Liquid and solid layers in a thermal deep learning machine

TL;DR

This paper introduces a thermal deep learning machine model inspired by statistical mechanics, revealing a phase structure of liquid and solid layers in neural networks that explains generalization and parameter constraints.

Contribution

It establishes a physical model of deep learning with a Hamiltonian, demonstrating phase transitions and layer-specific dynamics through numerical experiments.

Findings

Identification of liquid and solid phases in neural network parameters

Layer-specific dynamics show hierarchical and structureless behavior

Phase diagram correlates network depth with phase states

Abstract

Based on deep neural networks (DNNs), deep learning has been successfully applied to many problems, but its mechanism is still not well understood -- especially the reason why over-parametrized DNNs can generalize. A recent statistical mechanics theory on supervised learning by a prototypical multi-layer perceptron (MLP) on some artificial learning scenarios predicts that adjustable parameters of over-parametrized MLPs become strongly constrained by the training data close to the input/output boundaries, while the parameters in the center remain largely free, giving rise to a solid-liquid-solid structure. Here we establish this picture, through numerical experiments on benchmark real-world data using a thermal deep learning machine that explores the phase space of the synaptic weights and neurons. The supervised training is implemented by a GPU-accelerated molecular dynamics algorithm, which operates at very low temperatures, and the trained machine exhibits good generalization ability in the test. Global and layer-specific dynamics, with complex non-equilibrium aging behavior, are characterized by time-dependent auto-correlation and replica-correlation functions. Our analyses reveal that the design space of the parameters in the liquid and solid layers are respectively structureless and hierarchical. Our main results are summarized by a data storage ratio -- network depth phase diagram with liquid and solid phases. The proposed thermal machine, which is a physical model with a well-defined Hamiltonian, that reduces to MLP in the zero-temperature limit, can serve as a starting point for physically interpretable deep learning.