Loss Functions and Operators Generated by f-Divergences

Vincent Roulet, Tianlin Liu, Nino Vieillard, Michael Eli Sander, Mathieu Blondel
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:52110-52138, 2025.

Abstract

The logistic loss (a.k.a. cross-entropy loss) is one of the most popular loss functions used for multiclass classification. It is also the loss function of choice for next-token prediction in language modeling. It is associated with the Kullback-Leibler (KL) divergence and the softargmax operator. In this work, we propose to construct new convex loss functions based on $f$-divergences. Our loss functions generalize the logistic loss in two directions: i) by replacing the KL divergence with $f$-divergences and ii) by allowing non-uniform reference measures. We instantiate our framework for numerous $f$-divergences, recovering existing losses and creating new ones. By analogy with the logistic loss, the loss function generated by an $f$-divergence is associated with an operator, that we dub $f$-softargmax. We derive a novel parallelizable bisection algorithm for computing the $f$-softargmax associated with any $f$-divergence. On the empirical side, one of the goals of this paper is to determine the effectiveness of loss functions beyond the classical cross-entropy in a language model setting, including on pre-training, post-training (SFT) and distillation. We show that the loss function generated by the $\alpha$-divergence (which is equivalent to Tsallis $\alpha$-negentropy in the case of unit reference measures) with $\alpha=1.5$ performs well across several tasks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-roulet25a, title = {Loss Functions and Operators Generated by f-Divergences}, author = {Roulet, Vincent and Liu, Tianlin and Vieillard, Nino and Sander, Michael Eli and Blondel, Mathieu}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {52110--52138}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/roulet25a/roulet25a.pdf}, url = {https://proceedings.mlr.press/v267/roulet25a.html}, abstract = {The logistic loss (a.k.a. cross-entropy loss) is one of the most popular loss functions used for multiclass classification. It is also the loss function of choice for next-token prediction in language modeling. It is associated with the Kullback-Leibler (KL) divergence and the softargmax operator. In this work, we propose to construct new convex loss functions based on $f$-divergences. Our loss functions generalize the logistic loss in two directions: i) by replacing the KL divergence with $f$-divergences and ii) by allowing non-uniform reference measures. We instantiate our framework for numerous $f$-divergences, recovering existing losses and creating new ones. By analogy with the logistic loss, the loss function generated by an $f$-divergence is associated with an operator, that we dub $f$-softargmax. We derive a novel parallelizable bisection algorithm for computing the $f$-softargmax associated with any $f$-divergence. On the empirical side, one of the goals of this paper is to determine the effectiveness of loss functions beyond the classical cross-entropy in a language model setting, including on pre-training, post-training (SFT) and distillation. We show that the loss function generated by the $\alpha$-divergence (which is equivalent to Tsallis $\alpha$-negentropy in the case of unit reference measures) with $\alpha=1.5$ performs well across several tasks.} }
Endnote
%0 Conference Paper %T Loss Functions and Operators Generated by f-Divergences %A Vincent Roulet %A Tianlin Liu %A Nino Vieillard %A Michael Eli Sander %A Mathieu Blondel %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-roulet25a %I PMLR %P 52110--52138 %U https://proceedings.mlr.press/v267/roulet25a.html %V 267 %X The logistic loss (a.k.a. cross-entropy loss) is one of the most popular loss functions used for multiclass classification. It is also the loss function of choice for next-token prediction in language modeling. It is associated with the Kullback-Leibler (KL) divergence and the softargmax operator. In this work, we propose to construct new convex loss functions based on $f$-divergences. Our loss functions generalize the logistic loss in two directions: i) by replacing the KL divergence with $f$-divergences and ii) by allowing non-uniform reference measures. We instantiate our framework for numerous $f$-divergences, recovering existing losses and creating new ones. By analogy with the logistic loss, the loss function generated by an $f$-divergence is associated with an operator, that we dub $f$-softargmax. We derive a novel parallelizable bisection algorithm for computing the $f$-softargmax associated with any $f$-divergence. On the empirical side, one of the goals of this paper is to determine the effectiveness of loss functions beyond the classical cross-entropy in a language model setting, including on pre-training, post-training (SFT) and distillation. We show that the loss function generated by the $\alpha$-divergence (which is equivalent to Tsallis $\alpha$-negentropy in the case of unit reference measures) with $\alpha=1.5$ performs well across several tasks.
APA
Roulet, V., Liu, T., Vieillard, N., Sander, M.E. & Blondel, M.. (2025). Loss Functions and Operators Generated by f-Divergences. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:52110-52138 Available from https://proceedings.mlr.press/v267/roulet25a.html.

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