Generalised Mixability, Constant Regret, and Bayesian Updating
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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 Φ-mixability where the Bregman divergence D_Φ 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.
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