# Statistical Guarantees for Variational Autoencoders using PAC-Bayesian Theory

arXiv: https://arxiv.org/abs/2310.04935 · HTML: https://mlsanity.com/paper/2310.04935/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 61 references that are in the map.
- First posted on arXiv 7 Oct 2023.

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

- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Generating Sentences from a Continuous Space](https://mlsanity.com/paper/1511.06349/) (2015)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [A Primer on PAC-Bayesian Learning](https://mlsanity.com/paper/1901.05353/) (2019)
- [Tighter risk certificates for neural networks](https://mlsanity.com/paper/2007.12911/) (2020)
- [PAC-Bayesian Theory Meets Bayesian Inference](https://mlsanity.com/paper/1605.08636/) (2016)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [Entropy-SGD optimizes the prior of a PAC-Bayes bound: Generalization properties of Entropy-SGD and data-dependent priors](https://mlsanity.com/paper/1712.09376/) (2017)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [Meta-Learning by Adjusting Priors Based on Extended PAC-Bayes Theory](https://mlsanity.com/paper/1711.01244/) (2017)
- [PAC-Bayes Analysis Beyond the Usual Bounds](https://mlsanity.com/paper/2006.13057/) (2020)
- [A Strongly Quasiconvex PAC-Bayesian Bound](https://mlsanity.com/paper/1608.05610/) (2016)
- [On the Discrimination-Generalization Tradeoff in GANs](https://mlsanity.com/paper/1711.02771/) (2017)
- [Some Theoretical Insights into Wasserstein GANs](https://mlsanity.com/paper/2006.02682/) (2020)
- [Statistical guarantees for generative models without domination](https://mlsanity.com/paper/2010.09237/) (2020)
- [Nonparametric Density Estimation under Adversarial Losses](https://mlsanity.com/paper/1805.08836/) (2018)
- [Online PAC-Bayes Learning](https://mlsanity.com/paper/2206.00024/) (2022)
- [On PAC-Bayesian reconstruction guarantees for VAEs](https://mlsanity.com/paper/2202.11455/) (2022)
- [PAC-Bayes and Domain Adaptation](https://mlsanity.com/paper/1707.05712/) (2017)
- [Certifiably Robust Variational Autoencoders](https://mlsanity.com/paper/2102.07559/) (2021)
- [PAC-Bayesian Generalization Bounds for Adversarial Generative Models](https://mlsanity.com/paper/2302.08942/) (2023)
- [Statistical Regeneration Guarantees of the Wasserstein Autoencoder with Latent Space Consistency](https://mlsanity.com/paper/2110.03995/) (2021)

## Built on by (most cited first, 16 of 16)

- [A Note on the Convergence of Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2312.05989/) (2023)
- [Statistical Guarantees for Lifelong Reinforcement Learning using PAC-Bayes Theory](https://mlsanity.com/paper/2411.00401/) (2024)
- [Generalization in VAE and Diffusion Models: A Unified Information-Theoretic Analysis](https://mlsanity.com/paper/2506.00849/) (2025)
- [Variational excess risk bound for general state space models](https://mlsanity.com/paper/2312.09607/) (2023)
- [A Markov Random Field Multi-Modal Variational AutoEncoder](https://mlsanity.com/paper/2408.09576/) (2024)
- [Multi-Component VAE with Gaussian Markov Random Field](https://mlsanity.com/paper/2507.12165/) (2025)
- [Bound to Disagree: Generalization Bounds via Certifiable Surrogates](https://mlsanity.com/paper/2602.23128/) (2026)
- [Multi-View Majority Vote Learning Algorithms: Direct Minimization of PAC-Bayesian Bounds](https://mlsanity.com/paper/2411.06276/) (2024)
- [Multiview Representation Learning via Distributed Joint Latent Space Structuring](https://mlsanity.com/paper/2504.18455/) (2025)
- [PAC-Bayesian Reconstruction Guarantees for Time Series Variational Autoencoders](https://mlsanity.com/paper/2609.05212/) (2026)
- [Information-theoretic Generalization Analysis for VQ-VAEs: A Role of Latent Variables](https://mlsanity.com/paper/2505.19470/) (2025)
- [Deep Actor-Critics with Tight Risk Certificates](https://mlsanity.com/paper/2505.19682/) (2025)
- [Understanding Cross-Domain Adaptation in Low-Resource Topic Modeling](https://mlsanity.com/paper/2506.07453/) (2025)
- [Nonlinear Dimensionality Reduction Techniques for Bayesian Optimization](https://mlsanity.com/paper/2510.15435/) (2025)
- [Can Generative Artificial Intelligence Survive Data Contamination? Theoretical Guarantees under Contaminated Recursive Training](https://mlsanity.com/paper/2602.16065/) (2026)
- [Latent Inference-Time Guidance of Time Series Foundation Models](https://mlsanity.com/paper/2609.38058/) (2026)

## Abstract (arXiv)

Since their inception, Variational Autoencoders (VAEs) have become central in machine learning. Despite their widespread use, numerous questions regarding their theoretical properties remain open. Using PAC-Bayesian theory, this work develops statistical guarantees for VAEs. First, we derive the first PAC-Bayesian bound for posterior distributions conditioned on individual samples from the data-generating distribution. Then, we utilize this result to develop generalization guarantees for the VAE's reconstruction loss, as well as upper bounds on the distance between the input and the regenerated distributions. More importantly, we provide upper bounds on the Wasserstein distance between the input distribution and the distribution defined by the VAE's generative model.