Towards quantifying information flows: relative entropy in deep neural networks and the renormalization group

TL;DR

This paper explores the analogy between the renormalization group and deep neural networks by analyzing information flow through relative entropy, revealing similar monotonic behaviors and implications for machine learning and physics.

Contribution

It introduces a quantitative approach to compare RG and neural networks using relative entropy, linking information flow to physical and machine learning systems.

Findings

Relative entropy increases monotonically in both RG and neural networks.

The behavior supports the connection between relative entropy and the c-theorem.

Relative entropy is insensitive to phase transitions in the studied models.

Abstract

We investigate the analogy between the renormalization group (RG) and deep neural networks, wherein subsequent layers of neurons are analogous to successive steps along the RG. In particular, we quantify the flow of information by explicitly computing the relative entropy or Kullback-Leibler divergence in both the one- and two-dimensional Ising models under decimation RG, as well as in a feedforward neural network as a function of depth. We observe qualitatively identical behavior characterized by the monotonic increase to a parameter-dependent asymptotic value. On the quantum field theory side, the monotonic increase confirms the connection between the relative entropy and the c-theorem. For the neural networks, the asymptotic behavior may have implications for various information maximization methods in machine learning, as well as for disentangling compactness and generalizability. Furthermore, while both the two-dimensional Ising model and the random neural networks we consider exhibit non-trivial critical points, the relative entropy appears insensitive to the phase structure of either system. In this sense, more refined probes are required in order to fully elucidate the flow of information in these models.