Equilibrium and non-equilibrium regimes in the learning of restricted Boltzmann machines* *This article is an updated version of: Decelle A, Furtlehner C and Seoane B 2021 Equilibrium and non-equilibrium regimes in the learning of restricted Boltzmann machines Advances in Neural Information Processing Systems vol 34 ed M Ranzato, A Beygelzimer, Y Dauphin, P S Liang and J Wortman Vaughan (New

Abstract

Training restricted Boltzmann machines (RBMs) have been challenging for a long time due to the difficulty of precisely computing the log-likelihood gradient. Over the past few decades, many works have proposed more or less successful training recipes but without studying the crucial quantity of the problem: the mixing time, i.e. the number of Monte Carlo iterations needed to sample new configurations from a model. 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. Finally, we discuss how to exploit in practice both regimes depending on the task one aims to fulfill: (i) short k can be used to generate convincing samples in short learning times, (ii) large k (or increasingly large) is needed to learn the correct equilibrium distribution of the RBM. Finally, the existence of these two operational regimes seems to be a general property of energy based models trained via likelihood maximization.