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

Aurélien Decelle, Giulio Biroli, Beatriz Seoane, Dimitrios Bachtis

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.