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 $phi(cdot)$ parametrized by a $d times d$ weight matrix $mathbf W_*$, i.e., $f_*(mathbf x) = phi(mathbf W_* mathbf x)$. 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 $epsilon$ using $tilde O(d^2/epsilon^2)$ samples which is (near-)information theoretically optimal. Our results apply to a broad class of non-linear activation functions $phi$, 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.
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