When can two unsupervised learners achieve PAC separation?

In this paper we study a new restriction of the PAC learning framework, in which each label class is handled by an unsupervised learner that aims to fit an appropriate probability distribution to its own data. A hypothesis is derived by choosing, for any unlabeled instance, the label whose distribution assigns it the higher likelihood. The motivation for the new learning setting is that the general approach of fitting separate distributions to each label class, is often used in practice for classification problems. The set of probability distributions that is obtained is more useful than a collection of decision boundaries. A question that arises, however, is whether it is ever more tractable (in terms of computational complexity or sample-size required) to find a simple decision boundary than to divide the problem up into separate unsupervised learning problems and find appropriate distributions. Within the framework, we give algorithms for learning various simple geometric concept classes. In the boolean domain we show how to learn parity functions, and functions having a constant upper bound on the number of relevant attributes. These results distinguish the new setting from various other well-known restrictions of PAC-learning. We give an algorithm for learning monomials over input vectors generated by an unknown product distribution. The main open problem is whether monomials (or any other concept class) distinguish learnability in this framework from standard PAC-learnability