Information bottleneck (IB) is a method for extracting information from one random variable X that is relevant for predicting another random variable Y. To do so, IB identifies an intermediate "bottleneck" variable T that has low mutual information I(X;T) and high mutual information I(Y;T). The "IB curve" characterizes the set of bottleneck variables that achieve maximal I(Y;T) for a given I(X;T), and is typically explored by maximizing the "IB Lagrangian", I(Y;T) - βI(X;T). In some cases, Y is a deterministic function of X, including many classification problems in supervised learning where the output class Y is a deterministic function of the input X. We demonstrate three caveats when using IB in any situation where Y is a deterministic function of X: (1) the IB curve cannot be recovered by maximizing the IB Lagrangian for different values of β; (2) there are "uninteresting" trivial solutions at all points of the IB curve; and (3) for multi-layer classifiers that achieve low prediction error, different layers cannot exhibit a strict trade-off between compression and prediction, contrary to a recent proposal. We also show that when Y is a small perturbation away from being a deterministic function of X, these three caveats arise in an approximate way. To address problem (1), we propose a functional that, unlike the IB Lagrangian, can recover the IB curve in all cases. We demonstrate the three caveats on the MNIST dataset.
Information bottleneck behaves in surprising ways whenever the output is a deterministic function of the input. Argues that most real classification problems show such a deterministic relation between the class labels and the inputs X and explores several issues that result from such pathologies. Explores issues that arise when applying information bottlenext concepts to deterministic supervised learning models The authors clarify several counter-intuitive behaviors of the information bottleneck method for supervised learning of a deterministic rule.