  Dropout is a widely-used regularization technique, often required to obtain
state-of-the-art for a number of architectures. This work demonstrates that
dropout introduces two distinct but entangled regularization effects: an
explicit effect (also studied in prior work) which occurs since dropout
modifies the expected training objective, and, perhaps surprisingly, an
additional implicit effect from the stochasticity in the dropout training
update. This implicit regularization effect is analogous to the effect of
stochasticity in small mini-batch stochastic gradient descent. We disentangle
these two effects through controlled experiments. We then derive analytic
simplifications which characterize each effect in terms of the derivatives of
the model and the loss, for deep neural networks. We demonstrate these
simplified, analytic regularizers accurately capture the important aspects of
dropout, showing they faithfully replace dropout in practice.
