How (Not) To Train Your Neural Network Using the Information Bottleneck Principle

In this theory paper, we investigate training deep neural networks (DNNs) for classification via minimizing the information bottleneck (IB) functional. We show that, even if the joint distribution between continuous feature variables and the discrete class variable is known, the resulting optimization problem suffers from two severe issues: First, for deterministic DNNs, the IB functional is infinite for almost all weight matrices, making the optimization problem ill-posed. Second, the invariance of the IB functional under bijections prevents it from capturing desirable properties for classification, such as robustness, architectural simplicity, and simplicity of the learned representation. We argue that these issues are partly resolved for stochastic DNNs, DNNs that include a (hard or soft) decision rule, or by replacing the IB functional with related, but more wellbehaved cost functions. We conclude that recent successes reported about training DNNs using the IB framework must be attributed to such solutions. As a side effect, our results imply limitations of the IB framework for the analysis of DNNs.