# Statistical Error Bounds for GANs with Nonlinear Objective Functionals

arXiv: https://arxiv.org/abs/2406.16834 · HTML: https://mlsanity.com/paper/2406.16834/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 36 references that are in the map.
- First posted on arXiv 24 Jun 2024.

## Builds on (most cited first, 22 of 22)

- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [Understanding the Limitations of Variational Mutual Information Estimators](https://mlsanity.com/paper/1910.06222/) (2019)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [An error analysis of generative adversarial networks for learning distributions](https://mlsanity.com/paper/2105.13010/) (2021)
- [Approximation and Convergence Properties of Generative Adversarial Learning](https://mlsanity.com/paper/1705.08991/) (2017)
- [Some Theoretical Insights into Wasserstein GANs](https://mlsanity.com/paper/2006.02682/) (2020)
- [Bridging the Gap Between f-GANs and Wasserstein GANs](https://mlsanity.com/paper/1910.09779/) (2019)
- [Structure-preserving GANs](https://mlsanity.com/paper/2202.01129/) (2022)
- [A Convex Duality Framework for GANs](https://mlsanity.com/paper/1810.11740/) (2018)
- [Optimizing Variational Representations of Divergences and Accelerating their Statistical Estimation](https://mlsanity.com/paper/2006.08781/) (2020)
- [Sample Complexity of Probability Divergences under Group Symmetry](https://mlsanity.com/paper/2302.01915/) (2023)
- [Formulation and properties of a divergence used to compare probability measures without absolute continuity](https://mlsanity.com/paper/1911.07422/) (2019)
- [On the Statistical Properties of Generative Adversarial Models for Low Intrinsic Data Dimension](https://mlsanity.com/paper/2401.15801/) (2024)
- [The Inductive Bias of Restricted f-GANs](https://mlsanity.com/paper/1809.04542/) (2018)
- [Statistical Guarantees of Group-Invariant GANs](https://mlsanity.com/paper/2305.13517/) (2023)

## Built on by (most cited first, 2 of 2)

- [Concentration Inequalities for Stochastic Optimization of Unbounded Objective Functions with Application to Denoising Score Matching](https://mlsanity.com/paper/2502.08628/) (2025)
- [Statistical Guarantees for Distributionally Robust Optimization with Optimal Transport and OT-Regularized Divergences](https://mlsanity.com/paper/2603.27871/) (2026)

## Abstract (arXiv)

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 α-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. 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.