  We study minimax convergence rates of nonparametric density estimation under
a large class of loss functions called "adversarial losses", which, besides
classical $\mathcal{L}^p$ losses, includes maximum mean discrepancy (MMD),
Wasserstein distance, and total variation distance. These losses are closely
related to the losses encoded by discriminator networks in generative
adversarial networks (GANs). In a general framework, we study how the choice of
loss and the assumed smoothness of the underlying density together determine
the minimax rate. We also discuss implications for training GANs based on deep
ReLU networks, and more general connections to learning implicit generative
models in a minimax statistical sense.
