Paper:
Published as a conference paper at ICLR 2018 IMPROVING THE IMPROVED TRAINING OF WASSERSTEIN GANS: A CONSISTENCY TERM AND ITS DUAL EFFECT
Despite being impactful on a variety of problems and applications, the generative adversarial nets (GANs) are remarkably difficult to train. This issue is formally analyzed by Arjovsky & Bottou (2017), who also propose an alternative direction to avoid the caveats in the minmax two-player training of GANs. The corresponding algorithm, called Wasserstein GAN (WGAN), hinges on the 1-Lipschitz continuity of the discriminator. In this paper, we propose a novel approach to enforcing the Lipschitz continuity in the training procedure of WGANs. Our approach seamlessly connects WGAN with one of the recent semi-supervised learning methods. As a result, it gives rise to not only better photo-realistic samples than the previous methods but also state-of-the-art semi-supervised learning results. In particular, our approach gives rise to the inception score of more than 5.0 with only 1,000 CIFAR-10 images and is the first that exceeds the accuracy of 90% on the CIFAR-10 dataset using only 4,000 labeled images, to the best of our knowledge.

Review:
Summary:
The paper proposes a new regularizer for wgans, to be combined with the traditional gradient penalty. The theoretical motivation is bleak, and the analysis contains some important mistakes. The results are very good, as noticed by the comments, the fact that the method is also less susceptible to overfitting is also an important result, though this might be purely due to dropout. One of the main problems is that the largest dataset used is CIFAR, which is small. Experiments on something like bedrooms or imagenet would make the paper much stronger. 
If the authors fix the theoretical analysis and add evidence in a larger dataset I will raise the score.
Detailed comments:
- The motivation of 1.2 and the sentence "Arguably, it is fairly safe to limit our scope to the manifold that supports the real data distribution P_r and its surrounding regions" are incredibly wrong. First of all, it should be noted that the duality uses 1-Lip in the entire space between Pr and Pg, not in Pr alone. If the manifolds are not extremely close (such as in the beginning of training), then the discriminator can be almost exactly 1 in the real data, and 0 on the fake. Thus the discriminator would be almost exactly constant (0-Lip) near the real manifold, but will fail to be 1-lip in the decision boundary, this is where interpolations fix this issue. See Figure 2 of the wgan paper for example, in this simple example an almost perfect discriminator would have almost 0 penalty.
- In the 'Potential caveats' section, the implication that 1-Lip may not be enforced in non-examined samples is checkable by an easy experiment, which is to look for samples that have gradients of the critic wrt the input with norm > 1. I performed the exp in figure 8 and saw that by taking a slightly higher lambda, one reaches gradients that are as close to 1 as with ct-gan. Since ct-gan uses an extra regularizer, I think the authors need some stronger evidence to support the claim that ct-gan better battles this 'potential caveat'.
- It's important to realize that the CT regularizer with M' = 1 (1-Lip constraint) will only be positive for an almost 1-Lip function if x and x' are sampled when x - x' has a very similar direction than the gradient at x. This is very hard in high dimensional spaces, and when I implemented a CT regularizer indeed the ration of eq (4) was quite less than the norm of the gradient. It would be useful to plot the value of the CT regularizer (the eq 4 version) as the training iterations progresses. Thus the CT regularizer works as an overall Lipschitz penalty, as opposed to penalizing having more than 1 for the Lipschitz constant. This difference is non-trivial and should be discussed.
- Line 11 of the algorithm is missing L^(i) inside the sum.
- One shouldn't use MNIST for anything else than deliberately testing an overfitting problem. Figure 4 is thus relevant, but the semi-supervised results of MNIST or the sample quality experiments give hardly any evidence to support the method.
- The overfitting result is very important, but one should disambiguate this from being due to dropout. Comparing with wgangp + dropout is thus important in this experiment.
- The authors should provide experiments in at least one larger dataset like bedrooms or imagenet (not faces, which is known to be very easy). This would strengthen the paper quite a bit.