Learning a Restricted Boltzmann Machine using biased Monte Carlo sampling

Restricted Boltzmann Machines are simple and powerful generative models capable of encoding any complex dataset. Despite all their advantages, in practice, trainings are often unstable, and it is hard to assess their quality because dynamics are hampered by extremely slow time-dependencies. This situation becomes critical when dealing with low-dimensional clustered datasets, where the time needed to sample ergodically the trained models becomes computationally prohibitive. In this work, we show that this divergence of Monte Carlo mixing times is related to a phase coexistence phenomenon, similar to that encountered in Physics in the vicinity of a first order phase transition. We show that sampling the equilibrium distribution via Markov Chain Monte Carlo can be dramatically accelerated using biased sampling techniques, in particular, the Tethered Monte Carlo method (TMC). This sampling technique solves efficiently the problem of evaluating the quality of a given trained model and the generation of new samples in reasonable times. In addition, we show that this sampling technique can be exploited to improve the computation of the log-likelihood gradient during the training too, which produces dramatic improvements when training RBMs with artificial clustered datasets. When dealing with real low-dimensional datasets, this new training procedure fits RBM models with significantly faster relaxational dynamics than those obtained with standard PCD recipes. We also show that TMC sampling can be used to recover free-energy profile of the RBM, which turns out to be extremely useful to compute the probability distribution of a given model and to improve the generation of new decorrelated samples on slow PCD trained models.