On the Complexity of Function Learning
AbstractThe 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 range such as 
$$\mathbb{N}$$
or
$$\mathbb{R}$$
. 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 binary 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 involve 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 learning algorithm for learning convex piecewise linear functions from 
$$\mathbb{R}^d $$
 into 
$$\mathbb{R}$$
. Previously, the class of linear functions from 
$$\mathbb{R}^d $$
 into 
$$\mathbb{R}$$
 was the only class of functions with multidimensional domain that was known to be learnable within the rigorous framework of a formal model for online learning.Finally we give a sufficient condition for an arbitrary class 
$$\mathcal{F}$$
 of functions from 
$$\mathbb{R}$$
 into 
$$\mathbb{R}$$
 that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from 
$$\mathcal{F}$$
. This allows us to exhibit a number of further nontrivial classes of functions from 
$$\mathbb{R}$$
 into 
$$\mathbb{R}$$
 for which there exist efficient learning algorithms.