Computing the Shattering Coefficient of Supervised Learning Algorithms

TL;DR

This paper analyzes the Shattering coefficient within Statistical Learning Theory, providing a proof for Hilbert spaces and discussing its implications for learning guarantees in supervised algorithms.

Contribution

It offers a proof of the Shattering coefficient for Hilbert spaces and explores its impact on the uniform convergence guarantees in supervised learning.

Findings

Shattering coefficient has polynomial growth in Hilbert spaces.

Provides theoretical bounds for learning guarantees.

Enhances understanding of convergence conditions in supervised learning.

Abstract

The Statistical Learning Theory (SLT) provides the theoretical guarantees for supervised machine learning based on the Empirical Risk Minimization Principle (ERMP). Such principle defines an upper bound to ensure the uniform convergence of the empirical risk Remp(f), i.e., the error measured on a given data sample, to the expected value of risk R(f) (a.k.a. actual risk), which depends on the Joint Probability Distribution P(X x Y) mapping input examples x in X to class labels y in Y. The uniform convergence is only ensured when the Shattering coefficient N(F,2n) has a polynomial growing behavior. This paper proves the Shattering coefficient for any Hilbert space H containing the input space X and discusses its effects in terms of learning guarantees for supervised machine algorithms.