Bounds on the rate of convergence of learning processes based on random sets and set‐valued probability

Abstract

Purpose -Bounds on the rate of convergence of learning processes based on random samples and probability are one of the essential components of statistical learning theory (SLT). The constructive distribution-independent bounds on generalization are the cornerstone of constructing support vector machines. Random sets and set-valued probability are important extensions of random variables and probability, respectively. The paper aims to address these issues. Design/methodology/approach -In this study, the bounds on the rate of convergence of learning processes based on random sets and set-valued probability are discussed. First, the Hoeffding inequality is enhanced based on random sets, and then making use of the key theorem the non-constructive distribution-dependent bounds of learning machines based on random sets in set-valued probability space are revisited. Second, some properties of random sets and set-valued probability are discussed. Findings -In the sequel, the concepts of the annealed entropy, the growth function, and VC dimension of a set of random sets are presented. Finally, the paper establishes the VC dimension theory of SLT based on random sets and set-valued probability, and then develops the constructive distribution-independent bounds on the rate of uniform convergence of learning processes. It shows that such bounds are important to the analysis of the generalization abilities of learning machines. Originality/value -SLT is considered at present as one of the fundamental theories about small statistical learning.