Measure Theoretic Approach to Nonuniform Learnability

TL;DR

This paper redefines nonuniform learnability using measure theory, introduces the Generalize Measure Learnability algorithm, and explores its complexity and consistency, extending the theoretical understanding of learning frameworks.

Contribution

It presents a measure theoretic redefinition of nonuniform learnability and introduces a new algorithm with complexity bounds and applicability to countable hypothesis classes.

Findings

GML achieves statistical consistency on countable hypothesis classes.

The measure theoretic approach generalizes previous learnability frameworks.

Sample and computational complexity bounds are established for GML.

Abstract

An earlier introduced characterization of nonuniform learnability that allows the sample size to depend on the hypothesis to which the learner is compared has been redefined using the measure theoretic approach. Where nonuniform learnability is a strict relaxation of the Probably Approximately Correct framework. Introduction of a new algorithm, Generalize Measure Learnability framework, to implement this approach with the study of its sample and computational complexity bounds. Like the Minimum Description Length principle, this approach can be regarded as an explication of Occam razor. Furthermore, many situations were presented, Hypothesis Classes that are countable where we can apply the GML framework, which we can learn to use the GML scheme and can achieve statistical consistency.