Networks (GANs), where a discriminator is trained to distinguish between a real-life distribution and a synthetic distribution.

I do not completely understand (apart for some parts of the proofs) why refer to these functions as Graph-based. Boolean k-ary functions may be thought of as hyper-graphs. The definition shouldn't be unusual and it will be clarified to avoid any possible This is completely analogous to the standard empirical distribution for hypotheses classes. It might be helpful to summarise, ..., some basic properties of this new notion of VC dimension... ..., is there a Sauer-Shelah type upper bound on the size of the class in terms of the graph VC dimension? VC dimension entail small graph VC dimension). Shelah Lemma for graph VC dimension, indeed this is noteworthy and we should discuss this in the main text.

Neural Information Processing Systems http://nips.cc/

A basic question in learning theory is to identify if two distributions are identical when we have access only to examples sampled from the distributions. This basic task is considered, for example, in the context of Generative Adversarial Networks (GANs), where a discriminator is trained to distinguish between a real-life distribution and a synthetic distribution. % Classically, we use a hypothesis class $H$ and claim that the two distributions are distinct if for some $h\in H$ the expected value on the two distributions is (significantly) different. Our starting point is the following fundamental problem: "is having the hypothesis dependent on more than a single random example beneficial". To address this challenge we define $k$-ary based discriminators, which have a family of Boolean $k$-ary functions $\mathcal{G}$. Each function $g\in \mathcal{G}$ naturally defines a hyper-graph, indicating whether a given hyper-edge exists. A function $g\in \mathcal{G}$ distinguishes between two distributions, if the expected value of $g$, on a $k$-tuple of i.i.d examples, on the two distributions is (significantly) different. We study the expressiveness of families of $k$-ary functions, compared to the classical hypothesis class $H$, which is $k=1$. We show a separation in expressiveness of $k+1$-ary versus $k$-ary functions. This demonstrate the great benefit of having $k\geq 2$ as distinguishers. For $k\geq 2$ we introduce a notion similar to the VC-dimension, and show that it controls the sample complexity. We proceed and provide upper and lower bounds as a function of our extended notion of VC-dimension.