Risk Bounds for the Majority Vote: From a PAC-Bayesian Analysis to a Learning Algorithm

TL;DR

This paper introduces the C-bound for majority votes in binary classification, providing a PAC-Bayesian framework to estimate it from training data, and proposes the MinCq algorithm that minimizes this bound, achieving competitive results.

Contribution

It develops a new risk bound called the C-bound, offers a PAC-Bayesian analysis to estimate it, and introduces the MinCq learning algorithm based on minimizing this bound.

Findings

The C-bound effectively captures the quality and disagreement of voters.

MinCq achieves state-of-the-art performance compared to AdaBoost and SVM.

The analysis provides new PAC-Bayesian bounds, some without Kullback-Leibler divergence.

Abstract

We propose an extensive analysis of the behavior of majority votes in binary classification. In particular, we introduce a risk bound for majority votes, called the C-bound, that takes into account the average quality of the voters and their average disagreement. We also propose an extensive PAC-Bayesian analysis that shows how the C-bound can be estimated from various observations contained in the training data. The analysis intends to be self-contained and can be used as introductory material to PAC-Bayesian statistical learning theory. It starts from a general PAC-Bayesian perspective and ends with uncommon PAC-Bayesian bounds. Some of these bounds contain no Kullback-Leibler divergence and others allow kernel functions to be used as voters (via the sample compression setting). Finally, out of the analysis, we propose the MinCq learning algorithm that basically minimizes the C-bound. MinCq reduces to a simple quadratic program. Aside from being theoretically grounded, MinCq achieves state-of-the-art performance, as shown in our extensive empirical comparison with both AdaBoost and the Support Vector Machine.