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 $\G$.

Originality: The paper introduces a new notion of VC-dimension for k-ary Boolean functions that controls the sample complexity of graph-based discriminators. I do not think that such a notion of VC-dimension has been considered in the literature. Quality: The main results and the proofs generally seem to be correct. All the main claims in the paper are supported with complete proofs or references to proofs in the literature. I did not check all the proofs in detail; I read the proofs of Theorem 2 and Lemma 1 and they seem to be correct (but please check for typos; some of them are mentioned below under "Minor Comments").

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 \G .

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 $\G$.