# Divergence Triangle for Joint Training of Generator Model, Energy-based Model, and Inference Model

arXiv: https://arxiv.org/abs/1812.10907 · HTML: https://mlsanity.com/paper/1812.10907/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Dec 2018.

## Built on by (most cited first, 30 of 34)

- [Latent Diffusion Energy-Based Model for Interpretable Text Modeling](https://mlsanity.com/paper/2206.05895/) (2022)
- [A Tale of Two Flows: Cooperative Learning of Langevin Flow and Normalizing Flow Toward Energy-Based Model](https://mlsanity.com/paper/2205.06924/) (2022)
- [Abstract Spatial-Temporal Reasoning via Probabilistic Abduction and Execution](https://mlsanity.com/paper/2103.14230/) (2021)
- [JEM++: Improved Techniques for Training JEM](https://mlsanity.com/paper/2109.09032/) (2021)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [TEA: Test-time Energy Adaptation](https://mlsanity.com/paper/2311.14402/) (2023)
- [Learning Energy-Based Models by Cooperative Diffusion Recovery Likelihood](https://mlsanity.com/paper/2309.05153/) (2023)
- [Energy-Based Models for Anomaly Detection: A Manifold Diffusion Recovery Approach](https://mlsanity.com/paper/2310.18677/) (2023)
- [Learning Probabilistic Models from Generator Latent Spaces with Hat EBM](https://mlsanity.com/paper/2210.16486/) (2022)
- [Bounds all around: training energy-based models with bidirectional bounds](https://mlsanity.com/paper/2111.00929/) (2021)
- [Learning Joint Latent Space EBM Prior Model for Multi-layer Generator](https://mlsanity.com/paper/2306.06323/) (2023)
- [Learning Energy-based Model via Dual-MCMC Teaching](https://mlsanity.com/paper/2312.02469/) (2023)
- [Improving Adversarial Energy-Based Model via Diffusion Process](https://mlsanity.com/paper/2403.01666/) (2024)
- [Inferential Wasserstein Generative Adversarial Networks](https://mlsanity.com/paper/2109.05652/) (2021)
- [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)
- [Progressive Test Time Energy Adaptation for Medical Image Segmentation](https://mlsanity.com/paper/2503.16616/) (2025)
- [Learning Latent Energy-Based Models via Interacting Particle Langevin Dynamics](https://mlsanity.com/paper/2510.12311/) (2025)
- [A Hybrid of Generative and Discriminative Models Based on the Gaussian-coupled Softmax Layer](https://mlsanity.com/paper/2305.05912/) (2023)
- [Maximum Entropy Inverse Reinforcement Learning of Diffusion Models with Energy-Based Models](https://mlsanity.com/paper/2407.00626/) (2024)
- [Conjugate Energy-Based Models](https://mlsanity.com/paper/2106.13798/) (2021)
- [Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching](https://mlsanity.com/paper/2504.16262/) (2025)
- [Learning Energy-based Variational Latent Prior for VAEs](https://mlsanity.com/paper/2510.00260/) (2025)
- [Progressive Energy-Based Cooperative Learning for Multi-Domain Image-to-Image Translation](https://mlsanity.com/paper/2306.14448/) (2023)
- [MITA: Bridging the Gap between Model and Data for Test-time Adaptation](https://mlsanity.com/paper/2410.09398/) (2024)
- [Deep Consensus Learning](https://mlsanity.com/paper/2103.08475/) (2021)
- [EBM Life Cycle: MCMC Strategies for Synthesis, Defense, and Density Modeling](https://mlsanity.com/paper/2205.12243/) (2022)
- [EBMs vs. CL: Exploring Self-Supervised Visual Pretraining for Visual Question Answering](https://mlsanity.com/paper/2206.14355/) (2022)
- [STANLEY: Stochastic Gradient Anisotropic Langevin Dynamics for Learning Energy-Based Models](https://mlsanity.com/paper/2310.12667/) (2023)
- [Generalized Contrastive Divergence: Joint Training of Energy-Based Model and Diffusion Model through Inverse Reinforcement Learning](https://mlsanity.com/paper/2312.03397/) (2023)

## Abstract (arXiv)

This paper proposes the divergence triangle as a framework for joint training of generator model, energy-based model and inference model. The divergence triangle is a compact and symmetric (anti-symmetric) objective function that seamlessly integrates variational learning, adversarial learning, wake-sleep algorithm, and contrastive divergence in a unified probabilistic formulation. This unification makes the processes of sampling, inference, energy evaluation readily available without the need for costly Markov chain Monte Carlo methods. Our experiments demonstrate that the divergence triangle is capable of learning (1) an energy-based model with well-formed energy landscape, (2) direct sampling in the form of a generator network, and (3) feed-forward inference that faithfully reconstructs observed as well as synthesized data. The divergence triangle is a robust training method that can learn from incomplete data.