Exact Learning Composed Classes with a Small Number of Mistakes

The Composition Lemma is one of the strongest tools for learning complex classes. It shows that if a class is learnable then composing the class with a class of polynomial number of concepts gives a learnable class. In this paper we extend the Composition Lemma as follows: we show that composing an attribute efficient learnable class with a learnable class with polynomial shatter coefficient gives a learnable class. This result extends many results in the literature and gives polynomial learning algorithms for new classes.