text: This paper is in some sense a "position paper,"  giving a framework for thinking about the loss functions implicitly used by the generator of GAN-type models.  It advocates thinking about the loss in a way similar to how it is considered in structured prediction.  It also proposes that approximating the dual formulation of various divergences with functions from a parametric class, as is typically done in GAN-type setups, is not only more tractable (computationally and in sample complexity) than the full nonparametric estimation, but also gives a better actual loss. Overall, I like the argument here, and think that it is a useful framework for thinking about these things.  My main concern is that the practical contribution on top of Liu et al. (2017) might be somewhat limited. A few small points: - f-divergences can actually be nonparametrically estimated purely from samples, e.g. with the k-nearest neighbor estimator of https://arxiv.org/abs/1411.2045, or (for certain f-divergences) the kernel density based estimator of https://arxiv.org/abs/1402.2966.  These are unlikely to lead to a practical learning algorithm,  but could be mentioned in Table 1. - The discussion of MMD in the end of section 3.1 is a little off.  MMD is fundamentally defined by the kernel choice;  Dziugaite et al. (2015) only demonstrated that the Gaussian RBF kernel is a poor choice for MNIST modeling,  while the samples of Li et al. (2015) simply by using a mixture of Gaussian kernels were much better.  No reasonable fixed kernel is likely to yield good results on a harder image modeling problem,  but that is a slightly different message than the one this paragraph conveys. - It would be interesting to replicate the analysis of Danihelka et al. (2017) on the Thin-8 dataset.  This might help clarify which of the undesirable effects observed in the VAE model here are due to likelihood, and which due to other aspects of VAEs (like the use of the lower bound).