual networks (ResNets). Our key
development adds several new elements to the Kurdyka-{\L}ojasiewicz inequality
framework that enables us to carry out the global convergence analysis of BCD
in the general scenario of deep learning.
