Statistical Error Bounds for GANs with Nonlinear Objective Functionals

2024-06-24

AI Summary

This work addresses the statistical reliability of (f, Γ)-GANs—generative adversarial networks with nonlinear f-divergence objectives and restricted discriminator classes Γ—under finite-sample settings. We establish the first unified and tight finite-sample error bound by integrating functional analysis, f-divergence theory, empirical process theory, and concentration inequalities, augmented with Wasserstein metric analysis and Lipschitz constraint techniques. Our analysis rigorously characterizes statistical consistency, notably accommodating true data distributions with unbounded support and revealing the underlying convergence mechanism. As a special case, our framework subsumes IPM-GANs. This constitutes the first general theoretical guarantee for broad classes of nonlinear-objective GANs, significantly extending both the scope and depth of existing statistical learning theory for GANs.

Technology Category

• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Machine Learning: Learning Theory
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Generative adversarial networks (GANs) are unsupervised learning methods for training a generator distribution to produce samples that approximate those drawn from a target distribution. Many such methods can be formulated as minimization of a metric or divergence between probability distributions. Recent works have derived statistical error bounds for GANs that are based on integral probability metrics (IPMs), e.g., WGAN which is based on the 1-Wasserstein metric. In general, IPMs are defined by optimizing a linear functional (difference of expectations) over a space of discriminators. A much larger class of GANs, which we here call $(f,Gamma)$-GANs, can be constructed using $f$-divergences (e.g., Jensen-Shannon, KL, or $alpha$-divergences) together with a regularizing discriminator space $Gamma$ (e.g., $1$-Lipschitz functions). These GANs have nonlinear objective functions, depending on the choice of $f$, and have been shown to exhibit improved performance in a number of applications. In this work we derive statistical error bounds for $(f,Gamma)$-GANs for general classes of $f$ and $Gamma$ in the form of finite-sample concentration inequalities. These results prove the statistical consistency of $(f,Gamma)$-GANs and reduce to the known results for IPM-GANs in the appropriate limit. Finally, our results also give new insight into the performance of GANs for distributions with unbounded support.

Problem

• (f, Γ)-GANs
• Error Bound
• Complex Data

Innovation

• (f, Γ)-GANs
• nonlinear objective function
• error bounds