Paper:
Improved Training of Wasserstein GANs
Generative Adversarial Networks (GANs) are powerful generative models, but suffer from training instability. The recently proposed Wasserstein GAN (WGAN) makes progress toward stable training of GANs, but sometimes can still generate only poor samples or fail to converge. We find that these problems are often due to the use of weight clipping in WGAN to enforce a Lipschitz constraint on the critic, which can lead to undesired behavior. We propose an alternative to clipping weights: penalize the norm of gradient of the critic with respect to its input. Our proposed method performs better than standard WGAN and enables stable training of a wide variety of GAN architectures with almost no hyperparameter tuning, including 101-layer ResNets and language models with continuous generators. We also achieve high quality generations on CIFAR-10 and LSUN bedrooms. †

Review:
+ good results
+ pretty pictures
- exposition could be improved
- claims are not backed up
- minor technical contribution
- insufficient evaluation
Let me first say that the visual results of this paper are great. The proposed algorithm produces pretty pictures of bedrooms and tiny cifar images.
Unfortunately the rest of the paper needs to be significantly improved for acceptance at NIPS. The exposition and structure of the paper is below NIPS standards. For example
L3: "..., but can still generate low-quality samples or fail to converge in some settings." I'm assuming the authors mean "can only..."?
l6: "pathological behavior". Pathological seems the wrong word here...
Sections 3 and 4 should be flipped. This way the exposition doesn't need to refer to a method that was not introduced yet.
Secondly, the paper is full of claims that are not backed up. The reviewer recommends the authors to either remove these claims or experimentally or theoretically back them up. Here are a few:
 L101: Proof that only simple functions have gradient norm almost everywhere, note that almost everywhere is important here. There can be exponentially many parts of the landscape that are disconnected by gradient norm!=1 parts.
If a proof is not possible, please remove this claim.
 L 144: Two sided penalty works better. Please show this in experiments.
The actual technical contribution seems like a small modification of WGAN. Box constraint is relaxed to a l2 norm on gradients.
Finally the evaluation is insufficient. The paper shows some pretty generation results for various hyper-parameter setting, but doesn't show any concrete numeric evaluation, apart from the easy to cheat and fairly meaningless inception score. The authors should perform a through user study to validate their approach, especially in light of a slim technical contribution.
Minor:
How is (3) efficiently evaluated? Did the authors have to modify specific deep learning packages to compute the gradient of the gradient? How is this second derivative defined for ReLUs?
Post rebuttal:
The authors answered some of my concerns in the rebuttal. I'm glad the authors promised to improve the writing and remove the misleading claims. However the rebuttal didn't fully convince me, see below.
I'm still a bit hesitant to accept the argument that prettier pictures are better, the authors rebuttal didn't change much in that regard. I'm sure some image filter, or denoising CNN could equally improve the image quality. I would strongly suggest the authors to find a way to quantitatively evaluate their method (other than the inception score).
I'm also fairly certain that the argument about the second derivative of ReLU's is wrong. The second derivative of a relu is not defined, as far as I know. The authors mention it's all zero, but that would make it a linear function (and a relu is not linear)... The reviewers had an extensive discussion on this and we agree that for ReLU networks the objective is not continuous, hence the derivative of the objective does not point down hill at all times. There are a few ways the authors should address this:
 1. Try twice differentiable non-linearities (elu, tanh or log(1+exp(x)).
 2. Show that treating the ReLU as a linear function (second derivative of zero) does give a good enough gradient estimate, for example through finite differences, or by verifying how often the "gradient" points down hill.