The Wasserstein probability metric has received much attention from the machine learning community. Unlike the Kullback-Leibler divergence, which strictly measures change in probability, the Wasserstein metric reflects the underlying geometry between outcomes. The value of being sensitive to this geometry has been demonstrated, among others, in ordinal regression and generative modelling, and most recently in reinforcement learning. In this paper we describe three natural properties of probability divergences that we believe reflect requirements from machine learning: sum invariance, scale sensitivity, and unbiased sample gradients. The Wasserstein metric possesses the first two properties but, unlike the Kullback-Leibler divergence, does not possess the third. We provide empirical evidence suggesting this is a serious issue in practice. Leveraging insights from probabilistic forecasting we propose an alternative to the Wasserstein metric, the Cramér distance. We show that the Cramér distance possesses all three desired properties, combining the best of the Wasserstein and Kullback-Leibler divergences. We give empirical results on a number of domains comparing these three divergences. To illustrate the practical relevance of the Cramér distance we design a new algorithm, the Cramér Generative Adversarial Network (GAN), and show that it has a number of desirable properties over the related Wasserstein GAN.
The manuscript proposes to use the Cramer distance as a measure between distributions (acting as a loss) when optimizing
an objective function using stochastic gradient descent (SGD). Cramer distance is a Bregman divergence and is a member of the Lp family of divergences.  Here a "distance" means a symmetric divergence measure that satisfies the relaxed triangle inequality. The motivation for using the Cramer distance is that it has unbiased sample gradients while still enjoying some other properties such as scale sensitivity and sum invariant. The authors also proof that for the Bernoulli distribution, there is a lower bound independent of the sample size for the deviation between the gradient of the Cramer distance, and the expectation of the estimated gradient of the Cramer distance. Then, the multivariate case of the Cramer distance, called the energy distance, is also briefly presented. The paper closes with some experiments on ordinal regression using neural networks and training GANs using the Cramer distance. 
In general, the manuscript is well written and the ideas are smoothly presented. While the manuscript gives some interesting insights, I find that the contribution could have been explained in a more broader sense, with a stronger compelling message.
Some remarks and questions:
1.	The KL divergence considered here is sum invariant but not scale sensitive, and has unbiased sample gradients. The 
	authors are considering here the standard (asymmetric) KL divergence (sec. 2.1). Is it the case that losing scale
	sensitivity make the KL divergence insensitive to the geometry of the outcomes? or is it due to the fact the KL 
	divergence is not symmetric? or ?
2.	The main argument for the paper is that the simple sample-based estimate for the gradient using the Wasserstein 
	metric is a biased estimate for the true gradient of the Wasserstein distance, and hence it is not favored with
	SGD-type algorithms. Are there any other estimators in the literature for the gradient of the Wasserstein distance?
	Was this issue overlooked in the literature?
3.	I am not sure if a biased estimate for the gradient will lead to a ``wrong minimum'' in an energy space that has 
	infinitely many local minima.  Of course one should use an unbiased estimate for the gradient whenever this is possible.
	However, even when this is possible, there is no guarantee that this will consistently lead to deeper and ``better''
	minima, and there is no guarantee as well that these deep local minima reflect meaningful results.
4.	To what extent can one generalize theorem 1 to other probability distributions (continuous and discrete) and to the 
	multivariate cases as well?
5.	I also don't think that the example given in sec. 4.2 and depicted in Fig. 1 is the best and simplest way to illustrate
	the benefit of Cramer distance over Wasserstein. Similarly, the experiments for the multivariate case using GANs and
	Neural Networks do not really deliver tangible, concrete and conclusive results. Partly, these results are very  
       qualitative, which can be understood within the context of GANs. However, the authors could have used other       
       models/algorithms where they can obtain concrete quantitative results (for this type of contribution). In addition, 
	such sophisticated models (with various hyper-parameters) can mask the true benefit for the Cramer distance, and can 
	also mask the extent of how good/poor is the sample estimate for the Wasserstein gradient.