# EBMs Trained with Maximum Likelihood are Generator Models Trained with a Self-adverserial Loss

arXiv: https://arxiv.org/abs/2102.11757 · HTML: https://mlsanity.com/paper/2102.11757/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 35 references that are in the map.
- First posted on arXiv 23 Feb 2021.

## Builds on (most cited first, 26 of 26)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium](https://mlsanity.com/paper/1706.08500/) (2017)
- [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)
- [Learning Energy-Based Models by Diffusion Recovery Likelihood](https://mlsanity.com/paper/2012.08125/) (2020)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [VAEBM: A Symbiosis between Variational Autoencoders and Energy-based Models](https://mlsanity.com/paper/2010.00654/) (2020)
- [Residual Energy-Based Models for Text Generation](https://mlsanity.com/paper/2004.11714/) (2020)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Flow Contrastive Estimation of Energy-Based Models](https://mlsanity.com/paper/1912.00589/) (2019)
- [Your GAN is Secretly an Energy-based Model and You Should use Discriminator Driven Latent Sampling](https://mlsanity.com/paper/2003.06060/) (2020)
- [Refining Deep Generative Models via Discriminator Gradient Flow](https://mlsanity.com/paper/2012.00780/) (2020)
- [Learning Energy-Based Model with Variational Auto-Encoder as Amortized Sampler](https://mlsanity.com/paper/2012.14936/) (2020)
- [Monge-Amp`ere Flow for Generative Modeling](https://mlsanity.com/paper/1809.10188/) (2018)
- [Training Deep Energy-Based Models with f-Divergence Minimization](https://mlsanity.com/paper/2003.03463/) (2020)
- [Learning Energy-Based Models in High-Dimensional Spaces with Multi-scale Denoising Score Matching](https://mlsanity.com/paper/1910.07762/) (2019)
- [Generative VoxelNet: Learning Energy-Based Models for 3D Shape Synthesis and Analysis](https://mlsanity.com/paper/2012.13522/) (2020)
- [Discriminator Contrastive Divergence: Semi-Amortized Generative Modeling by Exploring Energy of the Discriminator](https://mlsanity.com/paper/2004.01704/) (2020)
- [Exponential Tilting of Generative Models: Improving Sample Quality by Training and Sampling from Latent Energy](https://mlsanity.com/paper/2006.08100/) (2020)

## Built on by (most cited first, 2 of 2)

- [Learning Energy-based Model via Dual-MCMC Teaching](https://mlsanity.com/paper/2312.02469/) (2023)
- [Restoration based Generative Models](https://mlsanity.com/paper/2303.05456/) (2023)

## Abstract (arXiv)

Maximum likelihood estimation is widely used in training Energy-based models (EBMs). Training requires samples from an unnormalized distribution, which is usually intractable, and in practice, these are obtained by MCMC algorithms such as Langevin dynamics. However, since MCMC in high-dimensional space converges extremely slowly, the current understanding of maximum likelihood training, which assumes approximate samples from the model can be drawn, is problematic. In this paper, we try to understand this training procedure by replacing Langevin dynamics with deterministic solutions of the associated gradient descent ODE. Doing so allows us to study the density induced by the dynamics (if the dynamics are invertible), and connect with GANs by treating the dynamics as generator models, the initial values as latent variables and the loss as optimizing a critic defined by the very same energy that determines the generator through its gradient. Hence the term - self-adversarial loss. We show that reintroducing the noise in the dynamics does not lead to a qualitative change in the behavior, and merely reduces the quality of the generator. We thus show that EBM training is effectively a self-adversarial procedure rather than maximum likelihood estimation.