  We uncover a fairly general principle in online learning: If regret can be
(approximately) expressed as a function of certain "sufficient statistics" for
the data sequence, then there exists a special Burkholder function that 1) can
be used algorithmically to achieve the regret bound and 2) only depends on
these sufficient statistics, not the entire data sequence, so that the online
strategy is only required to keep the sufficient statistics in memory. This
characterization is achieved by bringing the full power of the Burkholder
Method --- originally developed for certifying probabilistic martingale
inequalities --- to bear on the online learning setting.
  To demonstrate the scope and effectiveness of the Burkholder method, we
develop a novel online strategy for matrix prediction that attains a regret
bound corresponding to the variance term in matrix concentration inequalities.
We also present a linear-time/space prediction strategy for parameter free
supervised learning with linear classes and general smooth norms.
