# How to Train Your Energy-Based Models

arXiv: https://arxiv.org/abs/2101.03288 · HTML: https://mlsanity.com/paper/2101.03288/

## Key facts

- Cited by 274 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 85 references that are in the map.
- First posted on arXiv 9 Jan 2021.

## Builds on (most cited first, 30 of 43)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One](https://mlsanity.com/paper/1912.03263/) (2019)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Learning Gradient Fields for Shape Generation](https://mlsanity.com/paper/2008.06520/) (2020)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [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)
- [Improved Contrastive Divergence Training of Energy Based Models](https://mlsanity.com/paper/2012.01316/) (2020)
- [Residual Energy-Based Models for Text Generation](https://mlsanity.com/paper/2004.11714/) (2020)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Flow Contrastive Estimation of Energy-Based Models](https://mlsanity.com/paper/1912.00589/) (2019)

## Built on by (most cited first, 30 of 274)

- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Implicit Behavioral Cloning](https://mlsanity.com/paper/2109.00137/) (2021)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Pre-training Molecular Graph Representation with 3D Geometry](https://mlsanity.com/paper/2110.07728/) (2021)
- [A Survey on Generative Diffusion Model](https://mlsanity.com/paper/2209.02646/) (2022)
- [Restoring Vision in Adverse Weather Conditions with Patch-Based Denoising Diffusion Models](https://mlsanity.com/paper/2207.14626/) (2022)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Recursive Introspection: Teaching Language Model Agents How to Self-Improve](https://mlsanity.com/paper/2407.18219/) (2024)
- [A Survey of Imitation Learning: Algorithms, Recent Developments, and Challenges](https://mlsanity.com/paper/2309.02473/) (2023)
- [Learning to Efficiently Sample from Diffusion Probabilistic Models](https://mlsanity.com/paper/2106.03802/) (2021)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [Generative Flow Networks for Discrete Probabilistic Modeling](https://mlsanity.com/paper/2202.01361/) (2022)
- [GraphEBM: Molecular Graph Generation with Energy-Based Models](https://mlsanity.com/paper/2102.00546/) (2021)
- [A Complete Survey on Generative AI (AIGC): Is ChatGPT from GPT-4 to GPT-5 All You Need?](https://mlsanity.com/paper/2303.11717/) (2023)
- [The Ingredients for Robotic Diffusion Transformers](https://mlsanity.com/paper/2410.10088/) (2024)
- [Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamics](https://mlsanity.com/paper/2302.00600/) (2023)
- [To Compress or Not to Compress- Self-Supervised Learning and Information Theory: A Review](https://mlsanity.com/paper/2304.09355/) (2023)
- [Efficient Online Reinforcement Learning for Diffusion Policy](https://mlsanity.com/paper/2502.00361/) (2025)
- [Molecular Geometry Pretraining with SE(3)-Invariant Denoising Distance Matching](https://mlsanity.com/paper/2206.13602/) (2022)
- [MolGenSurvey: A Systematic Survey in Machine Learning Models for Molecule Design](https://mlsanity.com/paper/2203.14500/) (2022)
- [Controllable and Compositional Generation with Latent-Space Energy-Based Models](https://mlsanity.com/paper/2110.10873/) (2021)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [A Text-guided Protein Design Framework](https://mlsanity.com/paper/2302.04611/) (2023)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)

## Abstract (arXiv)

Energy-Based Models (EBMs), also known as non-normalized probabilistic models, specify probability density or mass functions up to an unknown normalizing constant. Unlike most other probabilistic models, EBMs do not place a restriction on the tractability of the normalizing constant, thus are more flexible to parameterize and can model a more expressive family of probability distributions. However, the unknown normalizing constant of EBMs makes training particularly difficult. Our goal is to provide a friendly introduction to modern approaches for EBM training. We start by explaining maximum likelihood training with Markov chain Monte Carlo (MCMC), and proceed to elaborate on MCMC-free approaches, including Score Matching (SM) and Noise Constrastive Estimation (NCE). We highlight theoretical connections among these three approaches, and end with a brief survey on alternative training methods, which are still under active research. Our tutorial is targeted at an audience with basic understanding of generative models who want to apply EBMs or start a research project in this direction.