Fast, accurate training and sampling of Restricted Boltzmann Machines

Thanks to their simple architecture, Restricted Boltzmann Machines (RBMs) are powerful tools for modeling complex systems and extracting interpretable insights from data. However, training RBMs, as other energy-based models, on highly structured data poses a major challenge, as effective training relies on mixing the Markov chain Monte Carlo simulations used to estimate the gradient. This process is often hindered by multiple second-order phase transitions and the associated critical slowdown. In this paper, we present an innovative method in which the principal directions of the dataset are integrated into a low-rank RBM through a convex optimization procedure. This approach enables efficient sampling of the equilibrium measure via a static Monte Carlo process. By starting the standard training process with a model that already accurately represents the main modes of the data, we bypass the initial phase transitions. Our results show that this strategy successfully trains RBMs to capture the full diversity of data in datasets where previous methods fail. Furthermore, we use the training trajectories to propose a new sampling method, {\em parallel trajectory tempering}, which allows us to sample the equilibrium measure of the trained model much faster than previous optimized MCMC approaches and a better estimation of the log-likelihood. We illustrate the success of the training method on several highly structured datasets.

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

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.

Inferring effective couplings with Restricted Boltzmann Machines

Generative models offer a direct way to model complex data. Among them, energy-based models provide us with a neural network model that aims to accurately reproduce all statistical correlations observed in the data at the level of the Boltzmann weight of the model. However, one challenge is to understand the physical interpretation of such models. In this study, we propose a simple solution by implementing a direct mapping between the energy function of the Restricted Boltzmann Machine and an effective Ising spin Hamiltonian that includes high-order interactions between spins. This mapping includes interactions of all possible orders, going beyond the conventional pairwise interactions typically considered in the inverse Ising approach, and allowing the description of complex datasets. Earlier works attempted to achieve this goal, but the proposed mappings did not do properly treat the complexity of the problem or did not contain direct prescriptions for practical application. To validate our method, we performed several controlled numerical experiments where we trained the RBMs using equilibrium samples of predefined models containing local external fields, two-body and three-body interactions in various low-dimensional topologies. The results demonstrate the effectiveness of our proposed approach in learning the correct interaction network and pave the way for its application in modeling interesting datasets. We also evaluate the quality of the inferred model based on different training methods.

The Copycat Perceptron: Smashing Barriers Through Collective Learning

We characterize the equilibrium properties of a model of $y$ coupled binary perceptrons in the teacher-student scenario, subject to a suitable learning rule, with an explicit ferromagnetic coupling proportional to the Hamming distance between the students' weights. In contrast to recent works, we analyze a more general setting in which a thermal noise is present that affects the generalization performance of each student. Specifically, in the presence of a nonzero temperature, which assigns nonzero probability to configurations that misclassify samples with respect to the teacher's prescription, we find that the coupling of replicas leads to a shift of the phase diagram to smaller values of $\alpha$: This suggests that the free energy landscape gets smoother around the solution with good generalization (i.e., the teacher) at a fixed fraction of reviewed examples, which allows local update algorithms such as Simulated Annealing to reach the solution before the dynamics gets frozen. Finally, from a learning perspective, these results suggest that more students (in this case, with the same amount of data) are able to learn the same rule when coupled together with a smaller amount of data.

Fast and Functional Structured Data Generators Rooted in Out-of-Equilibrium Physics

In this study, we address the challenge of using energy-based models to produce high-quality, label-specific data in complex structured datasets, such as population genetics, RNA or protein sequences data. Traditional training methods encounter difficulties due to inefficient Markov chain Monte Carlo mixing, which affects the diversity of synthetic data and increases generation times. To address these issues, we use a novel training algorithm that exploits non-equilibrium effects. This approach, applied on the Restricted Boltzmann Machine, improves the model's ability to correctly classify samples and generate high-quality synthetic data in only a few sampling steps. The effectiveness of this method is demonstrated by its successful application to four different types of data: handwritten digits, mutations of human genomes classified by continental origin, functionally characterized sequences of an enzyme protein family, and homologous RNA sequences from specific taxonomies.

Unsupervised hierarchical clustering using the learning dynamics of RBMs

Datasets in the real world are often complex and to some degree hierarchical, with groups and sub-groups of data sharing common characteristics at different levels of abstraction. Understanding and uncovering the hidden structure of these datasets is an important task that has many practical applications. To address this challenge, we present a new and general method for building relational data trees by exploiting the learning dynamics of the Restricted Boltzmann Machine (RBM). Our method is based on the mean-field approach, derived from the Plefka expansion, and developed in the context of disordered systems. It is designed to be easily interpretable. We tested our method in an artificially created hierarchical dataset and on three different real-world datasets (images of digits, mutations in the human genome, and a homologous family of proteins). The method is able to automatically identify the hierarchical structure of the data. This could be useful in the study of homologous protein sequences, where the relationships between proteins are critical for understanding their function and evolution.

Explaining the effects of non-convergent sampling in the training of Energy-Based Models

In this paper, we quantify the impact of using non-convergent Markov chains to train Energy-Based models (EBMs). In particular, we show analytically that EBMs trained with non-persistent short runs to estimate the gradient can perfectly reproduce a set of empirical statistics of the data, not at the level of the equilibrium measure, but through a precise dynamical process. Our results provide a first-principles explanation for the observations of recent works proposing the strategy of using short runs starting from random initial conditions as an efficient way to generate high-quality samples in EBMs, and lay the groundwork for using EBMs as diffusion models. After explaining this effect in generic EBMs, we analyze two solvable models in which the effect of the non-convergent sampling in the trained parameters can be described in detail. Finally, we test these predictions numerically on the Boltzmann machine.

Thermodynamics of bidirectional associative memories

In this paper we investigate the equilibrium properties of bidirectional associative memories (BAMs). Introduced by Kosko in 1988 as a generalization of the Hopfield model to a bipartite structure, the simplest architecture is defined by two layers of neurons, with synaptic connections only between units of different layers: even without internal connections within each layer, information storage and retrieval are still possible through the reverberation of neural activities passing from one layer to another. We characterize the computational capabilities of a stochastic extension of this model in the thermodynamic limit, by applying rigorous techniques from statistical physics. A detailed picture of the phase diagram at the replica symmetric level is provided, both at finite temperature and in the noiseless regime. An analytical and numerical inspection of the transition curves (namely critical lines splitting the various modes of operation of the machine) is carried out as the control parameters - noise, load and asymmetry between the two layer sizes - are tuned. In particular, with a finite asymmetry between the two layers, it is shown how the BAM can store information more efficiently than the Hopfield model by requiring less parameters to encode a fixed number of patterns. Comparisons are made with numerical simulations of neural dynamics. Finally, a low-load analysis is carried out to explain the retrieval mechanism in the BAM by analogy with two interacting Hopfield models. A potential equivalence with two coupled Restricted Boltmzann Machines is also discussed.

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.

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 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 $k$s can be used to generate convincing samples in short times, (ii) large $k$ (or increasingly large) must be used to learn the correct equilibrium distribution of the RBM.