TR00-050 Authors: Peter Auer, Philip M. Long, Gerhard J. Woeginger

Publication: 13th July 2000 16:24

Downloads: 1913

Keywords:

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 range such

as N or 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 R^d into R.

Previously, the class of linear functions from R^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 learning.

Finally we give a sufficient condition for an

arbitrary class F of functions from R into R

that allows us to learn the class of all functions that can

be written as the pointwise maximum of k functions from F.

This allows us to exhibit a number of further nontrivial

classes of functions from R into R for which there exist

efficient learning algorithms.