Minimax Optimality (Probably) Doesn't Imply Distribution Learning for GANs

TL;DR

This paper demonstrates that even achieving minimax optimality in Wasserstein GANs does not necessarily imply the ability to learn the true data distribution, under cryptographic assumptions, highlighting fundamental limitations.

Contribution

It establishes a theoretical connection between GANs and pseudorandom generators, showing minimax optimality is insufficient for distribution learning.

Findings

Existence of generators far from target distribution undetectable by polynomial Lipschitz discriminators.

Minimax optimality does not guarantee distribution learning under cryptographic assumptions.

Deep connection between GANs and pseudorandom generators revealed.

Abstract

Arguably the most fundamental question in the theory of generative adversarial networks (GANs) is to understand to what extent GANs can actually learn the underlying distribution. Theoretical and empirical evidence suggests local optimality of the empirical training objective is insufficient. Yet, it does not rule out the possibility that achieving a true population minimax optimal solution might imply distribution learning. In this paper, we show that standard cryptographic assumptions imply that this stronger condition is still insufficient. Namely, we show that if local pseudorandom generators (PRGs) exist, then for a large family of natural continuous target distributions, there are ReLU network generators of constant depth and polynomial size which take Gaussian random seeds so that (i) the output is far in Wasserstein distance from the target distribution, but (ii) no polynomially large Lipschitz discriminator ReLU network can detect this. This implies that even achieving a population minimax optimal solution to the Wasserstein GAN objective is likely insufficient for distribution learning in the usual statistical sense. Our techniques reveal a deep connection between GANs and PRGs, which we believe will lead to further insights into the computational landscape of GANs.