Generative Adversarial Networks (GANs) have shown great results in accurately modeling complex distributions, but their training is known to be difficult due to instabilities caused by a challenging minimax optimization problem. This is especially troublesome given the lack of an evaluation metric that can reliably detect non-convergent behaviors. We leverage the notion of duality gap from game theory in order to propose a novel convergence metric for GANs that has low computational cost. We verify the validity of the proposed metric for various test scenarios commonly used in the literature.
In this paper, the authors proposed the duality gap as the criterion for evaluating the training of GAN. To justify the proposed criterion, the authors designed empirical experiments on both synthetic and real-world datasets to demonstrate the ability of the duality gap for detecting divergence, mode collapse, sample equality, as well as the generalization to other application domains besides image generation. Comparing with the existing criteria, e.g., FID and INC, the duality gap shows better ability and computational efficiency. 
However, the paper ignores rich literatures in optimization that uses the duality gap as the criterion for characterizing the convergence of algorithms for min-max saddle point problem, e.g., [1]. In fact, in optimization community, using duality gap to screening the convergence on saddle point problem is a common knowledge. [1] even provides the finite-step convergence rate when the saddle point problem is convex-concave. This paper is only introducing that into machine learning community. Therefore, the novelty of the paper seems not enough. 
Secondly, the duality gap is only able to screen the optimization convergence and the solution quality w.r.t. **the same objective**. It is not valid to compare different GANs with different losses function using the duality gap. Theoretically, for any loss function derived from some divergences, e.g. [2], the global optimal solution  can always achieve zero duality gap. In other words, for different GANs, with different objectives, the duality gap cannot distinguish which one is better. In such sense, the title is very misleading. 
Thirdly, how the evaluate such criterion in practice in GAN scenario is not clearly explained. Considering the neural network parametrization of both the generator and discriminator, the argmax_v M(u, v) and argmin_u M(u, v) is not tractable. Without the optimal solution, what is the meaning of the ``duality gap'' should be explained. What will happen if we only obtain the suboptimal solutions which themselves are model collapsed? Without such discussion in both theoretical and/or empirical aspects, I am not very convincing about the conclusion. 
Finally, if one follows the Fenchel dual view of GAN in [2, 3], the min-max is the variational form of some divergences, which the GANs are directly optimizing. It is straightforwardly to see the better min-max value is, the smaller divergence between generated samples and ground-truth is, and thus, the better quality of the generator is. The fact that min-max objective is indeed able to characterize the quality of generator is obvious and well-known. Otherwise, there is not need to use such objective in the optimization to train the model. 
[1] Nemirovski, A., Juditsky, A., Lan, G., and Shapiro, A. (2009). Robust stochastic approximation approach to stochastic programming. SIAM J. on Optimization, 19(4):1574–1609.
[2]  I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems (NIPS), pages 2672–2680, 2014.
[3] S. Nowozin, B. Cseke, and R. Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. arXiv:1606.00709, 2016