A Convenient Infinite Dimensional Framework for Generative Adversarial Learning
This work provides a theoretical foundation for understanding the convergence properties of GANs, which is crucial for researchers and practitioners developing and applying these models.
This paper proposes an infinite-dimensional theoretical framework for generative adversarial learning, assuming uniformly bounded and Hölder differentiable probability density functions. It shows that the Rosenblatt transformation induces an optimal generator and that the Jensen-Shannon divergence between the generated and data distributions converges to zero, with convergence rates provided under certain regularity assumptions.
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 ($C^{k,α}$) 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 $C^{k,α}$-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.