Mathematical Perspectives on Neural Networks

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

20 Inductive Principles of Statistics and Learning Theory

V. N. Vapnik

AT&T Bell Laboratories

The purpose of this chapter is to present learning theory from the point of view of fundamental problems of mathematical statistics. It is shown that estimation of a probability measure serves as a basis for any kind of learning procedure; thus, the learning methods rely on general inductive principles of measure estimation.

Analysis of the following two principles constitutes the main content of this chapter: the principle of minimization of the empirical risk (and its generalization) based on the theory of weak convergence of empirical probability measures, and the principle of stochastic regularization that implements the theory of strong convergence of estimators of a probability measure. These inductive principles are applied to various learning (estimation) problems including pattern recognition, regression, and estimation of probability density.


1. ESTIMATION OF A PROBABILITY MEASURE AND THE PROBLEM OF DEPENDENCY ESTIMATION

1.1. Introduction

The problem of the search for empirical dependencies viewed as a computational (rather than a general scientific) problem apparently originated in the 1960s, with the appearance of the theory of pattern recognition. Later on, however, it was discovered that well-known problems, such as estimation of regression based on sample data or interpretation of the data of indirect experiments (for example, estimation of probability density) in principle, form the same range of problems. It is required to obtain an approximation for a functional dependence whose properties are reflected in the given set of examples.

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