  The regularization and output consistency behavior of dropout and layer-wise
pretraining for learning deep networks have been fairly well studied. However,
our understanding of how the asymptotic convergence of backpropagation in deep
architectures is related to the structural properties of the network and other
design choices (like denoising and dropout rate) is less clear at this time. An
interesting question one may ask is whether the network architecture and input
data statistics may guide the choices of learning parameters and vice versa. In
this work, we explore the association between such structural, distributional
and learnability aspects vis-\`a-vis their interaction with parameter
convergence rates. We present a framework to address these questions based on
convergence of backpropagation for general nonconvex objectives using
first-order information. This analysis suggests an interesting relationship
between feature denoising and dropout. Building upon these results, we obtain a
setup that provides systematic guidance regarding the choice of learning
parameters and network sizes that achieve a certain level of convergence (in
the optimization sense) often mediated by statistical attributes of the inputs.
Our results are supported by a set of experimental evaluations as well as
independent empirical observations reported by other groups.
