The Dual Information Bottleneck
Authors: Zoe Piran, Naftali Tishby
Keywords: optimal prediction learning, exponential families, critical points, information theory
TL;DR: A new dual formulation of the Information Bottleneck, optimizing label prediction and preserving distributions of exponential form.
Abstract: The Information-Bottleneck (IB) framework suggests a general characterization of optimal representations in learning, and deep learning in particular. It is based on the optimal trade off between the representation complexity and accuracy, both of which are quantified by mutual information. The problem is solved by alternating projections between the encoder and decoder of the representation, which can be performed locally at each representation level. The framework, however, has practical drawbacks, in that mutual information is notoriously difficult to handle at high dimension, and only has closed form solutions in special cases. Further, because it aims to extract representations which are minimal sufficient statistics of the data with respect to the desired label, it does not necessarily optimize the actual prediction of unseen labels. Here we present a  formal dual problem to the IB which has several interesting properties. By switching the order in the KL-divergence between the representation decoder and data, the optimal decoder becomes the geometric rather than the arithmetic mean of the input points. While providing a good approximation to the original IB, it also preserves the form of exponential families, and optimizes the mutual information on the predicted label rather than the desired one. We also analyze the critical points of the dualIB and discuss their importance for the quality of this approach.