A Convex Duality Framework for GANs (1810.11740v1)

Abstract: Generative adversarial network (GAN) is a minimax game between a generator mimicking the true model and a discriminator distinguishing the samples produced by the generator from the real training samples. Given an unconstrained discriminator able to approximate any function, this game reduces to finding the generative model minimizing a divergence measure, e.g. the Jensen-Shannon (JS) divergence, to the data distribution. However, in practice the discriminator is constrained to be in a smaller class $\mathcal{F}$ such as neural nets. Then, a natural question is how the divergence minimization interpretation changes as we constrain $\mathcal{F}$. In this work, we address this question by developing a convex duality framework for analyzing GANs. For a convex set $\mathcal{F}$, this duality framework interprets the original GAN formulation as finding the generative model with minimum JS-divergence to the distributions penalized to match the moments of the data distribution, with the moments specified by the discriminators in $\mathcal{F}$. We show that this interpretation more generally holds for f-GAN and Wasserstein GAN. As a byproduct, we apply the duality framework to a hybrid of f-divergence and Wasserstein distance. Unlike the f-divergence, we prove that the proposed hybrid divergence changes continuously with the generative model, which suggests regularizing the discriminator's Lipschitz constant in f-GAN and vanilla GAN. We numerically evaluate the power of the suggested regularization schemes for improving GAN's training performance.

A Convex Duality Framework for GANs: An Expert Review

Generative Adversarial Networks (GANs) have emerged as a prominent machine learning technique for generating synthetic data that closely resembles real-world samples. The original formulation of GANs as introduced by Goodfellow et al. in 2014 frames the problem as a minimax game between a generator, which aims to produce data similar to the target distribution, and a discriminator, which endeavors to differentiate between the generator’s output and authentic data. Although theoretically appealing, practical implementations of GANs often diverge due to constraints imposed on the discriminator model, which is typically restricted to neural network architectures. This paper, "A Convex Duality Framework for GANs" by Farzan Farnia and David Tse, seeks to reconcile this discrepancy by developing a novel convex duality framework, providing deeper insights into GANs' divergence minimization interpretation in constrained settings.

Overview of Contributions

The paper addresses two critical questions: How does the interpretation of GANs as a divergence minimization process change when the discriminator set is constrained? Moreover, how can we leverage duality theory to develop a broader theoretical understanding of GAN variants? To this end, the authors introduce a framework that extends the interpretation of divergence measure minimization, prevalent in unconstrained GANs, to scenarios where the discriminator set belongs to a restricted, convex class.

The primary contributions include:

1. Convex Duality for GANs: The authors articulate a dual interpretation of the GAN problem by demonstrating that when the discriminator is confined to a convex function class, the generative model aims to minimize the Jensen-Shannon (JS) divergence, extended to distributions with discriminator-matching moment constraints. This interpretation is generalized for f-GANs and Wasserstein GANs.
2. Hybrid Divergence Measures: The paper introduces a formulation combining f-divergence and optimal transport costs, such as the first- and second-order Wasserstein distances, to form a hybrid loss metric that changes continuously with the generator, overcoming discontinuities traditionally observed in f-GANs.
3. Numerical Validation: Through empirical analysis on benchmark datasets like CelebA and LSUN, the paper illustrates the improved training stability offered by regularizing GANs with Lipschitz constraints and adversarial learning schemes.

Strong Numerical Claims and Insights

This convex duality framework reveals GAN’s training performance scope by deriving robust regularization schemes that ensure continuity of the divergence measure, mitigating mode collapse risks and enhancing convergence properties. The authors provide substantial numerical results validating the proposed hybrid divergence measures, showing significant improvements in sample generation quality using spectral normalization and adversarial training methods.

Implications and Speculative Developments

Practically, the duality framework guides the design of stable training procedures for GAN architectures, particularly when leveraging neural network-based discriminators. Regularizing the Lipschitz constants in GAN architectures suggests that carefully designed constraints can help stabilize GAN training, avoiding frequently reported pitfalls such as poor convergence and mode collapse.

Theoretically, this work enriches GAN literature by connecting convex duality and divergence minimization properties, offering a promising angle for future research exploring GAN generalization properties, optimization robustness, and ultimately enhancing the alignment between theoretical GAN formulations and empirical implementations.

As developments in adversarial training methods continue to unfold, including differential privacy and fairness considerations within GAN architectures, this paper sets a precedent for employing foundational mathematical principles to tackle emerging challenges in generative modeling.

Conclusion

Farnia and Tse's paper offers a rigorous exploration into the convex duality framework for GANs, presenting new theoretical insights and practical solutions for optimizing GAN performance in constrained settings. By establishing connections between GAN models and divergence measures, the work paves the way for continued innovations in adversarial learning methodologies, poised to refine the generative model landscape comprehensively.