[Paper Review] Convergence Problems with Generative Adversarial Networks (GANs)

Samuel A. Barnett|arXiv (Cornell University)|Jun 29, 2018

Generative Adversarial Networks and Image Synthesis29 references44 citations

TL;DR

This dissertation analyzes the convergence challenges of GANs from mathematical perspectives, detailing limitations of the original formulation and exploring topological and game-theoretic approaches to improve stability.

ABSTRACT

Generative adversarial networks (GANs) are a novel approach to generative modelling, a task whose goal it is to learn a distribution of real data points. They have often proved difficult to train: GANs are unlike many techniques in machine learning, in that they are best described as a two-player game between a discriminator and generator. This has yielded both unreliability in the training process, and a general lack of understanding as to how GANs converge, and if so, to what. The purpose of this dissertation is to provide an account of the theory of GANs suitable for the mathematician, highlighting both positive and negative results. This involves identifying the problems when training GANs, and how topological and game-theoretic perspectives of GANs have contributed to our understanding and improved our techniques in recent years.

Motivation & Objective

• Explain GANs and their training as a two-player game between generator and discriminator.
• Identify convergence problems such as failure to improve and mode collapse in MM-GANs.
• Introduce and analyze divergences and their role in training objectives.
• Explore topological and game-theoretic frameworks to improve convergence and stability.

Proposed method

• Formulate the idealised minimax GAN (IMM-GAN) objective and prove the optimal discriminator.
• Show the equivalence of IMM-GAN to minimizing JS divergence between p_r and p_G.
• Discuss divergences (KL, JS) and their properties relevant to GAN training.
• Introduce MM-GAN as a practical version with neural network parametrisations.
• Review convergence issues (failure to improve, vanishing gradients, mode collapse) and propose interpretations.

Experimental results

Research questions

• RQ1 Under what conditions does the IMM-GAN objective guarantee convergence to p_r = p_G?
• RQ2 How do divergences like KL and JS relate to GAN training dynamics and convergence?
• RQ3 What are the causes and manifestations of convergence problems such as mode collapse in MM-GANs?
• RQ4 Can alternative divergences or metrics (e.g., Wasserstein) improve training stability and convergence?
• RQ5 What game-theoretic insights can explain multiple equilibria and mixing strategies in GAN training?

Key findings

• The IMM-GAN objective converges to p_r = p_G if the optimal discriminator is used, with a minimum value of -log 4.
• JS divergence relates the IMM-GAN objective to a measure of difference between p_r and p_G, guiding convergence analysis.
• Gradient-based training can suffer from vanishing gradients and mode collapse, especially when the discriminator overfits.
• Wasserstein-based and IPM approaches offer alternative perspectives that may address JS-based convergence issues.
• Existence of equilibria in GANs does not guarantee convergence to those equilibria in practice, motivating broader game-theoretic considerations.