Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

18 Probably Approximately Correct Learning and Decision-Theoretic Generalizations

David Haussler University of California, Santa Cruz


PART 1: OVERVIEW OF THE PROBABLY APPROXIMATELY CORRECT LEARNING FRAMEWORK

SUMMARY OF PART 1

Here we survey some recent theoretical results on the efficiency of machine learning algorithms. The main tool described is the notion of probably approximately correct (PAC) learning, introduced by Valiant. We define this learning model and then look at some of the results obtained in it. We then consider some criticisms of the PAC model and the extensions proposed to address these criticisms. Finally, we look briefly at other models recently proposed in computational learning theory.


1. INTRODUCTION

It is a dangerous thing to try to formalize an enterprise as complex and varied as machine learning so that it can be subjected to rigorous mathematical analysis. To be tractable, a formal model must be simple. Thus, inevitably, most people will feel that important aspects of the activity have been left out of the theory. Of course, they will be right. Therefore, it is not advisable to present a theory of machine learning as having reduced the entire field to its bare essentials. All that can be hoped for is that some aspects of the phenomenon are brought more clearly into focus using the tools of mathematical analysis and that perhaps a few new insights are gained. It is in this light that we wish to discuss the results obtained in the last few years in what is now called (probably approximately correct) (PAC) learning theory ( Angluin 1988).

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