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

TL;DR

This paper unifies various latent variable generative model objectives through a Lagrangian duality framework, analyzing their trade-offs and proposing a dual optimization method for Pareto optimal solutions.

Contribution

It reveals that many existing objectives are dual functions of a single primal problem and introduces a dual optimization approach to improve model training.

Findings

Unified understanding of multiple objectives via Lagrangian duality

Characterization of statistical and computational trade-offs

Proposed dual optimization method achieves Pareto optimality

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 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.