  We provide a general mechanism to design online learning algorithms based on
a minimax analysis within a drifting-games framework. Different online learning
settings (Hedge, multi-armed bandit problems and online convex optimization)
are studied by converting into various kinds of drifting games. The original
minimax analysis for drifting games is then used and generalized by applying a
series of relaxations, starting from choosing a convex surrogate of the 0-1
loss function. With different choices of surrogates, we not only recover
existing algorithms, but also propose new algorithms that are totally
parameter-free and enjoy other useful properties. Moreover, our drifting-games
framework naturally allows us to study high probability bounds without
resorting to any concentration results, and also a generalized notion of regret
that measures how good the algorithm is compared to all but the top small
fraction of candidates. Finally, we translate our new Hedge algorithm into a
new adaptive boosting algorithm that is computationally faster as shown in
experiments, since it ignores a large number of examples on each round.
