  We extend the theory of boosting for regression problems to the online
learning setting. Generalizing from the batch setting for boosting, the notion
of a weak learning algorithm is modeled as an online learning algorithm with
linear loss functions that competes with a base class of regression functions,
while a strong learning algorithm is an online learning algorithm with convex
loss functions that competes with a larger class of regression functions. Our
main result is an online gradient boosting algorithm which converts a weak
online learning algorithm into a strong one where the larger class of functions
is the linear span of the base class. We also give a simpler boosting algorithm
that converts a weak online learning algorithm into a strong one where the
larger class of functions is the convex hull of the base class, and prove its
optimality.
