  The information bottleneck (IB) principle has been suggested as a way to
analyze deep neural networks. The learning dynamics are studied by inspecting
the mutual information (MI) between the hidden layers and the input and output.
Notably, separate fitting and compression phases during training have been
reported. This led to some controversy including claims that the observations
are not reproducible and strongly dependent on the type of activation function
used as well as on the way the MI is estimated. Our study confirms that
different ways of binning when computing the MI lead to qualitatively different
results, either supporting or refusing IB conjectures. To resolve the
controversy, we study the IB principle in settings where MI is non-trivial and
can be computed exactly. We monitor the dynamics of quantized neural networks,
that is, we discretize the whole deep learning system so that no approximation
is required when computing the MI. This allows us to quantify the information
flow without measurement errors. In this setting, we observed a fitting phase
for all layers and a compression phase for the output layer in all experiments;
the compression in the hidden layers was dependent on the type of activation
function. Our study shows that the initial IB results were not artifacts of
binning when computing the MI. However, the critical claim that the compression
phase may not be observed for some networks also holds true.
