Learning Representations for Neural Network-Based Classification Using the Information Bottleneck Principle

TL;DR

This paper critically examines the application of the information bottleneck principle to neural network training, revealing fundamental issues with deterministic models and suggesting solutions involving stochasticity or alternative cost functions.

Contribution

The paper identifies key limitations of the IB functional in deterministic DNNs and proposes that stochastic models or modified cost functions can overcome these issues.

Findings

IB functional is often ill-posed or non-differentiable for deterministic DNNs

Invariance of IB functional limits its ability to capture robustness and simplicity

Solutions include stochastic DNNs or alternative cost functions

Abstract

In this theory paper, we investigate training deep neural networks (DNNs) for classification via minimizing the information bottleneck (IB) functional. We show that the resulting optimization problem suffers from two severe issues: First, for deterministic DNNs, either the IB functional is infinite for almost all values of network parameters, making the optimization problem ill-posed, or it is piecewise constant, hence not admitting gradient-based optimization methods. Second, the invariance of the IB functional under bijections prevents it from capturing properties of the learned representation that are desirable for classification, such as robustness and simplicity. 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 well-behaved 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 indicate limitations of the IB framework for the analysis of DNNs. We also note that rather than trying to repair the inherent problems in the IB functional, a better approach may be to design regularizers on latent representation enforcing the desired properties directly.