A packing lemma for VCNk-dimension and learning high-dimensional data

Recently, the authors introduced the theory of high-arity PAC learning, which is well-suited for learning graphs, hypergraphs and relational structures. In the same initial work, the authors proved a high-arity analogue of the Fundamental Theorem of Statistical Learning that almost completely characterizes all notions of high-arity PAC learning in terms of a combinatorial dimension, called the Vapnik--Chervonenkis--Natarajan (VCNk) k-dimension, leaving as an open problem only the characterization of non-partite, non-agnostic high-arity PAC learnability. In this work, we complete this characterization by proving that non-partite non-agnostic high-arity PAC learnability implies a high-arity version of the Haussler packing property, which in turn implies finiteness of VCNk-dimension. This is done by obtaining direct proofs that classic PAC learnability implies classic Haussler packing property, which in turn implies finite Natarajan dimension and noticing that these direct proofs nicely lift to high-arity.