In most current formulations of adversarial training, the discriminators can be expressed as single-input operators, that is, the mapping they define is separable over observations. In this work, we argue that this property might help explain the infamous mode collapse phenomenon in adversarially-trained generative models. Inspired by discrepancy measures and two-sample tests between probability distributions, we propose distributional adversaries that operate on samples, i.e., on sets of multiple points drawn from a distribution, rather than on single observations. We show how they can be easily implemented on top of existing models. Various experimental results show that generators trained in combination with our distributional adversaries are much more stable and are remarkably less prone to mode collapse than traditional models trained with observation-wise prediction discriminators. In addition, the application of our framework to domain adaptation results in strong improvement over recent state-of-the-art.
The paper proposes to replace single-sample discriminators in adversarial training with discriminators that explicitly operate on distributions of examples, so as to incentivize the generator to cover the full distribution of the training data and not collapse to isolated modes. 
The idea of avoiding mode collapse by providing multiple samples to the discriminator is not new; the paper acknowledges prior work on minibatch discrimination but does not really describe the differences with previous work in any technical detail. Not being highly familiar with this literature, my reading is that the scheme in this paper grounds out into a somewhat different architecture than previous minibatch discriminators, with a nice interpretation in terms of a sample-based approximation to a neural mean embedding. However the paper does not provide any empirical evidence that their approach actually works better than previous approaches to minibatch discrimination. By comparing only to one-sample discriminators it leaves open the (a priori quite plausible) possibility that minibatch discrimination is generally a good idea but that other architectures might work equally well or better, i.e., the experiments do not demonstrate that the MMD machinery that forms the core of the paper has any real purchase.
The paper also proposes a two-sample objective DAN-2S, in which the discriminator is asked to classify two sets of samples as coming from the same or different distributions. This is an interesting approach, although empirically it does not appear to have any advantage over the simpler DAN-S -- do the authors agree with this interpretation? If so it is still a worthwhile negative result, but the paper should make this conclusion explicit. Alternately if there are cases when the two-sample test is actually recommended, that should be made explicit as well. 
Overall this paper seems borderline -- a nice theoretical story, grounding out into a simple architecture that does seem to work in practice (the domain adaptation results are promising), but with somewhat sloppy writing and experimentation that doesn't clearly demonstrate the value of the proposed approach. I hope the authors continue to improve the paper by comparing to other minibatch discrimination techniques. It would also be helpful to see value on a real-world task where mode collapse is explicitly seen as a problem (and/or to provide some intuition for why this would be the case in the Amazon reviews dataset). 
Specific comments:
- Eqn (2.2) is described as representing the limit of a converged discriminator, but it looks like this is just the general gradient of the objective --- where does D* enter into the picture?
- Fig 1: the label R is never explained; why not just use P_x?
- Section 5.1 "we use the pure distributional objective for DAN (i.e., setting λ != 0 in (3.5))" should this be λ = 0? 
- "Results" in the domain adaptation experiments are not clearly explained -- what do the reported numbers represent? (presumably accuracy(stddev) but the figure caption should say this). It is also silly to report accuracy to 2 decimal places when they are clearly not significant at that level.