Generative adversarial networks, or GANs, commonly display unstable behavior during training. In this work, we develop a principled theoretical framework for understanding the stability of various types of GANs. In particular, we derive conditions that guarantee eventual stationarity of the generator when it is trained with gradient descent, conditions that must be satisfied by the divergence that is minimized by the GAN and the generator's architecture. We find that existing GAN variants satisfy some, but not all, of these conditions. Using tools from convex analysis, optimal transport, and reproducing kernels, we construct a GAN that fulfills these conditions simultaneously. In the process, we explain and clarify the need for various existing GAN stabilization techniques, including Lipschitz constraints, gradient penalties, and smooth activation functions.
This paper provides a unified theoretical framework for regularizing GAN losses. It accounts for most regularization technics especially spectral normalization and gradient penalty and explains how those two methods are in fact complementary. So far this was only observed experimentally but without any theoretical insight. The result goes beyond that as the criterion could be applied to general convex cost functional.
The main general theorem is Theorem 1 which states 3 conditions on the optimal critic and 2 others on the generator. The paper is mainly concerned by the conditions on the optimal critic and show that the first 2 conditions can be achieved by the Spectral normalization, while the last one can be achieved by some gradient penalty.
The paper is clearly written, well structured and pleasant to read.
I have the following two remarks:
	- Proposition 8 provides a way to ensure condition 2 holds (beta-smoothness). It requires spectral normalization and smooth activation functions. In practice, while the spectral normalization is important, the choice of the activation is not in general 1-smooth (Leaky-relu for instance). Does it really matter in practice? 
	Some illustrative experiments could be beneficial to better understand what's happening.
	- Is it that hard to obtain generators that satisfy condition G1 and G2, it seems to be a natural consequence on the regularity of the mapping f? If that is the case, it might be worth better explaining how this is challenging.
	
Limitations: The paper considers only the setting where the optimal critic is reached and therefore it is still unclear if the analysis carries on to the training procedures used in practice (non-optimal critic). The authors recognize this limitation and leave it for future work.
Overall, I feel that the paper provides good insights on what regularization is important for training gans and why. For that reason, I think this paper should be accepted.
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Revision:
I think the paper provides a good theoretical contribution in terms of interpreting many of the tricks used for improving GAN training. In fact the paper also suggests some new regularization methods (prop 13 for conditions D3)  which would constrain the RKHS norm of the critic. The authors show how it is related to gradient penalty, in a particular case, but the result also suggests something more general. For instance [1], consider an abstract RKHS space containing deep networks and provide an upper-bound on the rkhs norm of such networks in terms of the spectral norm of their weights and a lower-bound in terms of its Lipschitz constant. 
I do agree with reviewer 1 that a better discussion of the connection to [2] should be included since that paper was interested in  ensuring weak continuity of the loss, which can be thought of as  a first requirement to get more regularity of the cost functional.
I still think the paper is worth being accepted and raised my score to 8 as I think the authors addressed the major concerns that were raised. 
[1] A. Bietti, G. Mialon, D. Chen, and J. Mairal. A Kernel Perspective for Regularizing Deep Neural Networks.
[2] Michael Arbel, Dougal Sutherland, Mikołaj Binkowski, and Arthur Gretton. On gradient regularizers for MMD GANs. In Advances in Neural Information Processing Systems, pp. 6700–6710, 2018.