  We introduce an efficient algorithmic framework for model selection in online
learning, also known as parameter-free online learning. Departing from previous
work, which has focused on highly structured function classes such as nested
balls in Hilbert space, we propose a generic meta-algorithm framework that
achieves online model selection oracle inequalities under minimal structural
assumptions. We give the first computationally efficient parameter-free
algorithms that work in arbitrary Banach spaces under mild smoothness
assumptions; previous results applied only to Hilbert spaces. We further derive
new oracle inequalities for matrix classes, non-nested convex sets, and
$\mathbb{R}^{d}$ with generic regularizers. Finally, we generalize these
results by providing oracle inequalities for arbitrary non-linear classes in
the online supervised learning model. These results are all derived through a
unified meta-algorithm scheme using a novel "multi-scale" algorithm for
prediction with expert advice based on random playout, which may be of
independent interest.
