A Theory of Usable Information Under Computational Constraints

TL;DR

This paper introduces a variational extension of Shannon's information theory called predictive -information that accounts for computational constraints, enabling better analysis of complex systems and neural networks.

Contribution

It develops a new framework for information that incorporates modeling power and computation, extending mutual information and enabling reliable high-dimensional estimation.

Findings

Predictive -information outperforms mutual information in structure learning.

It can be estimated reliably from high-dimensional data with PAC guarantees.

Demonstrates effectiveness in fair representation learning.

Abstract

We propose a new framework for reasoning about information in complex systems. Our foundation is based on a variational extension of Shannon's information theory that takes into account the modeling power and computational constraints of the observer. The resulting \emph{predictive $\mathcal{V}$-information} encompasses mutual information and other notions of informativeness such as the coefficient of determination. Unlike Shannon's mutual information and in violation of the data processing inequality, $\mathcal{V}$-information can be created through computation. This is consistent with deep neural networks extracting hierarchies of progressively more informative features in representation learning. Additionally, we show that by incorporating computational constraints, $\mathcal{V}$-information can be reliably estimated from data even in high dimensions with PAC-style guarantees. Empirically, we demonstrate predictive $\mathcal{V}$-information is more effective than mutual information for structure learning and fair representation learning.