Paper:
DISTRIBUTIONAL CONCAVITY REGULARIZATION FOR GANS
We propose Distributional Concavity (DC) regularization for Generative Adversarial Networks (GANs), a functional gradient-based method that promotes the entropy of the generator distribution and works against mode collapse. Our DC regularization is an easy-to-implement method that can be used in combination with the current state of the art methods like Spectral Normalization and Wasserstein GAN with gradient penalty to further improve the performance. We will not only show that our DC regularization can achieve highly competitive results on ILSVRC2012 and CIFAR datasets in terms of Inception score and Fréchet inception distance, but also provide a mathematical guarantee that our method can always increase the entropy of the generator distribution. We will also show an intimate theoretical connection between our method and the theory of optimal transport.

Review:
Summary:
This paper proposes distributional concavity regularization for GANs which encourages producing generator distributions with higher entropy. The paper motivates the proposed method as follows:
-       Using the concept of functional gradient, the paper interprets the update in the generator parameters as an update in the generator distribution
-       Given this functional gradient perspective, the paper proposes updating the generator distribution toward a target distribution which has *higher entropy and satisfies monoticity*
-       Then, the paper proves that this condition can be satisfied by ensuring that generator’s objective (L) is concave
-       Since it’s difficult to ensure concavity when parametrizing generators as deep neural networks, the paper proposes adding a simple penalty term that encourages the concavity of generator objective
Experiments confirm the validity the proposed approach. Interestingly, the paper shows that performance of multiple GAN variants can be improved with their proposed method on several image datasets
 
Strengths:
-   	The proposed method is very interesting and is based on sound theory
-   	Connection to optimal transport theory is also interesting
-   	In practice, the method is very simple to implement and seems to produce good results
 
Weaknesses:
-       Readability of the paper can be generally improved. I had to go over the paper many times to get the idea.
-       Figures should be provided with more detailed captions, which explain main result and providing context (e.g. explaining baselines).
 
Questions/Comments:
-       Equation (7) has typos (uses theta_old instead of theta in some places)
-       Section 4.1 (effect of monoticity) is a bit confusing. My understanding is that parameter update rule of equation (3) and (6) are equivalent, but you seem to use (6) there. Can you clarify what you do there and in general this experiment a bit more?
-       Comparing with entropy maximization method of EGAN (Dai et al, 2017) is a good idea, but I’m wondering if you can compare it on low dimensional settings (e.g. as in Fig 2). It is also not clear why increasing entropy with EGAN-VI is worse than baselines in Table 1.
 
Overall recommendation:
The paper is based on sound theory and provides very interesting perspective. The method seems to work in practice on a variety of experimental setting. Therefore, I recommend accepting it.