On the Difference Between the Information Bottleneck and the Deep Information Bottleneck

TL;DR

This paper analyzes the assumptions behind the Deep Variational Information Bottleneck, proposes a way to relax these assumptions, and interprets information bottleneck models within a unified graphical framework.

Contribution

It introduces a method to optimize a lower bound for mutual information that relaxes previous assumptions and unifies different IB models under a common graphical interpretation.

Findings

Requiring both Markov chains limits potential distributions.

A lower bound for I(T;Y) can be optimized with only one Markov chain.

Original and deep IB models are special cases of a fundamental IB model.

Abstract

Combining the Information Bottleneck model with deep learning by replacing mutual information terms with deep neural nets has proved successful in areas ranging from generative modelling to interpreting deep neural networks. In this paper, we revisit the Deep Variational Information Bottleneck and the assumptions needed for its derivation. The two assumed properties of the data $X$, $Y$ and their latent representation $T$ take the form of two Markov chains $T-X-Y$ and $X-T-Y$. Requiring both to hold during the optimisation process can be limiting for the set of potential joint distributions $P(X,Y,T)$. We therefore show how to circumvent this limitation by optimising a lower bound for $I(T;Y)$ for which only the latter Markov chain has to be satisfied. The actual mutual information consists of the lower bound which is optimised in DVIB and cognate models in practice and of two terms measuring how much the former requirement $T-X-Y$ is violated. Finally, we propose to interpret the family of information bottleneck models as directed graphical models and show that in this framework the original and deep information bottlenecks are special cases of a fundamental IB model.