Cauchy-Schwarz Divergence Information Bottleneck for Regression

TL;DR

This paper introduces a novel Cauchy-Schwarz divergence-based information bottleneck method for regression, which improves estimation, generalization, and robustness without relying on variational inference or distributional assumptions.

Contribution

It develops a new IB framework for regression using Cauchy-Schwarz divergence, avoiding variational approximations and enhancing robustness and performance.

Findings

Outperforms existing deep IB methods on six real-world regression tasks.

Achieves the best trade-off between prediction accuracy and compression.

Provides strong adversarial robustness guarantees.

Abstract

The information bottleneck (IB) approach is popular to improve the generalization, robustness and explainability of deep neural networks. Essentially, it aims to find a minimum sufficient representation $\mathbf{t}$ by striking a trade-off between a compression term $I(\mathbf{x};\mathbf{t})$ and a prediction term $I(y;\mathbf{t})$, where $I(\cdot;\cdot)$ refers to the mutual information (MI). MI is for the IB for the most part expressed in terms of the Kullback-Leibler (KL) divergence, which in the regression case corresponds to prediction based on mean squared error (MSE) loss with Gaussian assumption and compression approximated by variational inference. In this paper, we study the IB principle for the regression problem and develop a new way to parameterize the IB with deep neural networks by exploiting favorable properties of the Cauchy-Schwarz (CS) divergence. By doing so, we move away from MSE-based regression and ease estimation by avoiding variational approximations or distributional assumptions. We investigate the improved generalization ability of our proposed CS-IB and demonstrate strong adversarial robustness guarantees. We demonstrate its superior performance on six real-world regression tasks over other popular deep IB approaches. We additionally observe that the solutions discovered by CS-IB always achieve the best trade-off between prediction accuracy and compression ratio in the information plane. The code is available at \url{https://github.com/SJYuCNEL/Cauchy-Schwarz-Information-Bottleneck}.