[Paper Review] Learning in Implicit Generative Models

Shakir Mohamed, Balaji Lakshminarayanan|arXiv (Cornell University)|Oct 11, 2016

Generative Adversarial Networks and Image Synthesis47 references180 citations

TL;DR

The paper frames GANs within a broad likelihood-free, implicit-generative modeling paradigm and connects density-ratio, hypothesis testing, and multiple learning objectives across related methods.

ABSTRACT

Generative adversarial networks (GANs) provide an algorithmic framework for constructing generative models with several appealing properties: they do not require a likelihood function to be specified, only a generating procedure; they provide samples that are sharp and compelling; and they allow us to harness our knowledge of building highly accurate neural network classifiers. Here, we develop our understanding of GANs with the aim of forming a rich view of this growing area of machine learning---to build connections to the diverse set of statistical thinking on this topic, of which much can be gained by a mutual exchange of ideas. We frame GANs within the wider landscape of algorithms for learning in implicit generative models--models that only specify a stochastic procedure with which to generate data--and relate these ideas to modelling problems in related fields, such as econometrics and approximate Bayesian computation. We develop likelihood-free inference methods and highlight hypothesis testing as a principle for learning in implicit generative models, using which we are able to derive the objective function used by GANs, and many other related objectives. The testing viewpoint directs our focus to the general problem of density ratio estimation. There are four approaches for density ratio estimation, one of which is a solution using classifiers to distinguish real from generated data. Other approaches such as divergence minimisation and moment matching have also been explored in the GAN literature, and we synthesise these views to form an understanding in terms of the relationships between them and the wider literature, highlighting avenues for future exploration and cross-pollination.

Motivation & Objective

• Differentiate prescribed versus implicit probabilistic models and motivate implicit generative modeling for complex simulators.
• Develop likelihood-free inference via density comparisons to learn in implicit models without tractable likelihoods.
• Frame GANs and related methods through hypothesis testing and density-ratio estimation to unify diverse approaches.
• Explore connections between classification-based density ratios, divergence minimization, and moment/matching techniques.
• Discuss evaluation, training challenges, and cross-pollination with ABC and related frameworks.

Proposed method

• Class-probability estimation to compute density ratios by training a discriminator distinguishing real from generated data.
• Divergence minimization (f-divergences) and their variational formulations to guide learning.
• Ratio matching and least-squares / Bregman divergence approaches to estimate density ratios and drive generator learning.
• Moment matching and kernel-based or MMD-style objectives to compare distributions via test statistics.
• Bi-level optimization combining ratio (discriminator) loss and generator loss, with various practical loss choices and training considerations.
• Discussion of alternative losses (e.g., Wasserstein) and their implications for gradient behavior and training stability.

Experimental results

Research questions

• RQ1 How can we perform learning in implicit generative models without explicit likelihoods by comparing real and generated data?
• RQ2 What are the relationships among class-probability estimation, divergence minimization, ratio matching, and moment matching in this context?
• RQ3 How can density ratios serve as a central tool (discriminator) to guide both inference and generator optimization?
• RQ4 What are effective loss functions and training strategies to stabilize learning in implicit models and GANs?
• RQ5 How do these approaches relate to and inform methods in ABC and other likelihood-free frameworks?

Key findings

• Density-ratio estimation via a classifier provides a practical and central mechanism for likelihood-free learning in implicit models.
• Divergence minimization and class-probability estimation are two sides of the same framework for comparing p* and q.
• Different loss families (Bernoulli, Brier, KL, f-divergences) yield equivalent fixed points but differ in training dynamics and gradients.
• Ratio matching and Bregman divergences offer unifying perspectives that connect GANs, f-GANs, and LSIF-style methods.
• Moment matching and MMD-based approaches complement density-ratio methods and relate to ABC and optimal transport concepts.
• Training stability and gradient quality motivate using alternative objectives (e.g., Wasserstein) beyond the standard GAN loss.