On Fast Dropout and its Applicability to Recurrent Networks
Authors: Justin Bayer, Christian Osendorfer, Sebastian Urban, Nutan Chen, Daniela Korhammer, Patrick van der Smagt
ICLR 2014

Abstract: Recurrent Neural Networks (RNNs) are rich models for the processing of sequential data. Recent work on advancing the state of the art has been focused on the optimization or modelling of RNNs, mostly motivated by adressing the problems of the vanishing and exploding gradients.
The control of overfitting has seen considerably less attention.
This paper contributes to that by analyzing fast dropout, a recent regularization method for generalized linear models and neural networks from a back-propagation inspired perspective. We show that fast dropout implements a quadratic form of an adaptive, per-parameter regularizer, which rewards large weights in the light of underfitting, penalizes them for overconfident predictions and vanishes at minima of an unregularized training loss. One consequence of this is the absense of a global weight attractor, which is particularly appealing for RNNs, since the dynamics are not biased towards a certain regime. We positively test the hypothesis that this improves the performance of RNNs on four musical data sets and a natural language processing (NLP) task, on which we achieve state of the art results.


Peer review:
[Review]:
The paper applies the fast dropout of Wang and Manning to RNNs, obtaining excellent results
on two standard RNN datasets.  This work also provieds an interesting analysis of fast droupt,
presenting it as an explicit regularization.

Pros:  The results are good.    The paper convincingly shows that fast dropout is effective
for training RNNs.   I like the equation E[dJ/da * da/dw_i] = dJ/dE[a] * dE[a]/dw_i + dJ/dV[a] * dV[a]/dw_i.
It helps reason about stochastic systems such as this.  

Cons:  The novelty lies almost entirely in the new analysis of dropout.   I also wish the analysis could tell
us more about dropout. The analysis establishes that the variance may increase or decrease in response to delta_a.
But what is it?  That's the most interesting part.  We need to know what happens to delta_a in order to be able
to say nontrivial things about dropout.  Right now, the analysis says, it all boils down to delta_a.  But how 
can we relate delta_a to situations that occur in practice?  I.e., can you say, delta_a is positive in situtaion a,
and negative in situations b?   Perhaps the analysis already allows for such conclusions, but I don't
see them, and so it would be nice to inculde something like that.  

Also, it is worth stating that a global regularizer may not exist, and the contribution of dropout to the update
may not be the derivative of any gradient.

There are a few typos:
Page 6, sampling:   hat a = E[a] + s * sqrt{V[a]}, and not E[a] + s * V[a] as written in the text.  I worry
that this may have caused an error in the remanider of the sampling section, but I am not sure because I didn't
understand it fully.

Likewies, Eq. 1 should be E[a], not V[a].