Information Bottleneck Analysis of Deep Neural Networks via Lossy
  Compression
Ivan Butakov Aleksander Tolmachev Sofia Malanchuk Anna Neopryatnaya Alexey Frolov Kirill Andreev
The Information Bottleneck (IB) principle offers an information-theoretic
framework for analyzing the training process of deep neural networks (DNNs).
Its essence lies in tracking the dynamics of two mutual information (MI)
values: one between the hidden layer and the class label, and the other between
the hidden layer and the DNN input. According to the hypothesis put forth by
Shwartz-Ziv and Tishby (2017), the training process consists of two distinct
phases: fitting and compression. The latter phase is believed to account for
the good generalization performance exhibited by DNNs. Due to the challenging
nature of estimating MI between high-dimensional random vectors, this
hypothesis has only been verified for toy NNs or specific types of NNs, such as
quantized NNs and dropout NNs. In this paper, we introduce a comprehensive
framework for conducting IB analysis of general NNs. Our approach leverages the
stochastic NN method proposed by Goldfeld et al. (2019) and incorporates a
compression step to overcome the obstacles associated with high dimensionality.
In other words, we estimate the MI between the compressed representations of
high-dimensional random vectors. The proposed method is supported by both
theoretical and practical justifications. Notably, we demonstrate the accuracy
of our estimator through synthetic experiments featuring predefined MI values.
Finally, we perform IB analysis on a close-to-real-scale convolutional DNN,
which reveals new features of the MI dynamics.