Cascade of phase transitions in the training of Energy-based models

TL;DR

This paper explores the training dynamics of energy-based models, specifically RBMs, revealing a cascade of phase transitions during learning that correspond to the progressive encoding of data modes, validated through analytical and empirical methods.

Contribution

The study uncovers a series of phase transitions in RBM training, linking them to the learning of principal data modes, and validates the findings with real dataset experiments and a mean-field scaling hypothesis.

Findings

Identifies a cascade of phase transitions during RBM training.

Shows phase transitions correspond to learning principal data modes.

Validates analytical results with experiments on real datasets.

Abstract

In this paper, we investigate the feature encoding process in a prototypical energy-based generative model, the Restricted Boltzmann Machine (RBM). We start with an analytical investigation using simplified architectures and data structures, and end with numerical analysis of real trainings on real datasets. Our study tracks the evolution of the model's weight matrix through its singular value decomposition, revealing a series of phase transitions associated to a progressive learning of the principal modes of the empirical probability distribution. The model first learns the center of mass of the modes and then progressively resolve all modes through a cascade of phase transitions. We first describe this process analytically in a controlled setup that allows us to study analytically the training dynamics. We then validate our theoretical results by training the Bernoulli-Bernoulli RBM on real data sets. By using data sets of increasing dimension, we show that learning indeed leads to sharp phase transitions in the high-dimensional limit. Moreover, we propose and test a mean-field finite-size scaling hypothesis. This shows that the first phase transition is in the same universality class of the one we studied analytically, and which is reminiscent of the mean-field paramagnetic-to-ferromagnetic phase transition.