Majority-of-Three: The Simplest Optimal Learner?

TL;DR

This paper demonstrates that a simple majority-of-three ERM classifiers achieves optimal expected error bounds in PAC learning, offering a simpler alternative to more complex algorithms.

Contribution

It proves that the majority vote of three ERM classifiers is optimal in expectation and nearly optimal with high probability, simplifying prior complex methods.

Findings

Achieves optimal in-expectation error bounds.

Provides near-optimal high-probability error bounds.

Suggests potential for full high-probability optimality.

Abstract

Developing an optimal PAC learning algorithm in the realizable setting, where empirical risk minimization (ERM) is suboptimal, was a major open problem in learning theory for decades. The problem was finally resolved by Hanneke a few years ago. Unfortunately, Hanneke's algorithm is quite complex as it returns the majority vote of many ERM classifiers that are trained on carefully selected subsets of the data. It is thus a natural goal to determine the simplest algorithm that is optimal. In this work we study the arguably simplest algorithm that could be optimal: returning the majority vote of three ERM classifiers. We show that this algorithm achieves the optimal in-expectation bound on its error which is provably unattainable by a single ERM classifier. Furthermore, we prove a near-optimal high-probability bound on this algorithm's error. We conjecture that a better analysis will prove that this algorithm is in fact optimal in the high-probability regime.