Calm Composite Losses: Being Improper Yet Proper Composite

Han Bao, Nontawat Charoenphakdee
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:2800-2808, 2025.

Abstract

Strict proper losses are fundamental loss functions inducing classifiers capable of estimating class probabilities. While practitioners have devised many loss functions, their properness is often unverified. In this paper, we identify several losses as improper, calling into question the validity of class probability estimates derived from their simplex-projected outputs. Nevertheless, we show that these losses are strictly proper composite with appropriate link functions, allowing predictions to be mapped into true class probabilities. We invent the calmness condition, which we prove suffices to identify that a loss has a strictly proper composite representation, and provide the general form of the inverse link. To further understand proper composite losses, we explore proper composite losses through the framework of property elicitation, revealing a connection between inverse link functions and Bregman projections. Numerical simulations are provided to demonstrate the behavior of proper composite losses and the effectiveness of the inverse link function.

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-bao25b, title = {Calm Composite Losses: Being Improper Yet Proper Composite}, author = {Bao, Han and Charoenphakdee, Nontawat}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {2800--2808}, year = {2025}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, volume = {258}, series = {Proceedings of Machine Learning Research}, month = {03--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v258/main/assets/bao25b/bao25b.pdf}, url = {https://proceedings.mlr.press/v258/bao25b.html}, abstract = {Strict proper losses are fundamental loss functions inducing classifiers capable of estimating class probabilities. While practitioners have devised many loss functions, their properness is often unverified. In this paper, we identify several losses as improper, calling into question the validity of class probability estimates derived from their simplex-projected outputs. Nevertheless, we show that these losses are strictly proper composite with appropriate link functions, allowing predictions to be mapped into true class probabilities. We invent the calmness condition, which we prove suffices to identify that a loss has a strictly proper composite representation, and provide the general form of the inverse link. To further understand proper composite losses, we explore proper composite losses through the framework of property elicitation, revealing a connection between inverse link functions and Bregman projections. Numerical simulations are provided to demonstrate the behavior of proper composite losses and the effectiveness of the inverse link function.} }
Endnote
%0 Conference Paper %T Calm Composite Losses: Being Improper Yet Proper Composite %A Han Bao %A Nontawat Charoenphakdee %B Proceedings of The 28th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2025 %E Yingzhen Li %E Stephan Mandt %E Shipra Agrawal %E Emtiyaz Khan %F pmlr-v258-bao25b %I PMLR %P 2800--2808 %U https://proceedings.mlr.press/v258/bao25b.html %V 258 %X Strict proper losses are fundamental loss functions inducing classifiers capable of estimating class probabilities. While practitioners have devised many loss functions, their properness is often unverified. In this paper, we identify several losses as improper, calling into question the validity of class probability estimates derived from their simplex-projected outputs. Nevertheless, we show that these losses are strictly proper composite with appropriate link functions, allowing predictions to be mapped into true class probabilities. We invent the calmness condition, which we prove suffices to identify that a loss has a strictly proper composite representation, and provide the general form of the inverse link. To further understand proper composite losses, we explore proper composite losses through the framework of property elicitation, revealing a connection between inverse link functions and Bregman projections. Numerical simulations are provided to demonstrate the behavior of proper composite losses and the effectiveness of the inverse link function.
APA
Bao, H. & Charoenphakdee, N.. (2025). Calm Composite Losses: Being Improper Yet Proper Composite. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:2800-2808 Available from https://proceedings.mlr.press/v258/bao25b.html.

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