  Boosting is a popular way to derive powerful learners from simpler hypothesis
classes. Following previous work (Mason et al., 1999; Friedman, 2000) on
general boosting frameworks, we analyze gradient-based descent algorithms for
boosting with respect to any convex objective and introduce a new measure of
weak learner performance into this setting which generalizes existing work. We
present the weak to strong learning guarantees for the existing gradient
boosting work for strongly-smooth, strongly-convex objectives under this new
measure of performance, and also demonstrate that this work fails for
non-smooth objectives. To address this issue, we present new algorithms which
extend this boosting approach to arbitrary convex loss functions and give
corresponding weak to strong convergence results. In addition, we demonstrate
experimental results that support our analysis and demonstrate the need for the
new algorithms we present.
