# The Geometry of Mixability

## TL;DR

This paper offers a geometric perspective on mixable loss functions, characterizing their properties through differential geometry and superprediction sets, which unifies binary and multi-class cases.

## Contribution

It introduces a geometric characterization of mixability for proper loss functions using superprediction sets, providing a coordinate-free framework that unifies binary and multi-class scenarios.

## Key findings

- Superprediction sets slide freely inside the log loss superprediction set for mixability.
- The geometric approach applies under general differentiability assumptions.
- Reconciliation of previous results for binary and multi-class cases.

## Abstract

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 $\ell$ is $\eta$-mixable if and only if the superpredition set $\textrm{spr}(\eta \ell)$ of the scaled loss function $\eta \ell$ slides freely inside the superprediction set $\textrm{spr}(\ell_{\log})$ of the log loss $\ell_{\log}$, under fairly general assumptions on the differentiability of $\ell$. 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.