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Statistical Error Bounds for GANs with Nonlinear Objective Functionals

Jeremiah Birrell

TL;DR

This work addresses statistical guarantees for GANs built on nonlinear objective functionals via the $(f,\Gamma)$-GAN framework, which unifies $f$-divergences with a discriminator space to produce nonlinear objectives. It introduces the generalized cumulant generating function $\Lambda_f^P$ and derives finite-sample concentration inequalities for the GAN estimator using novel Rademacher-complexity analyses, without requiring compact support. A key contribution is an error decomposition that separates optimization, discriminator-approximation, and statistical errors, with explicit bounds that depend on terms such as $\mathcal{R}_{\widetilde{\Gamma},Q,n}$ and $\Delta_{f,m}$. An additional result provides finite-second-moment-based bounds for Rademacher complexities under unbounded support, enabling applicability to heavy-tailed data and neural-networks, and linking to the IPM-GAN case in the linear-$f^*$ limit.

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 $(f,Γ)$-GANs, can be constructed using $f$-divergences (e.g., Jensen-Shannon, KL, or $α$-divergences) together with a regularizing discriminator space $Γ$ (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 $(f,Γ)$-GANs for general classes of $f$ and $Γ$ in the form of finite-sample concentration inequalities. These results prove the statistical consistency of $(f,Γ)$-GANs and reduce to the known results for IPM-GANs in the appropriate limit. Our results use novel Rademacher complexity bounds which provide new insight into the performance of IPM-GANs for distributions with unbounded support and have application to statistical learning tasks beyond GANs.

Statistical Error Bounds for GANs with Nonlinear Objective Functionals

TL;DR

This work addresses statistical guarantees for GANs built on nonlinear objective functionals via the -GAN framework, which unifies -divergences with a discriminator space to produce nonlinear objectives. It introduces the generalized cumulant generating function and derives finite-sample concentration inequalities for the GAN estimator using novel Rademacher-complexity analyses, without requiring compact support. A key contribution is an error decomposition that separates optimization, discriminator-approximation, and statistical errors, with explicit bounds that depend on terms such as and . An additional result provides finite-second-moment-based bounds for Rademacher complexities under unbounded support, enabling applicability to heavy-tailed data and neural-networks, and linking to the IPM-GAN case in the linear- limit.

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 -GANs, can be constructed using -divergences (e.g., Jensen-Shannon, KL, or -divergences) together with a regularizing discriminator space (e.g., -Lipschitz functions). These GANs have nonlinear objective functions, depending on the choice of , and have been shown to exhibit improved performance in a number of applications. In this work we derive statistical error bounds for -GANs for general classes of and in the form of finite-sample concentration inequalities. These results prove the statistical consistency of -GANs and reduce to the known results for IPM-GANs in the appropriate limit. Our results use novel Rademacher complexity bounds which provide new insight into the performance of IPM-GANs for distributions with unbounded support and have application to statistical learning tasks beyond GANs.

Paper Structure

This paper contains 15 sections, 20 theorems, 109 equations, 1 table.

Table of Contents

  1. Introduction
  2. Summary of Results
  3. Comparison with Generalization Bounds for the JS-GAN
  4. Properties of $\Lambda_f^P$
  5. Error Bounds for $(f,\Gamma)$-GANs
  6. $(f,\Gamma)$-GAN Error Decomposition
  7. Concentration Inequalities for $(f,\Gamma)$-GANs
  8. Rademacher Complexity Bounds for Distributions with Unbounded Support
  9. Conclusion
  10. Properties of $f$ and $f^*$
  11. Properties of the $(f,\Gamma)$-Divergences
  12. Rademacher Complexity and Uniform Law of Large Numbers
  13. Proofs of Important Lemmas
  14. Concentration Inequalities for the Estimation of $(f,\Gamma)$-Divergences
  15. Concentration Inequalities for Reverse $(f,\Gamma)$-GANs

Key Result

Lemma 2.1

Let $f\in\mathcal{F}_1(a,b)$ with $a\geq 0$, $P$ be a probability measure on $\mathcal{X}$, and $h\in\mathcal{M}_b(\mathcal{X})$ with $\alpha\leq h\leq\beta$. Then:

Theorems & Definitions (46)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Corollary 2.6
  • Lemma 2.7
  • Remark 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 36 more