## A Mathematical Theory of Generalization: Part II

**David H. Wolpert***Theoretical Division, Center for Nonlinear Studies,**Los Alamos National Laboratory, Los Alamos, NM, 87545, USA*

#### Abstract

The problem of how best to generalize from a given learning set of input--output examples is central to the fields of neural nets, statistics, approximation theory, and artificial intelligence. This series of papers investigates this problem from within an abstract and model-independent framework and then tests some of the resultant concepts in real-world situations. In this abstract framework a generalizer is completely specified by a certain countably infinite set of functions, so the mathematics of generalization becomes an investigation into candidate sets of criteria governing the behavior of that infinite set of functions. In the first paper of this series the foundations of this mathematics are spelled out and some relatively simple generalization criteria are investigated. Elsewhere the real-world generalizing of systems constructed with these generalization criteria in mind have been favorably compared to neural nets for several real generalization problems, including Sejnowski's problem of reading aloud. This leads to the conclusion that (current) neural nets in fact constitute a poor means of generalizing. In the second of this pair of papers other sets of criteria, more sophisticated than those criteria embodied in this first series of papers, are investigated. Generalizers meeting these more sophisticated criteria can readily be approximated on computers. Some of these approximations employ network structures built via an evolutionary process. A preliminary and favorable investigation into the generalization behavior of these approximations finishes the second paper of this series.