Higher dimensional PAC - learning and VC - dimension

The VC-dimension (Vapnic-Chervonenkis dimension) was introduced in 1970’s related to computational learning theory, combinatorics, and model theory which is a branch of mathematical logic. In fact, it is well known that for given class C, PAC-learnability of C, the finiteness of VC-dimension of C, and the dependency (which is a notion in model theory) of a formula defines C are essentially the same nowadays. One of the most important lemma around VC-dimension is so-called Sauer-Shelah lemma that claims the shatter function is bounded by a polynomial of degree d where d is the VC-dimension of the class. In 2009, Shelah introduced the notion of n-dependence that is a generalization of dependence motivated by an application of model theory to algebras. Recently one of the authors found a natural generalization of VC-dimension, V Cn-dimension, and Sauer-Shelah lemma for higher dimensional spaces by using combinatorics of hypergraphs, especially Zarankiewicz number from extremal graph theory, during the study of n-dependence. (See arXiv:1411.0120 for the details.) Hence it is natural to ask that if there is any generalization of PAC-learnability corresponding to V Cn-dimension. In this talk, we introduce PACn-learning and report some results we found around it. Actually, we have a proof that PACn-learnability implies the finiteness of V Cn-dimension, however the converse is not known so far. The most difficult part of the converse is that the existence of ε-nets is no longer true for higher dimensional case, though it is the key lemma for one-dimensional case.