Generalized mixability via entropic duality

Mixability is a property of a loss which characterizes when constant regret is possible in the game of prediction with expert advice. We show that a key property of mixability generalizes, and the exp and log operations present in the usual theory are not as special as one might have thought. In doing so we introduce a more general notion of φ-mixability where φ is a general entropy (i.e., any convex function on probabilities). We show how a property shared by the convex dual of any such entropy yields a natural algorithm (the minimizer of a regret bound) which, analogous to the classical Aggregating Algorithm, is guaranteed a constant regret when used with φ-mixable losses. We characterize which φ have non-trivial φ-mixable losses and relate φ-mixability and its associated Aggregating Algorithm to potential-based methods, a Blackwell-like condition, mirror descent, and risk measures from finance. We also define a notion of "dominance" between different entropies in terms of bounds they guarantee and conjecture that classical mixability gives optimal bounds, for which we provide some supporting empirical evidence.