Low-Dimension-to-High-Dimension Generalization and Its Implications for Length Generalization

Yang Chen, Long Yang, Yitao Liang, Zhouchen Lin
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:9566-9589, 2025.

Abstract

Low-Dimension-to-High-Dimension (LDHD) generalization, a subset of Out-of-Distribution (OOD) generalization, involves training on a low-dimensional subspace and testing in a high-dimensional space. Assuming instances are generated from latent variables reflecting problem scale, LDHD generalization captures the inherent scaling challenge of length generalization. We theoretically show that LDHD generalization is unattainable without appropriate inductive bias. Focusing on Boolean functions, we demonstrate that different architectures trained with (S)GD converge to min-degree interpolators w.r.t. different linearly independent sets, achieving LDHD generalization only when the target function aligns with this bias. From the perspective of LDHD generalization for length generalization, we explain the success of CoT in restructuring latent space for improved LDHD generalization. We further propose a principle for designing position embeddings to address both LDHD generalization and data format nuisances separately. Following the principle, we introduce RPE-Square, a novel embedding that enhances RPE to better handle data formats.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-chen25cb, title = {Low-Dimension-to-High-Dimension Generalization and Its Implications for Length Generalization}, author = {Chen, Yang and Yang, Long and Liang, Yitao and Lin, Zhouchen}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {9566--9589}, 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/chen25cb/chen25cb.pdf}, url = {https://proceedings.mlr.press/v267/chen25cb.html}, abstract = {Low-Dimension-to-High-Dimension (LDHD) generalization, a subset of Out-of-Distribution (OOD) generalization, involves training on a low-dimensional subspace and testing in a high-dimensional space. Assuming instances are generated from latent variables reflecting problem scale, LDHD generalization captures the inherent scaling challenge of length generalization. We theoretically show that LDHD generalization is unattainable without appropriate inductive bias. Focusing on Boolean functions, we demonstrate that different architectures trained with (S)GD converge to min-degree interpolators w.r.t. different linearly independent sets, achieving LDHD generalization only when the target function aligns with this bias. From the perspective of LDHD generalization for length generalization, we explain the success of CoT in restructuring latent space for improved LDHD generalization. We further propose a principle for designing position embeddings to address both LDHD generalization and data format nuisances separately. Following the principle, we introduce RPE-Square, a novel embedding that enhances RPE to better handle data formats.} }
Endnote
%0 Conference Paper %T Low-Dimension-to-High-Dimension Generalization and Its Implications for Length Generalization %A Yang Chen %A Long Yang %A Yitao Liang %A Zhouchen Lin %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-chen25cb %I PMLR %P 9566--9589 %U https://proceedings.mlr.press/v267/chen25cb.html %V 267 %X Low-Dimension-to-High-Dimension (LDHD) generalization, a subset of Out-of-Distribution (OOD) generalization, involves training on a low-dimensional subspace and testing in a high-dimensional space. Assuming instances are generated from latent variables reflecting problem scale, LDHD generalization captures the inherent scaling challenge of length generalization. We theoretically show that LDHD generalization is unattainable without appropriate inductive bias. Focusing on Boolean functions, we demonstrate that different architectures trained with (S)GD converge to min-degree interpolators w.r.t. different linearly independent sets, achieving LDHD generalization only when the target function aligns with this bias. From the perspective of LDHD generalization for length generalization, we explain the success of CoT in restructuring latent space for improved LDHD generalization. We further propose a principle for designing position embeddings to address both LDHD generalization and data format nuisances separately. Following the principle, we introduce RPE-Square, a novel embedding that enhances RPE to better handle data formats.
APA
Chen, Y., Yang, L., Liang, Y. & Lin, Z.. (2025). Low-Dimension-to-High-Dimension Generalization and Its Implications for Length Generalization. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:9566-9589 Available from https://proceedings.mlr.press/v267/chen25cb.html.

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