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

arXiv: https://arxiv.org/abs/2301.09428 · HTML: https://mlsanity.com/paper/2301.09428/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 53 references that are in the map.
- First posted on arXiv 23 Jan 2023.

## Builds on (most cited first, 20 of 20)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (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)
- [Maximum Entropy Generators for Energy-Based Models](https://mlsanity.com/paper/1901.08508/) (2019)
- [A Tale of Two Flows: Cooperative Learning of Langevin Flow and Normalizing Flow Toward Energy-Based Model](https://mlsanity.com/paper/2205.06924/) (2022)
- [Inverse statistical problems: from the inverse Ising problem to data science](https://mlsanity.com/paper/1702.01522/) (2017)
- [Deep Directed Generative Models with Energy-Based Probability Estimation](https://mlsanity.com/paper/1606.03439/) (2016)
- [Restricted Boltzmann Machine, recent advances and mean-field theory](https://mlsanity.com/paper/2011.11307/) (2020)
- [Equilibrium and non-Equilibrium regimes in the learning of Restricted Boltzmann Machines](https://mlsanity.com/paper/2105.13889/) (2021)
- [MCMC Should Mix: Learning Energy-Based Model with Neural Transport Latent Space MCMC](https://mlsanity.com/paper/2006.06897/) (2020)
- [Thermodynamics of Restricted Boltzmann Machines and related learning dynamics](https://mlsanity.com/paper/1803.01960/) (2018)
- [Gaussian-Bernoulli RBMs Without Tears](https://mlsanity.com/paper/2210.10318/) (2022)
- [Learning a Restricted Boltzmann Machine using biased Monte Carlo sampling](https://mlsanity.com/paper/2206.01310/) (2022)
- [Learning FRAME Models Using CNN Filters](https://mlsanity.com/paper/1509.08379/) (2015)
- [Robust Multi-Output Learning with Highly Incomplete Data via Restricted Boltzmann Machines](https://mlsanity.com/paper/1912.09382/) (2019)
- [Layerwise Systematic Scan: Deep Boltzmann Machines and Beyond](https://mlsanity.com/paper/1705.05154/) (2017)

## Built on by (most cited first, 12 of 12)

- [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)
- [Fast and Functional Structured Data Generators Rooted in Out-of-Equilibrium Physics](https://mlsanity.com/paper/2307.06797/) (2023)
- [Unsupervised hierarchical clustering using the learning dynamics of RBMs](https://mlsanity.com/paper/2302.01851/) (2023)
- [Parallel Learning by Multitasking Neural Networks](https://mlsanity.com/paper/2308.04106/) (2023)
- [Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching](https://mlsanity.com/paper/2504.16262/) (2025)
- [Pseudo-likelihood produces associative memories able to generalize, even for asymmetric couplings](https://mlsanity.com/paper/2507.05147/) (2025)
- [Absolute abstraction: a renormalisation group approach](https://mlsanity.com/paper/2407.01656/) (2024)
- [Emergence of Nonequilibrium Latent Cycles in Unsupervised Generative Modeling](https://mlsanity.com/paper/2512.11415/) (2025)
- [Distributional simplicity bias and effective convexity in Energy Based Models](https://mlsanity.com/paper/2605.07844/) (2026)
- [Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering](https://mlsanity.com/paper/2607.27077/) (2026)
- [Learning Graph Energies for Finite-Time Generation](https://mlsanity.com/paper/2610.19946/) (2026)

## Abstract (arXiv)

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 a ConvNet EBM and a Boltzmann machine.