# Generalized Mixability via Entropic Duality

arXiv: https://arxiv.org/abs/1406.6130 · HTML: https://mlsanity.com/paper/1406.6130/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2014.

## Built on by (most cited first, 3 of 3)

- [Generalized Linear Bandits: Almost Optimal Regret with One-Pass Update](https://mlsanity.com/paper/2507.11847/) (2025)
- [The Geometry of Mixability](https://mlsanity.com/paper/2302.11905/) (2023)
- [Lower-Bounded Proper Losses for Weakly Supervised Classification](https://mlsanity.com/paper/2103.02893/) (2021)

## Abstract (arXiv)

Mixability is a property of a loss which characterizes when fast convergence 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 this 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 precisely which Phi have Phi-mixable losses and put forward a number of conjectures about the optimality and relationships between different choices of entropy.