Equilibrium and non-Equilibrium regimes in the learning of Restricted Boltzmann Machines

Training Restricted Boltzmann Machines (RBMs) has been challenging for a long

time due to the difficulty of computing precisely the log-likelihood gradient. Over

the past decades, many works have proposed more or less successful training

recipes but without studying the crucial quantity of the problem: the mixing time.

In this work, we show that this mixing time plays a crucial role

in the dynamics and stability of the trained model, and that RBMs operate in two

well-defined regimes, namely equilibrium and out-of-equilibrium, depending on

the interplay between this mixing time of the model and the number of steps, k,

used to approximate the gradient. We further show empirically that this mixing

time increases with the learning, which often implies a transition from one regime

to another as soon as k becomes smaller than this time. In particular, we show that

using the popular k (persistent) contrastive divergence approaches, with k small,

the dynamics of the learned model are extremely slow and often dominated by

strong out-of-equilibrium effects. On the contrary, RBMs trained in equilibrium

display faster dynamics, and a smooth convergence to dataset-like configurations

during the sampling.