The VC-dimension of a class of multiples of the primes, and a connection to AdaBoost

TL;DR

This paper investigates the VC-dimension of classes of multiples of primes and integers, explores their connection to prime counting functions, and analyzes AdaBoost's behavior when applied to prime-identifying functions.

Contribution

It establishes the VC-dimension for prime-related classes and links it to prime counting, also deriving limit theorems for AdaBoost's empirical risk minimization on these functions.

Findings

VC-dimension of prime multiple classes characterized

Connections between VC-dimension and prime counting functions demonstrated

Limit theorems for AdaBoost weights and risk behavior established

Abstract

We discuss the VC-dimension of a class of multiples of integers and primes (equivalently indicator functions) and demonstrate connections to prime counting functions. Additionally, we prove limit theorems for the behavior of an empirical risk minimization rule as well as the weights assigned to the output hypothesis in AdaBoost for these "prime-identifying" indicator functions, when we sample $m_n$ i.i.d. points uniformly from the integers $\{2, \dots, n\}$.