Abstract
In recent years, generative adversarial networks (GANs) have demonstrated impressive experimental results while there are only a few works that foster statistical learning theory for GANs. In this work, we propose an infinite dimensional theoretical framework for generative adversarial learning. We assume that the probability density functions of the underlying measure are uniformly bounded, k-times α-Hölder differentiable () and uniformly bounded away from zero. Under these assumptions, we show that the Rosenblatt transformation induces an optimal generator, which is realizable in the hypothesis space of -generators. With a consistent definition of the hypothesis space of discriminators, we further show that the Jensen-Shannon divergence between the distribution induced by the generator from the adversarial learning procedure and the data generating distribution converges to zero. Under certain regularity assumptions on the density of the data generating process, we also provide rates of convergence based on chaining and concentration.
Funding Statement
H. A. acknowledges financial support from Bergisch Smart Mobility funded by the Ministry of Economic Affairs, Innovation, Digitalization and Energy of the State of North Rhine-Westphalia.
Acknowledgments
The authors thank the anonymous referee and Miriam Ackermann for her help in inproving the typoscript. H. G. would like to thank Philipp Petersen for interesting discussions. We also thank the anonymous referee for many useful hints that helped to improve the paper.
Citation
Hayk Asatryan. Hanno Gottschalk. Marieke Lippert. Matthias Rottmann. "A convenient infinite dimensional framework for generative adversarial learning." Electron. J. Statist. 17 (1) 391 - 428, 2023. https://doi.org/10.1214/23-EJS2104
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