# Bounds all around: training energy-based models with bidirectional bounds

arXiv: https://arxiv.org/abs/2111.00929 · HTML: https://mlsanity.com/paper/2111.00929/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 65 references that are in the map.
- First posted on arXiv 1 Nov 2021.

## Builds on (most cited first, 28 of 28)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Deep Anomaly Detection with Outlier Exposure](https://mlsanity.com/paper/1812.04606/) (2018)
- [Likelihood Ratios for Out-of-Distribution Detection](https://mlsanity.com/paper/1906.02845/) (2019)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)
- [Your GAN is Secretly an Energy-based Model and You Should use Discriminator Driven Latent Sampling](https://mlsanity.com/paper/2003.06060/) (2020)
- [No MCMC for me: Amortized sampling for fast and stable training of energy-based models](https://mlsanity.com/paper/2010.04230/) (2020)
- [Cooperative Training of Descriptor and Generator Networks](https://mlsanity.com/paper/1609.09408/) (2016)
- [Gradient Estimators for Implicit Models](https://mlsanity.com/paper/1705.07107/) (2017)
- [Hierarchical VAEs Know What They Don't Know](https://mlsanity.com/paper/2102.08248/) (2021)
- [Learning Descriptor Networks for 3D Shape Synthesis and Analysis](https://mlsanity.com/paper/1804.00586/) (2018)
- [Maximum Entropy Generators for Energy-Based Models](https://mlsanity.com/paper/1901.08508/) (2019)
- [A Spectral Approach to Gradient Estimation for Implicit Distributions](https://mlsanity.com/paper/1806.02925/) (2018)
- [Learning Energy-Based Model with Variational Auto-Encoder as Amortized Sampler](https://mlsanity.com/paper/2012.14936/) (2020)
- [Uncertainty in the Variational Information Bottleneck](https://mlsanity.com/paper/1807.00906/) (2018)
- [The Geometry of Deep Generative Image Models and its Applications](https://mlsanity.com/paper/2101.06006/) (2021)
- [Deep Directed Generative Models with Energy-Based Probability Estimation](https://mlsanity.com/paper/1606.03439/) (2016)
- [Divergence Triangle for Joint Training of Generator Model, Energy-based Model, and Inference Model](https://mlsanity.com/paper/1812.10907/) (2018)
- [Regularized Autoencoders via Relaxed Injective Probability Flow](https://mlsanity.com/paper/2002.08927/) (2020)
- [A deep learning theory for neural networks grounded in physics](https://mlsanity.com/paper/2103.09985/) (2021)
- [Accurate and Conservative Estimates of MRF Log-likelihood using Reverse Annealing](https://mlsanity.com/paper/1412.8566/) (2014)
- [Calibrating Energy-based Generative Adversarial Networks](https://mlsanity.com/paper/1702.01691/) (2017)
- [Generative Adversarial Networks as Variational Training of Energy Based Models](https://mlsanity.com/paper/1611.01799/) (2016)
- [Boltzmann machines and energy-based models](https://mlsanity.com/paper/1708.06008/) (2017)

## Built on by (most cited first, 10 of 10)

- [Energy-based learning algorithms for analog computing: a comparative study](https://mlsanity.com/paper/2312.15103/) (2023)
- [Improving Adversarial Energy-Based Model via Diffusion Process](https://mlsanity.com/paper/2403.01666/) (2024)
- [Bi-level Doubly Variational Learning for Energy-based Latent Variable Models](https://mlsanity.com/paper/2203.14702/) (2022)
- [Learning Unnormalized Statistical Models via Compositional Optimization](https://mlsanity.com/paper/2306.07485/) (2023)
- [Implicit Variational Inference for High-Dimensional Posteriors](https://mlsanity.com/paper/2310.06643/) (2023)
- [Combating Mode Collapse in GANs via Manifold Entropy Estimation](https://mlsanity.com/paper/2208.12055/) (2022)
- [Maximum Entropy Inverse Reinforcement Learning of Diffusion Models with Energy-Based Models](https://mlsanity.com/paper/2407.00626/) (2024)
- [How to Construct Energy for Images? Denoising Autoencoder Can Be Energy Based Model](https://mlsanity.com/paper/2303.03887/) (2023)
- [Learning Energy-Based Models by Self-normalising the Likelihood](https://mlsanity.com/paper/2503.07021/) (2025)
- [Exploring bidirectional bounds for minimax-training of Energy-based models](https://mlsanity.com/paper/2506.04609/) (2025)

## Abstract (arXiv)

Energy-based models (EBMs) provide an elegant framework for density estimation, but they are notoriously difficult to train. Recent work has established links to generative adversarial networks, where the EBM is trained through a minimax game with a variational value function. We propose a bidirectional bound on the EBM log-likelihood, such that we maximize a lower bound and minimize an upper bound when solving the minimax game. We link one bound to a gradient penalty that stabilizes training, thereby providing grounding for best engineering practice. To evaluate the bounds we develop a new and efficient estimator of the Jacobi-determinant of the EBM generator. We demonstrate that these developments significantly stabilize training and yield high-quality density estimation and sample generation.