Maximum Multiscale Entropy and Neural Network Regularization

TL;DR

This paper extends the maximum entropy principle to multiscale settings, linking it to renormalization concepts and demonstrating improved neural network regularization in teacher-student models.

Contribution

It introduces a multiscale entropy framework, generalizes maximum entropy distributions to multiple scales, and applies these ideas to enhance neural network regularization.

Findings

Multiscale entropy maximization relates to renormalization group procedures.

Gaussian distributions are preserved under decimation transformations.

Multiscale Gibbs posteriors outperform single-scale ones in neural network scenarios.

Abstract

A well-known result across information theory, machine learning, and statistical physics shows that the maximum entropy distribution under a mean constraint has an exponential form called the Gibbs-Boltzmann distribution. This is used for instance in density estimation or to achieve excess risk bounds derived from single-scale entropy regularizers (Xu-Raginsky '17). This paper investigates a generalization of these results to a multiscale setting. We present different ways of generalizing the maximum entropy result by incorporating the notion of scale. For different entropies and arbitrary scale transformations, it is shown that the distribution maximizing a multiscale entropy is characterized by a procedure which has an analogy to the renormalization group procedure in statistical physics. For the case of decimation transformation, it is further shown that this distribution is Gaussian whenever the optimal single-scale distribution is Gaussian. This is then applied to neural networks, and it is shown that in a teacher-student scenario, the multiscale Gibbs posterior can achieve a smaller excess risk than the single-scale Gibbs posterior.