# Consistency Conditions for Differentiable Surrogate Losses

arXiv: https://arxiv.org/abs/2505.13760 · HTML: https://mlsanity.com/paper/2505.13760/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 29 references that are in the map.
- First posted on arXiv 19 May 2025.

## Builds on (most cited first, 10 of 10)

- [On the Consistency of Ordinal Regression Methods](https://mlsanity.com/paper/1408.2327/) (2014)
- [Convex Calibration Dimension for Multiclass Loss Matrices](https://mlsanity.com/paper/1408.2764/) (2014)
- [An Embedding Framework for Consistent Polyhedral Surrogates](https://mlsanity.com/paper/1907.07330/) (2019)
- [Structured Prediction with Projection Oracles](https://mlsanity.com/paper/1910.11369/) (2019)
- [An Embedding Framework for the Design and Analysis of Consistent Polyhedral Surrogates](https://mlsanity.com/paper/2206.14707/) (2022)
- [Weston-Watkins Hinge Loss and Ordered Partitions](https://mlsanity.com/paper/2006.07346/) (2020)
- [Consistent Polyhedral Surrogates for Top-k Classification and Variants](https://mlsanity.com/paper/2207.08873/) (2022)
- [Consistent Algorithms for Multiclass Classification with a Reject Option](https://mlsanity.com/paper/1505.04137/) (2015)
- [The Structured Abstain Problem and the Lov'asz Hinge](https://mlsanity.com/paper/2203.08645/) (2022)
- [The Geometry and Calculus of Losses](https://mlsanity.com/paper/2209.00238/) (2022)

## Built on by (most cited first, 1 of 1)

- [Smoothed Elicitation Complexity for Approximate Gamma-calibration of Discrete Classification Tasks](https://mlsanity.com/paper/2605.23017/) (2026)

## Abstract (arXiv)

The statistical consistency of surrogate losses for discrete prediction tasks is often checked via the condition of calibration. However, directly verifying calibration can be arduous. Recent work shows that for polyhedral surrogates, a less arduous condition, indirect elicitation (IE), is still equivalent to calibration. We give the first results of this type for non-polyhedral surrogates, specifically the class of convex differentiable losses. We first prove that under mild conditions, IE and calibration are equivalent for one-dimensional losses in this class. We construct a counter-example that shows that this equivalence fails in higher dimensions. This motivates the introduction of strong IE, a strengthened form of IE that is equally easy to verify. We establish that strong IE implies calibration for differentiable surrogates and is both necessary and sufficient for strongly convex, differentiable surrogates. Finally, we apply these results to a range of problems to demonstrate the power of IE and strong IE for designing and analyzing consistent differentiable surrogates.