# The Geometry of Mixability

arXiv: https://arxiv.org/abs/2302.11905 · HTML: https://mlsanity.com/paper/2302.11905/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 23 references that are in the map.
- First posted on arXiv 23 Feb 2023.

## Builds on (most cited first, 5 of 5)

- [Fast rates in statistical and online learning](https://mlsanity.com/paper/1507.02592/) (2015)
- [The fundamental nature of the log loss function](https://mlsanity.com/paper/1502.06254/) (2015)
- [The Geometry and Calculus of Losses](https://mlsanity.com/paper/2209.00238/) (2022)
- [Constant Regret, Generalized Mixability, and Mirror Descent](https://mlsanity.com/paper/1802.06965/) (2018)
- [Generalized Mixability via Entropic Duality](https://mlsanity.com/paper/1406.6130/) (2014)

## Built on by (most cited first, 2 of 2)

- [An Axiomatic Approach to Loss Aggregation and an Adapted Aggregating Algorithm](https://mlsanity.com/paper/2406.02292/) (2024)
- [Comparing Corrupted Constrained Learning Problems](https://mlsanity.com/paper/2608.25745/) (2026)

## Abstract (arXiv)

Mixable loss functions are of fundamental importance in the context of prediction with expert advice in the online setting since they characterize fast learning rates. By re-interpreting properness from the point of view of differential geometry, we provide a simple geometric characterization of mixability for the binary and multi-class cases: a proper loss function ℓ is η-mixable if and only if the superpredition set textrmspr(η ℓ) of the scaled loss function η ℓ slides freely inside the superprediction set textrmspr(ℓlog) of the log loss ℓlog, under fairly general assumptions on the differentiability of ℓ. Our approach provides a way to treat some concepts concerning loss functions (like properness) in a ''coordinate-free'' manner and reconciles previous results obtained for mixable loss functions for the binary and the multi-class cases.