Universal Scaling Laws of Absorbing Phase Transitions in Artificial Deep Neural Networks

TL;DR

This paper reveals that deep neural networks exhibit universal scaling laws akin to phase transitions in physics, providing new insights into their critical behavior and implications for optimal training.

Contribution

It establishes a connection between neural network dynamics and non-equilibrium phase transitions, classifies networks into universality classes, and links criticality to generalization performance.

Findings

Neural networks operate near phase boundaries exhibiting universal scaling laws.

Multilayer perceptrons and convolutional networks belong to mean-field and directed percolation classes.

Hyperparameter tuning at the phase boundary influences generalization.

Abstract

We demonstrate that conventional artificial deep neural networks operating near the phase boundary of the signal propagation dynamics, also known as the edge of chaos, exhibit universal scaling laws of absorbing phase transitions in non-equilibrium statistical mechanics. We exploit the fully deterministic nature of the propagation dynamics to elucidate an analogy between a signal collapse in the neural networks and an absorbing state (a state that the system can enter but cannot escape from). Our numerical results indicate that the multilayer perceptrons and the convolutional neural networks belong to the mean-field and the directed percolation universality classes, respectively. Also, the finite-size scaling is successfully applied, suggesting a potential connection to the depth-width trade-off in deep learning. Furthermore, our analysis of the training dynamics under the gradient descent reveals that hyperparameter tuning to the phase boundary is necessary but insufficient for achieving optimal generalization in deep networks. Remarkably, nonuniversal metric factors associated with the scaling laws are shown to play a significant role in concretizing the above observations. These findings highlight the usefulness of the notion of criticality for analyzing the behavior of artificial deep neural networks and offer new insights toward a unified understanding of the essential relationship between criticality and intelligence.