# LegendreTron: Uprising Proper Multiclass Loss Learning

arXiv: https://arxiv.org/abs/2301.11695 · HTML: https://mlsanity.com/paper/2301.11695/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 48 references that are in the map.
- First posted on arXiv 27 Jan 2023.

## Builds on (most cited first, 14 of 14)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [A Tunable Loss Function for Robust Classification: Calibration, Landscape, and Generalization](https://mlsanity.com/paper/1906.02314/) (2019)
- [Learning Surrogate Losses](https://mlsanity.com/paper/1905.10108/) (2019)
- [Being Properly Improper](https://mlsanity.com/paper/2106.09920/) (2021)
- [SILVar: Single Index Latent Variable Models](https://mlsanity.com/paper/1705.03536/) (2017)
- [Supervised Learning: No Loss No Cry](https://mlsanity.com/paper/2002.03555/) (2020)
- [A Unified Framework of Surrogate Loss by Refactoring and Interpolation](https://mlsanity.com/paper/2007.13870/) (2020)
- [Monte Carlo Information Geometry: The dually flat case](https://mlsanity.com/paper/1803.07225/) (2018)
- [All your loss are belong to Bayes](https://mlsanity.com/paper/2006.04633/) (2020)
- [Learning Effective Loss Functions Efficiently](https://mlsanity.com/paper/1907.00103/) (2019)

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

- [Brenier Isotonic Regression](https://mlsanity.com/paper/2603.10452/) (2026)
- [Structured Proper Loss Geometries for Multiclass Classification: Theory and Controlled Empirical Evaluation](https://mlsanity.com/paper/2606.29471/) (2026)

## Abstract (arXiv)

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as properness, which asserts that Bayes' rule is optimal. Recent works have sought to learn losses and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps ℝ to [0,1] to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between ℝC-1 and the projected probability simplex tildeDeltaC-1 by using monotonicity of gradients of convex functions. We present sc LegendreTron as a novel and practical method that jointly learns proper canonical losses and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a t-test at 99% significance on all datasets with greater than 10 classes.