Accelerated Sampling with Stacked Restricted Boltzmann Machines

Sampling complex distributions is an important but difficult objective in various fields, including physics, chemistry, and statistics. An improvement of standard Monte Carlo (MC) methods, intensively used in particular in the context of disordered systems, is Parallel Tempering, also called replica exchange MC, in which a sequence of MC Markov chains at decreasing temperatures are run in parallel and can swap their configurations. In this work we apply the ideas of parallel tempering in the context of restricted Boltzmann machines (RBM), a paradigm of unsupervised architectures, capable to learn complex, multimodal distributions. Inspired by Deep Tempering, an approach introduced for deep belief networks, we show how to learn on top of the first RBM a stack of nested RBMs, using the representations of a RBM as ’data’ for the next one along the stack. In our Stacked Tempering approach the hidden configurations of a machine can be exchanged with the visible configurations of the next one in the stack. Replica exchanges between the different RBMs is facilitated by the increasingly clustered representations learnt by deeper RBMs, allowing for fast transitions between the different modes of the data distribution. Analytical calculations of mixing times in a simplified theoretical setting shed light on why Stacked Tempering works, and how hyperparameters, such as the aspect ratios of the RBMs and weight regularization should be chosen. We illustrate the efficiency of the Stacked Tempering method with respect to standard and replica exchange MC on several datasets: MNIST, in-silico Lattice Proteins, and the 2D-Ising model.