  The success of deep architectures is at least in part attributed to the
layer-by-layer unsupervised pre-training that initializes the network. Various
papers have reported extensive empirical analysis focusing on the design and
implementation of good pre-training procedures. However, an understanding
pertaining to the consistency of parameter estimates, the convergence of
learning procedures and the sample size estimates is still unavailable in the
literature. In this work, we study pre-training in classical and distributed
denoising autoencoders with these goals in mind. We show that the gradient
converges at the rate of $\frac{1}{\sqrt{N}}$ and has a sub-linear dependence
on the size of the autoencoder network. In a distributed setting where disjoint
sections of the whole network are pre-trained synchronously, we show that the
convergence improves by at least $\tau^{3/4}$, where $\tau$ corresponds to the
size of the sections. We provide a broad set of experiments to empirically
evaluate the suggested behavior.
