  We show a principled way of deriving online learning algorithms from a
minimax analysis. Various upper bounds on the minimax value, previously thought
to be non-constructive, are shown to yield algorithms. This allows us to
seamlessly recover known methods and to derive new ones. Our framework also
captures such "unorthodox" methods as Follow the Perturbed Leader and the R^2
forecaster. We emphasize that understanding the inherent complexity of the
learning problem leads to the development of algorithms.
  We define local sequential Rademacher complexities and associated algorithms
that allow us to obtain faster rates in online learning, similarly to
statistical learning theory. Based on these localized complexities we build a
general adaptive method that can take advantage of the suboptimality of the
observed sequence.
  We present a number of new algorithms, including a family of randomized
methods that use the idea of a "random playout". Several new versions of the
Follow-the-Perturbed-Leader algorithms are presented, as well as methods based
on the Littlestone's dimension, efficient methods for matrix completion with
trace norm, and algorithms for the problems of transductive learning and
prediction with static experts.
