Generalised Mixability, Constant Regret, and Bayesian Updating

Mixability of a loss is known to characterise when constant regret bounds are achievable in games of prediction with expert advice through the use of the aggregating algorithm [Vovk, 2001]. We provide a new interpretation of mixability via convex analysis that highlights the role of the Kullback-Leibler divergence in its definition. This naturally generalises to what we call Φ-mixability where the Bregman divergence DΦ replaces the KL divergence. We prove that losses that are Φ-mixable also enjoy constant regret bounds via a generalised aggregating algorithm that is similar to mirror descent.