Function Learning from Interpolation

In the problem of learning a real-valued function from examples, in an extension of the ‘PAC’ model, a learner sees a sequence of values of an unknown function at a number of randomly chosen points. On the basis of these examples, the learner chooses a function—called a hypothesis—from some class H of hypotheses, with the aim that the learner's hypothesis is close to the target function on future random examples. In this paper we require that, for most training samples, with high probability the absolute difference between the values of the learner's hypothesis and the target function on a random point is small. A natural learning algorithm to consider is one that chooses a function in H that is close to the target function on the training examples. This, together with the success criterion described above, leads to the definition of a statistical property which we would wish a class of functions to possess.