Bounding Embeddings of VC Classes into Maximum Classes

One of the earliest conjectures in computational learning theory-the Sample Compression conjecture-asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes---those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving the conjecture is by embedding general VC classes into maximum classes without super-linear increase to their VC dimensions, as such embeddings would extend the known compression schemes to all VC classes. We show that maximum classes can be characterised by a local-connectivity property of the graph obtained by viewing the class as a cubical complex. This geometric characterisation of maximum VC classes is applied to prove a negative embedding result which demonstrates VC-d classes that cannot be embedded in any maximum class of VC dimension lower than 2d. On the other hand, we show that every VC-d class C embeds in a VC-(d+D) maximum class where D is the deficiency of C, i.e., the difference between the cardinalities of a maximum VC-d class and of C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible results on embedding into maximum classes. For some special classes of Boolean functions, relationships with maximum classes are investigated. Finally we give a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum classes for smallest k.Comment: 22 pages, 2 figure

The unstable formula theorem revisited

We first prove that Littlestone classes, those which model theorists call stable, characterize learnability in a new statistical model: a learner in this new setting outputs the same hypothesis, up to measure zero, with probability one, after a uniformly bounded number of revisions. This fills a certain gap in the literature, and sets the stage for an approximation theorem characterizing Littlestone classes in terms of a range of learning models, by analogy to definability of types in model theory. We then give a complete analogue of Shelah's celebrated (and perhaps a priori untranslatable) Unstable Formula Theorem in the learning setting, with algorithmic arguments taking the place of the infinite