  Deep neural networks are over-parameterized, which implies that the number of
parameters are much larger than the number of samples used to train the
network. Even in such a regime deep architectures do not overfit. This
phenomenon is an active area of research and many theories have been proposed
trying to understand this peculiar observation. These include the Vapnik
Chervonenkis (VC) dimension bounds and Rademacher complexity bounds which show
that the capacity of the network is characterized by the norm of weights rather
than the number of parameters. However, the effect of input noise on these
measures for shallow and deep architectures has not been studied. In this
paper, we analyze the effects of various regularization schemes on the
complexity of a neural network which we characterize with the loss, $L_2$ norm
of the weights, Rademacher complexities (Directly Approximately Regularizing
Complexity-DARC1), VC dimension based Low Complexity Neural Network (LCNN) when
subject to varying degrees of Gaussian input noise. We show that $L_2$
regularization leads to a simpler hypothesis class and better generalization
followed by DARC1 regularizer, both for shallow as well as deeper
architectures. Jacobian regularizer works well for shallow architectures with
high level of input noises. Spectral normalization attains highest test set
accuracies both for shallow and deeper architectures. We also show that Dropout
alone does not perform well in presence of input noise. Finally, we show that
deeper architectures are robust to input noise as opposed to their shallow
counterparts.
