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

arXiv: https://arxiv.org/abs/2105.13889 · HTML: https://mlsanity.com/paper/2105.13889/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 66 references that are in the map.
- First posted on arXiv 28 May 2021.

## Builds on (most cited first, 18 of 18)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model](https://mlsanity.com/paper/1904.09770/) (2019)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [Equivalence of restricted Boltzmann machines and tensor network states](https://mlsanity.com/paper/1701.04831/) (2017)
- [Thermodynamics of Restricted Boltzmann Machines and related learning dynamics](https://mlsanity.com/paper/1803.01960/) (2018)
- [Accurate and Conservative Estimates of MRF Log-likelihood using Reverse Annealing](https://mlsanity.com/paper/1412.8566/) (2014)
- [Restricted Boltzmann Machines: Introduction and Review](https://mlsanity.com/paper/1806.07066/) (2018)
- [Deep Learning the Ising Model Near Criticality](https://mlsanity.com/paper/1708.04622/) (2017)
- ['Place-cell' emergence and learning of invariant data with restricted Boltzmann machines: breaking and dynamical restoration of continuous symmetries in the weight space](https://mlsanity.com/paper/1912.12942/) (2019)
- [Sparse generative modeling via parameter-reduction of Boltzmann machines: application to protein-sequence families](https://mlsanity.com/paper/2011.11259/) (2020)
- [The Accuracy of Restricted Boltzmann Machine Models of Ising Systems](https://mlsanity.com/paper/2004.12867/) (2020)
- [Population-Contrastive-Divergence: Does Consistency help with RBM training?](https://mlsanity.com/paper/1510.01624/) (2015)
- [Weighted Contrastive Divergence](https://mlsanity.com/paper/1801.02567/) (2018)
- [Efficient Learning of Restricted Boltzmann Machines Using Covariance Estimates](https://mlsanity.com/paper/1810.10777/) (2018)

## Built on by (most cited first, 27 of 27)

- [Explaining the effects of non-convergent sampling in the training of Energy-Based Models](https://mlsanity.com/paper/2301.09428/) (2023)
- [Cascade of phase transitions in the training of Energy-based models](https://mlsanity.com/paper/2405.14689/) (2024)
- [Learning a Restricted Boltzmann Machine using biased Monte Carlo sampling](https://mlsanity.com/paper/2206.01310/) (2022)
- [Disentangling representations in Restricted Boltzmann Machines without adversaries](https://mlsanity.com/paper/2206.11600/) (2022)
- [Phase transitions in the mini-batch size for sparse and dense two-layer neural networks](https://mlsanity.com/paper/2305.06435/) (2023)
- [Fast training and sampling of Restricted Boltzmann Machines](https://mlsanity.com/paper/2405.15376/) (2024)
- [Inferring effective couplings with Restricted Boltzmann Machines](https://mlsanity.com/paper/2309.02292/) (2023)
- [An efficient probabilistic hardware architecture for diffusion-like models](https://mlsanity.com/paper/2510.23972/) (2025)
- [Fast and Functional Structured Data Generators Rooted in Out-of-Equilibrium Physics](https://mlsanity.com/paper/2307.06797/) (2023)
- [Inferring Higher-Order Couplings with Neural Networks](https://mlsanity.com/paper/2501.06108/) (2025)
- [Unsupervised hierarchical clustering using the learning dynamics of RBMs](https://mlsanity.com/paper/2302.01851/) (2023)
- [Conditioned quantum-assisted deep generative surrogate for particle-calorimeter interactions](https://mlsanity.com/paper/2410.22870/) (2024)
- [Interpretable machine learning of amino acid patterns in proteins: a statistical ensemble approach](https://mlsanity.com/paper/2303.15228/) (2023)
- [Fundamental operating regimes, hyper-parameter fine-tuning and glassiness: towards an interpretable replica-theory for trained restricted Boltzmann machines](https://mlsanity.com/paper/2406.09924/) (2024)
- [Neural Langevin Machine: a local asymmetric learning rule can be creative](https://mlsanity.com/paper/2506.23546/) (2025)
- [Three Learning Stages and Accuracy-Efficiency Tradeoff of Restricted Boltzmann Machines](https://mlsanity.com/paper/2209.00873/) (2022)
- [adabmDCA 2.0 -- a flexible but easy-to-use package for Direct Coupling Analysis](https://mlsanity.com/paper/2501.18456/) (2025)
- [A simple probabilistic neural network for machine understanding](https://mlsanity.com/paper/2210.13179/) (2022)
- [Generative modeling through internal high-dimensional chaotic activity](https://mlsanity.com/paper/2405.10822/) (2024)
- [On the role of non-linear latent features in bipartite generative neural networks](https://mlsanity.com/paper/2506.10552/) (2025)
- [Exploration of the generative capabilities of Boltzmann machines applied to social systems under the majority rule](https://mlsanity.com/paper/2607.23349/) (2026)
- [EBM Life Cycle: MCMC Strategies for Synthesis, Defense, and Density Modeling](https://mlsanity.com/paper/2205.12243/) (2022)
- [Absolute abstraction: a renormalisation group approach](https://mlsanity.com/paper/2407.01656/) (2024)
- [Exploring the Energy Landscape of RBMs: Reciprocal Space Insights into Bosons, Hierarchical Learning and Symmetry Breaking](https://mlsanity.com/paper/2503.21536/) (2025)
- [Diluting Restricted Boltzmann Machines](https://mlsanity.com/paper/2511.00648/) (2025)
- [Emergence of Nonequilibrium Latent Cycles in Unsupervised Generative Modeling](https://mlsanity.com/paper/2512.11415/) (2025)
- [Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering](https://mlsanity.com/paper/2607.27077/) (2026)

## Abstract (arXiv)

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 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.