Information Bottleneck Analysis of Deep Neural Networks via Lossy Compression

TL;DR

This paper introduces a new framework for analyzing deep neural networks using the Information Bottleneck principle, overcoming previous estimation challenges and revealing new insights into the MI dynamics of large-scale models.

Contribution

We develop a novel MI estimation method based on lossy compression and stochastic neural networks, enabling IB analysis of complex, real-world DNNs.

Findings

The proposed estimator accurately measures mutual information in synthetic experiments.

IB analysis reveals new MI dynamics features in convolutional neural networks.

The method overcomes high-dimensional MI estimation challenges in large-scale DNNs.

Abstract

The Information Bottleneck (IB) principle offers an information-theoretic framework for analyzing the training process of deep neural networks (DNNs). Its essence lies in tracking the dynamics of two mutual information (MI) values: between the hidden layer output and the DNN input/target. According to the hypothesis put forth by Shwartz-Ziv & Tishby (2017), the training process consists of two distinct phases: fitting and compression. The latter phase is believed to account for the good generalization performance exhibited by DNNs. Due to the challenging nature of estimating MI between high-dimensional random vectors, this hypothesis was only partially verified for NNs of tiny sizes or specific types, such as quantized NNs. In this paper, we introduce a framework for conducting IB analysis of general NNs. Our approach leverages the stochastic NN method proposed by Goldfeld et al. (2019) and incorporates a compression step to overcome the obstacles associated with high dimensionality. In other words, we estimate the MI between the compressed representations of high-dimensional random vectors. The proposed method is supported by both theoretical and practical justifications. Notably, we demonstrate the accuracy of our estimator through synthetic experiments featuring predefined MI values and comparison with MINE (Belghazi et al., 2018). Finally, we perform IB analysis on a close-to-real-scale convolutional DNN, which reveals new features of the MI dynamics.