We reconsider the training objective of Generative Adversarial Networks (GANs) from the mixed Nash Equilibria (NE) perspective. Inspired by the classical prox methods, we develop a novel algorithmic framework for GANs via an infinite-dimensional two-player game and prove rigorous convergence rates to the mixed NE. We then propose a principled procedure to reduce our novel prox methods to simple sampling routines, leading to practically efficient algorithms. Finally, we provide experimental evidence that our approach outperforms methods that seek pure strategy equilibria, such as SGD, Adam, and RMSProp, both in speed and quality.
This paper extends the mirror-descent and mirror-prox algorithms to infinite dimensional Banach spaces so that they can be applied to solve the mixed Nash equilibrium of the popular generative adversarial networks. The main technical results appear to be formal but straightforward extensions of existing techniques in finite dimensional spaces. A sample-based practical algorithm is proposed so that the infinite dimensional algorithms can still be computed. Experiments are a bit disappointing as the authors only used visual appeal as an evaluation criterion. (I understand why the authors chose to do so but as an algorithmic paper, resorting to an evaluation based on visual appeal is almost always unsatisfactory.)
Quality: The quality of this work is moderate. Quite strangely, the authors made a fundamental mistake at the very beginning: their definition of approximate mixed equilibrium  (page 2, Notation) is bizarre and different from those in previous work (such as Nemirovski's MP paper). Fortunately, this is perhaps only an oversight on the definition; the algorithms and theorems are for the correct definition anyways. Example: consider min_{-1<= x <= 1} max_{-1<=y<=1} xy. Should we call (x, 0) an (approximate) NE for any x??
Another major issue with this work is its relaxation into mixed NE. The "bilinarization" trick in Eq (5) goes back to Kantorovich (who perhaps deserves to be mentioned), and is a relaxation in general: we now have to use a mixture of generators. Since MD/MP is not sparse, in the end we must use a large number of mixtures of generators. This certainly will create some computational issues, and make comparison to pure NE methods unfair.
Clarity: The writing of this work is mostly easily to follow. However, the presentation of the technical results suffers from a real dilemma: On one hand, the authors completely ignored the technical difference between infinite dimensional Banach spaces and finite dimensional spaces. In fact, the authors never even formally defined the underlying Banach spaces. Another example, is the mapping G on page 3 continuous? wrt what topology? without such discussion what do you mean by Frechet derivative on page 4? when is the entropy function well-defined? when is the integral of exponential well-defined? Part of me totally understand that these technicalities are daunting and perhaps should not appear in the main text. On the other hand, aren't these technicalities the only "interesting and nontrivial" part of the extension to infinite dimensional spaces? If we do not care about such technicalities and can safely "assume they can be taken care of," then why is this work nontrivial? I do not see a way to resolve this dilemma here but suggest the authors consider maybe a different venue for such type of results.
Originality: The novelty of this work is limited. The extension of MD/MP to infinite dimensional spaces is mostly formal but straightforward. In fact I believe previous authors such as Nemirovski deliberately restrict to finite dimensional spaces not because of technical incapability but to avoid uninspiring technicalities. Some very related previous works were not mentioned at all:
-- Mirror Descent Learning in Continuous Games
-- Convex Games in Banach Spaces
-- On the Universality of Online Mirror Descent
The sample based algorithms are more interesting because they make the infinite dimensional extensions implementable. However, one can not say much about their convergence behavior at the moment.
Significance: The main results, although not difficult to obtain, can potentially be very useful in broadening our arsenal of tools for training GANs. The claim "resolving the longstanding problem that no provably convergent algorithm exists for general GAN" in the Abstract is disturbing, because the authors changed the definition of GAN and because the technical contributions of this work do not live up to that strong claim.