  In boosting, we aim to leverage multiple weak learners to produce a strong
learner. At the center of this paradigm lies the concept of building the strong
learner as a voting classifier, which outputs a weighted majority vote of the
weak learners. While many successful boosting algorithms, such as the iconic
AdaBoost, produce voting classifiers, their theoretical performance has long
remained sub-optimal: the best known bounds on the number of training examples
necessary for a voting classifier to obtain a given accuracy has so far always
contained at least two logarithmic factors above what is known to be achievable
by general weak-to-strong learners. In this work, we break this barrier by
proposing a randomized boosting algorithm that outputs voting classifiers whose
generalization error contains a single logarithmic dependency on the sample
size. We obtain this result by building a general framework that extends sample
compression methods to support randomized learning algorithms based on
sub-sampling.
