  Training deep belief networks (DBNs) requires optimizing a non-convex
function with an extremely large number of parameters. Naturally, existing
gradient descent (GD) based methods are prone to arbitrarily poor local minima.
In this paper, we rigorously show that such local minima can be avoided (upto
an approximation error) by using the dropout technique, a widely used heuristic
in this domain. In particular, we show that by randomly dropping a few nodes of
a one-hidden layer neural network, the training objective function, up to a
certain approximation error, decreases by a multiplicative factor.
  On the flip side, we show that for training convex empirical risk minimizers
(ERM), dropout in fact acts as a "stabilizer" or regularizer. That is, a simple
dropout based GD method for convex ERMs is stable in the face of arbitrary
changes to any one of the training points. Using the above assertion, we show
that dropout provides fast rates for generalization error in learning (convex)
generalized linear models (GLM). Moreover, using the above mentioned stability
properties of dropout, we design dropout based differentially private
algorithms for solving ERMs. The learned GLM thus, preserves privacy of each of
the individual training points while providing accurate predictions for new
test points. Finally, we empirically validate our stability assertions for
dropout in the context of convex ERMs and show that surprisingly, dropout
significantly outperforms (in terms of prediction accuracy) the L2
regularization based methods for several benchmark datasets.
