# The Information Autoencoding Family: A Lagrangian Perspective on Latent Variable Generative Models

arXiv: https://arxiv.org/abs/1806.06514 · HTML: https://mlsanity.com/paper/1806.06514/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2018.

## Built on by (most cited first, 5 of 5)

- [D2C: Diffusion-Denoising Models for Few-shot Conditional Generation](https://mlsanity.com/paper/2106.06819/) (2021)
- [MINDE: Mutual Information Neural Diffusion Estimation](https://mlsanity.com/paper/2310.09031/) (2023)
- [Information Theoretic Text-to-Image Alignment](https://mlsanity.com/paper/2405.20759/) (2024)
- [Meta-Inverse Reinforcement Learning for Mean Field Games via Probabilistic Context Variables](https://mlsanity.com/paper/2509.03845/) (2025)
- [Virtual Human Generative Model: Masked Modeling Approach for Learning Human Characteristics](https://mlsanity.com/paper/2306.10656/) (2023)

## Abstract (arXiv)

A large number of objectives have been proposed to train latent variable generative models. We show that many of them are Lagrangian dual functions of the same primal optimization problem. The primal problem optimizes the mutual information between latent and visible variables, subject to the constraints of accurately modeling the data distribution and performing correct amortized inference. By choosing to maximize or minimize mutual information, and choosing different Lagrange multipliers, we obtain different objectives including InfoGAN, ALI/BiGAN, ALICE, CycleGAN, beta-VAE, adversarial autoencoders, AVB, AS-VAE and InfoVAE. Based on this observation, we provide an exhaustive characterization of the statistical and computational trade-offs made by all the training objectives in this class of Lagrangian duals. Next, we propose a dual optimization method where we optimize model parameters as well as the Lagrange multipliers. This method achieves Pareto optimal solutions in terms of optimizing information and satisfying the constraints.