Generalized Mixability via Entropic Duality
textabstract 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 $\Phi$-mixability where $\Phi$ is a general 
 entropy (\ie, 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 $\Phi$-mixable losses. 
 We characterize which $\Phi$ have non-trivial $\Phi$-mixable losses and 
 relate $\Phi$-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.