Generalization on the Unseen, Logic Reasoning and Degree Curriculum

Emmanuel Abbe, Samy Bengio, Aryo Lotfi, Kevin Rizk
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:31-60, 2023.

Abstract

This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an ’extrapolating’ or ’reasoning’ learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky min-degree solutions. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-abbe23a, title = {Generalization on the Unseen, Logic Reasoning and Degree Curriculum}, author = {Abbe, Emmanuel and Bengio, Samy and Lotfi, Aryo and Rizk, Kevin}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {31--60}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/abbe23a/abbe23a.pdf}, url = {https://proceedings.mlr.press/v202/abbe23a.html}, abstract = {This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an ’extrapolating’ or ’reasoning’ learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky min-degree solutions. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.} }
Endnote
%0 Conference Paper %T Generalization on the Unseen, Logic Reasoning and Degree Curriculum %A Emmanuel Abbe %A Samy Bengio %A Aryo Lotfi %A Kevin Rizk %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-abbe23a %I PMLR %P 31--60 %U https://proceedings.mlr.press/v202/abbe23a.html %V 202 %X This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an ’extrapolating’ or ’reasoning’ learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky min-degree solutions. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.
APA
Abbe, E., Bengio, S., Lotfi, A. & Rizk, K.. (2023). Generalization on the Unseen, Logic Reasoning and Degree Curriculum. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:31-60 Available from https://proceedings.mlr.press/v202/abbe23a.html.

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