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 Vovk’s aggregating algorithm. 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 $\mathrm{\Phi }$-mixability where the Bregman divergence $D_{\mathrm{\Phi }}$ replaces the KL divergence. We prove that losses that are $\mathrm{\Phi }$-mixable also enjoy constant regret bounds via a generalised aggregating algorithm that is similar to mirror descent.