Learning curves theory for hierarchically compositional data with power-law distributed features

Francesco Cagnetta, Hyunmo Kang, Matthieu Wyart
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:6149-6164, 2025.

Abstract

Recent theories suggest that Neural Scaling Laws arise whenever the task is linearly decomposed into units that are power-law distributed. Alternatively, scaling laws also emerge when data exhibit a hierarchically compositional structure, as is thought to occur in language and images. To unify these views, we consider classification and next-token prediction tasks based on probabilistic context-free grammars—probabilistic models that generate data via a hierarchy of production rules. For classification, we show that having power-law distributed production rules results in a power-law learning curve with an exponent depending on the rules’ distribution and a large multiplicative constant that depends on the hierarchical structure. By contrast, for next-token prediction, the distribution of production rules controls the fine details of the learning curve, but not the exponent describing the large-scale behaviour.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-cagnetta25a, title = {Learning curves theory for hierarchically compositional data with power-law distributed features}, author = {Cagnetta, Francesco and Kang, Hyunmo and Wyart, Matthieu}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {6149--6164}, 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/cagnetta25a/cagnetta25a.pdf}, url = {https://proceedings.mlr.press/v267/cagnetta25a.html}, abstract = {Recent theories suggest that Neural Scaling Laws arise whenever the task is linearly decomposed into units that are power-law distributed. Alternatively, scaling laws also emerge when data exhibit a hierarchically compositional structure, as is thought to occur in language and images. To unify these views, we consider classification and next-token prediction tasks based on probabilistic context-free grammars—probabilistic models that generate data via a hierarchy of production rules. For classification, we show that having power-law distributed production rules results in a power-law learning curve with an exponent depending on the rules’ distribution and a large multiplicative constant that depends on the hierarchical structure. By contrast, for next-token prediction, the distribution of production rules controls the fine details of the learning curve, but not the exponent describing the large-scale behaviour.} }
Endnote
%0 Conference Paper %T Learning curves theory for hierarchically compositional data with power-law distributed features %A Francesco Cagnetta %A Hyunmo Kang %A Matthieu Wyart %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-cagnetta25a %I PMLR %P 6149--6164 %U https://proceedings.mlr.press/v267/cagnetta25a.html %V 267 %X Recent theories suggest that Neural Scaling Laws arise whenever the task is linearly decomposed into units that are power-law distributed. Alternatively, scaling laws also emerge when data exhibit a hierarchically compositional structure, as is thought to occur in language and images. To unify these views, we consider classification and next-token prediction tasks based on probabilistic context-free grammars—probabilistic models that generate data via a hierarchy of production rules. For classification, we show that having power-law distributed production rules results in a power-law learning curve with an exponent depending on the rules’ distribution and a large multiplicative constant that depends on the hierarchical structure. By contrast, for next-token prediction, the distribution of production rules controls the fine details of the learning curve, but not the exponent describing the large-scale behaviour.
APA
Cagnetta, F., Kang, H. & Wyart, M.. (2025). Learning curves theory for hierarchically compositional data with power-law distributed features. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:6149-6164 Available from https://proceedings.mlr.press/v267/cagnetta25a.html.

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