A Lagrangian Perspective on Latent Variable Generative Models

Shengjia Zhao, Jiaming Song, Stefano Ermon
Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence, PMLR R16:1030-1040, 2018.

Abstract

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 prob- lem. The primal problem optimizes the mutual information between latent and visible variables, subject to the constraints of accurately model- ing the data distribution and performing correct amortized inference. By choosing to maximize or minimize mutual information, and choosing different Lagrange multipliers, we obtain differ- ent objectives including InfoGAN, ALI/BiGAN, ALICE, CycleGAN, beta-VAE, adversarial au- toencoders, AVB, AS-VAE and InfoVAE. Based on this observation, we provide an exhaustive characterization of the statistical and computa- tional trade-offs made by all the training objec- tives in this class of Lagrangian duals. Next, we propose a dual optimization method where we optimize model parameters as well as the La- grange multipliers. This method achieves Pareto optimal solutions in terms of optimizing informa- tion and satisfying the constraints.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR16-zhao18c, title = {A {L}agrangian Perspective on Latent Variable Generative Models}, author = {Zhao, Shengjia and Song, Jiaming and Ermon, Stefano}, booktitle = {Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence}, pages = {1030--1040}, year = {2018}, editor = {Globerson, Amir and Silva, Ricardo}, volume = {R16}, series = {Proceedings of Machine Learning Research}, month = {06--10 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r16/main/assets/zhao18c/zhao18c.pdf}, url = {https://proceedings.mlr.press/r16/zhao18c.html}, abstract = {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 prob- lem. The primal problem optimizes the mutual information between latent and visible variables, subject to the constraints of accurately model- ing the data distribution and performing correct amortized inference. By choosing to maximize or minimize mutual information, and choosing different Lagrange multipliers, we obtain differ- ent objectives including InfoGAN, ALI/BiGAN, ALICE, CycleGAN, beta-VAE, adversarial au- toencoders, AVB, AS-VAE and InfoVAE. Based on this observation, we provide an exhaustive characterization of the statistical and computa- tional trade-offs made by all the training objec- tives in this class of Lagrangian duals. Next, we propose a dual optimization method where we optimize model parameters as well as the La- grange multipliers. This method achieves Pareto optimal solutions in terms of optimizing informa- tion and satisfying the constraints.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T A Lagrangian Perspective on Latent Variable Generative Models %A Shengjia Zhao %A Jiaming Song %A Stefano Ermon %B Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2018 %E Amir Globerson %E Ricardo Silva %F pmlr-vR16-zhao18c %I PMLR %P 1030--1040 %U https://proceedings.mlr.press/r16/zhao18c.html %V R16 %X 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 prob- lem. The primal problem optimizes the mutual information between latent and visible variables, subject to the constraints of accurately model- ing the data distribution and performing correct amortized inference. By choosing to maximize or minimize mutual information, and choosing different Lagrange multipliers, we obtain differ- ent objectives including InfoGAN, ALI/BiGAN, ALICE, CycleGAN, beta-VAE, adversarial au- toencoders, AVB, AS-VAE and InfoVAE. Based on this observation, we provide an exhaustive characterization of the statistical and computa- tional trade-offs made by all the training objec- tives in this class of Lagrangian duals. Next, we propose a dual optimization method where we optimize model parameters as well as the La- grange multipliers. This method achieves Pareto optimal solutions in terms of optimizing informa- tion and satisfying the constraints. %Z Reissued by PMLR on 04 October 2026.
APA
Zhao, S., Song, J. & Ermon, S.. (2018). A Lagrangian Perspective on Latent Variable Generative Models. Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R16:1030-1040 Available from https://proceedings.mlr.press/r16/zhao18c.html. Reissued by PMLR on 04 October 2026.

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