  We develop a novel family of algorithms for the online learning setting with
regret against any data sequence bounded by the empirical Rademacher complexity
of that sequence. To develop a general theory of when this type of adaptive
regret bound is achievable we establish a connection to the theory of
decoupling inequalities for martingales in Banach spaces. When the hypothesis
class is a set of linear functions bounded in some norm, such a regret bound is
achievable if and only if the norm satisfies certain decoupling inequalities
for martingales. Donald Burkholder's celebrated geometric characterization of
decoupling inequalities (1984) states that such an inequality holds if and only
if there exists a special function called a Burkholder function satisfying
certain restricted concavity properties. Our online learning algorithms are
efficient in terms of queries to this function.
  We realize our general theory by giving novel efficient algorithms for
classes including lp norms, Schatten p-norms, group norms, and reproducing
kernel Hilbert spaces. The empirical Rademacher complexity regret bound implies
--- when used in the i.i.d. setting --- a data-dependent complexity bound for
excess risk after online-to-batch conversion. To showcase the power of the
empirical Rademacher complexity regret bound, we derive improved rates for a
supervised learning generalization of the online learning with low rank experts
task and for the online matrix prediction task.
  In addition to obtaining tight data-dependent regret bounds, our algorithms
enjoy improved efficiency over previous techniques based on Rademacher
complexity, automatically work in the infinite horizon setting, and are
scale-free. To obtain such adaptive methods, we introduce novel machinery, and
the resulting algorithms are not based on the standard tools of online convex
optimization.
