  We study a class of deep neural networks with networks that form a directed
acyclic graph (DAG). For backpropagation defined by gradient descent with
adaptive momentum, we show weights converge for a large class of nonlinear
activation functions. The proof generalizes the results of Wu et al. (2008) who
showed convergence for a feed forward network with one hidden layer. For an
example of the effectiveness of DAG architectures, we describe an example of
compression through an autoencoder, and compare against sequential feed forward
networks under several metrics.
