# The Geometry and Calculus of Losses

arXiv: https://arxiv.org/abs/2209.00238 · HTML: https://mlsanity.com/paper/2209.00238/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 114 references that are in the map.
- First posted on arXiv 1 Sep 2022.

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

- [Constant Regret, Generalized Mixability, and Mirror Descent](https://mlsanity.com/paper/1802.06965/) (2018)
- [Information Processing Equalities and the Information-Risk Bridge](https://mlsanity.com/paper/2207.11987/) (2022)
- [Foundations of gauge and perspective duality](https://mlsanity.com/paper/1702.08649/) (2017)
- [Exp-Concavity of Proper Composite Losses](https://mlsanity.com/paper/1805.07737/) (2018)
- [A note on the K-epigraph](https://mlsanity.com/paper/2107.00117/) (2021)

## Built on by (most cited first, 7 of 7)

- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [Scoring Rules and Calibration for Imprecise Probabilities](https://mlsanity.com/paper/2410.23001/) (2024)
- [Proper losses regret at least 1/2-order](https://mlsanity.com/paper/2407.10417/) (2024)
- [The Geometry of Mixability](https://mlsanity.com/paper/2302.11905/) (2023)
- [An Axiomatic Approach to Loss Aggregation and an Adapted Aggregating Algorithm](https://mlsanity.com/paper/2406.02292/) (2024)
- [Consistency Conditions for Differentiable Surrogate Losses](https://mlsanity.com/paper/2505.13760/) (2025)
- [Comparing Corrupted Constrained Learning Problems](https://mlsanity.com/paper/2608.25745/) (2026)

## Abstract (arXiv)

Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are binary and multiclass classification and class probability estimation. Central to their definition is the choice of loss function, which is the means by which the quality of a solution is evaluated. In this paper we systematically develop the theory of loss functions for such problems from a novel perspective whose basic ingredients are convex sets with a particular structure. The loss function is defined as the subgradient of the support function of the convex set. It is consequently automatically proper (calibrated for probability estimation). This perspective provides three novel opportunities. It enables the development of a fundamental relationship between losses and (anti)-norms that appears to have not been noticed before. Second, it enables the development of a calculus of losses induced by the calculus of convex sets which allows the interpolation between different losses, and thus is a potential useful design tool for tailoring losses to particular problems. In doing this we build upon, and considerably extend existing results on M-sums of convex sets. Third, the perspective leads to a natural theory of ``polar'' loss functions, which are derived from the polar dual of the convex set defining the loss, and which form a natural universal substitution function for Vovk's aggregating algorithm.