  We construct an efficient recursive ensemble algorithm for the multi-class
classification problem, inspired by SAMME (Zhu, Zou, Rosset, and Hastie
(2009)). We strengthen the weak learnability condition in Zhu, Zou, Rosset, and
Hastie (2009) by requiring that the weak learnability condition holds for any
subset of labels with at least two elements. This condition is simpler to check
than many proposed alternatives (e.g., Mukherjee and Schapire (2013)). As
SAMME, our algorithm is reduced to the Adaptive Boosting algorithm (Schapire
and Freund (2012)) if the number of labels is two, and can be motivated as a
functional version of the steepest descending method to find an optimal
solution. In contrast to SAMME, our algorithm's final hypothesis converges to
the correct label with probability 1. For any number of labels, the probability
of misclassification vanishes exponentially as the training period increases.
The sum of the training error and an additional term, that depends only on the
sample size, bounds the generalization error of our algorithm as the Adaptive
Boosting algorithm.
