Convergence and Sample Complexity of SGD in GANs

We provide theoretical convergence guarantees on training Generative Adversarial Networks (GANs) via SGD. We consider learning a target distribution modeled by a 1-layer Generator network with a non-linear activation function φ(·) parametrized by a d × d weight matrix mathbf W_*, i.e., f_*(mathbf x) = φ(mathbf W_* mathbf x). Our main result is that by training the Generator together with a Discriminator according to the Stochastic Gradient Descent-Ascent iteration proposed by Goodfellow et al. yields a Generator distribution that approaches the target distribution of f_*. Specifically, we can learn the target distribution within total-variation distance ε using tilde O(d2/ε2) samples which is (near-)information theoretically optimal. Our results apply to a broad class of non-linear activation functions φ, including ReLUs and is enabled by a connection with truncated statistics and an appropriate design of the Discriminator network. Our approach relies on a bilevel optimization framework to show that vanilla SGDA works.