On the Complexity of Function Learning
Abstraet. The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger fange such as Nor IR. In particular relatively few results exist about the general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binaly branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization of the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not invoive any "learning" model). Another general structural result of this paper relates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient leaming algorithm for learning convex piecewise linear functions from Rd into IR. Previously, the class of linear functions from 1R d into R was the only class of functions with multi-dimensional domain that was known to be learnable within the rigorous framework of a formal model for on-line leaming. Finally we give a sufficient condition for an arbitrary class 5 ~ of functions from IR into R that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from 5 r. This allows us to exhibit a number of further nontrivial classes of functions from ~ into R for which there exist eflicient ]earning algorithms. We consider the c o m p l e x i t y of function learning in the m o s t c o m m o n nonprobabilistic m o d e l s o f on-line learning. In contrast to the rather well-developed theory for the special case of {0, 1}-valued functions (i.e., concepts), relatively little is k n o w n about general properties of optimal m i s t a k e b o u n d s (resp. loss b o u n d s) for learning classes of functions with larger ranges (e,g., real-valued functions). Furthermore, nontrivial positive learning results have so far b e e n achieved in these models …