# Symmetric Equilibrium Learning of VAEs

arXiv: https://arxiv.org/abs/2307.09883 · HTML: https://mlsanity.com/paper/2307.09883/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 42 references that are in the map.
- First posted on arXiv 19 Jul 2023.

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Ladder Variational Autoencoders](https://mlsanity.com/paper/1602.02282/) (2016)
- [Adversarially Learned Inference](https://mlsanity.com/paper/1606.00704/) (2016)
- [Adversarial Variational Bayes: Unifying Variational Autoencoders and Generative Adversarial Networks](https://mlsanity.com/paper/1701.04722/) (2017)
- [Reweighted Wake-Sleep](https://mlsanity.com/paper/1406.2751/) (2014)
- [Deep AutoRegressive Networks](https://mlsanity.com/paper/1310.8499/) (2013)
- [Variational Inference using Implicit Distributions](https://mlsanity.com/paper/1702.08235/) (2017)
- [MuProp: Unbiased Backpropagation for Stochastic Neural Networks](https://mlsanity.com/paper/1511.05176/) (2015)
- [Doubly Semi-Implicit Variational Inference](https://mlsanity.com/paper/1810.02789/) (2018)
- [VAE Approximation Error: ELBO and Exponential Families](https://mlsanity.com/paper/2102.09310/) (2021)
- [Adversarial Symmetric Variational Autoencoder](https://mlsanity.com/paper/1711.04915/) (2017)
- [Flexible and accurate inference and learning for deep generative models](https://mlsanity.com/paper/1805.11051/) (2018)
- [Pairwise Supervised Hashing with Bernoulli Variational Auto-Encoder and Self-Control Gradient Estimator](https://mlsanity.com/paper/2005.10477/) (2020)
- [GibbsNet: Iterative Adversarial Inference for Deep Graphical Models](https://mlsanity.com/paper/1712.04120/) (2017)
- [Amortised Learning by Wake-Sleep](https://mlsanity.com/paper/2002.09737/) (2020)
- [On the Generative Utility of Cyclic Conditionals](https://mlsanity.com/paper/2106.15962/) (2021)

## Built on by (most cited first, 3 of 3)

- [Generative AI for Urban Design: A Stepwise Approach Integrating Human Expertise with Multimodal Diffusion Models](https://mlsanity.com/paper/2505.24260/) (2025)
- [Game-theoretic distributed learning of generative models for heterogeneous data collections](https://mlsanity.com/paper/2511.01740/) (2025)
- [Deep Multivariate Models with Parametric Conditionals](https://mlsanity.com/paper/2602.01953/) (2026)

## Abstract (arXiv)

We view variational autoencoders (VAE) as decoder-encoder pairs, which map distributions in the data space to distributions in the latent space and vice versa. The standard learning approach for VAEs is the maximisation of the evidence lower bound (ELBO). It is asymmetric in that it aims at learning a latent variable model while using the encoder as an auxiliary means only. Moreover, it requires a closed form a-priori latent distribution. This limits its applicability in more complex scenarios, such as general semi-supervised learning and employing complex generative models as priors. We propose a Nash equilibrium learning approach, which is symmetric with respect to the encoder and decoder and allows learning VAEs in situations where both the data and the latent distributions are accessible only by sampling. The flexibility and simplicity of this approach allows its application to a wide range of learning scenarios and downstream tasks.