# Multiclass Classification Calibration Functions

arXiv: https://arxiv.org/abs/1609.06385 · HTML: https://mlsanity.com/paper/1609.06385/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Sep 2016.

## Built on by (most cited first, 13 of 13)

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- [Deep Minimax Classifiers for Imbalanced Datasets with a Small Number of Minority Samples](https://mlsanity.com/paper/2502.16948/) (2025)
- [Channel-Preserving Representation Alignment for EEG-to-Music Reconstruction](https://mlsanity.com/paper/2606.04040/) (2026)

## Abstract (arXiv)

In this paper we refine the process of computing calibration functions for a number of multiclass classification surrogate losses. Calibration functions are a powerful tool for easily converting bounds for the surrogate risk (which can be computed through well-known methods) into bounds for the true risk, the probability of making a mistake. They are particularly suitable in non-parametric settings, where the approximation error can be controlled, and provide tighter bounds than the common technique of upper-bounding the 0-1 loss by the surrogate loss. The abstract nature of the more sophisticated existing calibration function results requires calibration functions to be explicitly derived on a case-by-case basis, requiring repeated efforts whenever bounds for a new surrogate loss are required. We devise a streamlined analysis that simplifies the process of deriving calibration functions for a large number of surrogate losses that have been proposed in the literature. The effort of deriving calibration functions is then surmised in verifying, for a chosen surrogate loss, a small number of conditions that we introduce. As case studies, we recover existing calibration functions for the well-known loss of Lee et al. (2004), and also provide novel calibration functions for well-known losses, including the one-versus-all loss and the logistic regression loss, plus a number of other losses that have been shown to be classification-calibrated in the past, but for which no calibration function had been derived.