  The significance of the study of the theoretical and practical properties of
AdaBoost is unquestionable, given its simplicity, wide practical use, and
effectiveness on real-world datasets. Here we present a few open problems
regarding the behavior of "Optimal AdaBoost," a term coined by Rudin,
Daubechies, and Schapire in 2004 to label the simple version of the standard
AdaBoost algorithm in which the weak learner that AdaBoost uses always outputs
the weak classifier with lowest weighted error among the respective hypothesis
class of weak classifiers implicit in the weak learner. We concentrate on the
standard, "vanilla" version of Optimal AdaBoost for binary classification that
results from using an exponential-loss upper bound on the misclassification
training error. We present two types of open problems. One deals with general
weak hypotheses. The other deals with the particular case of decision stumps,
as often and commonly used in practice. Answers to the open problems can have
immediate significant impact to (1) cementing previously established results on
asymptotic convergence properties of Optimal AdaBoost, for finite datasets,
which in turn can be the start to any convergence-rate analysis; (2)
understanding the weak-hypotheses class of effective decision stumps generated
from data, which we have empirically observed to be significantly smaller than
the typically obtained class, as well as the effect on the weak learner's
running time and previously established improved bounds on the generalization
performance of Optimal AdaBoost classifiers; and (3) shedding some light on the
"self control" that AdaBoost tends to exhibit in practice.
