  We explain that the difficulties of training deep neural networks come from a
syndrome of three consistency issues. This paper describes our efforts in their
analysis and treatment. The first issue is the training speed inconsistency in
different layers. We propose to address it with an intuitive,
simple-to-implement, low footprint second-order method. The second issue is the
scale inconsistency between the layer inputs and the layer residuals. We
explain how second-order information provides favorable convenience in removing
this roadblock. The third and most challenging issue is the inconsistency in
residual propagation. Based on the fundamental theorem of linear algebra, we
provide a mathematical characterization of the famous vanishing gradient
problem. Thus, an important design principle for future optimization and neural
network design is derived. We conclude this paper with the construction of a
novel contractive neural network.
