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

Yang Chen, Long Yang, Yitao Liang, Zhouchen Lin

TL;DR

The paper introduces the Low-Dimension-to-High-Dimension generalization ($LDHD$) framework to formalize length generalization as a structured OOD problem, proving a No-Free-Lunch limitation without suitable inductive bias. It analyzes how model architectures, notably Random Feature Models with Projection ($RFMP$) and Position-Only Linear Attention with Advice ($PLAA$), induce min-degree interpolators with respect to problem-specific bases, constraining LDHD generalization to targets aligned with these biases. The work then translates these insights into practical techniques for sequence models, including Chain-of-Thought as latent-space augmentation and principled position-embedding design, introducing the RPE-Square embedding that effectively handles data-format nuisances and enhances length generalization on tasks such as unaligned copy and URF addition. Overall, the LDHD perspective provides a principled path for designing inductive biases and embeddings to enable robust length generalization in reasoning and transformer-based models.

Abstract

Low-Dimension-to-High-Dimension (LDHD) generalization is a special case of Out-of-Distribution (OOD) generalization, where the training data are restricted to a low-dimensional subspace of the high-dimensional testing space. Assuming that each instance is generated from a latent variable and the dimension of the latent variable reflects the problem scale, the inherent scaling challenge in length generalization can be captured by the LDHD generalization in the latent space. We theoretically demonstrate that LDHD generalization is generally unattainable without exploiting prior knowledge to provide appropriate inductive bias. Specifically, we explore LDHD generalization in Boolean functions. We verify that different architectures trained with (S)GD converge to \emph{min-degree interpolators w.r.t. different independent sets}. LDHD generalization is achievable if and only if the target function coincides with this inductive bias. Applying the insights from LDHD generalization to length generalization, we explain the effectiveness of CoT as changing the structure latent space to enable better LDHD generalization. We also propose a principle for position embedding design to handle both the inherent LDHD generalization and the nuisances such as the data format. Following the principle, we propose a novel position embedding called RPE-Square that remedies the RPE for dealing with the data format nuisance.

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

TL;DR

The paper introduces the Low-Dimension-to-High-Dimension generalization () framework to formalize length generalization as a structured OOD problem, proving a No-Free-Lunch limitation without suitable inductive bias. It analyzes how model architectures, notably Random Feature Models with Projection () and Position-Only Linear Attention with Advice (), induce min-degree interpolators with respect to problem-specific bases, constraining LDHD generalization to targets aligned with these biases. The work then translates these insights into practical techniques for sequence models, including Chain-of-Thought as latent-space augmentation and principled position-embedding design, introducing the RPE-Square embedding that effectively handles data-format nuisances and enhances length generalization on tasks such as unaligned copy and URF addition. Overall, the LDHD perspective provides a principled path for designing inductive biases and embeddings to enable robust length generalization in reasoning and transformer-based models.

Abstract

Low-Dimension-to-High-Dimension (LDHD) generalization is a special case of Out-of-Distribution (OOD) generalization, where the training data are restricted to a low-dimensional subspace of the high-dimensional testing space. Assuming that each instance is generated from a latent variable and the dimension of the latent variable reflects the problem scale, the inherent scaling challenge in length generalization can be captured by the LDHD generalization in the latent space. We theoretically demonstrate that LDHD generalization is generally unattainable without exploiting prior knowledge to provide appropriate inductive bias. Specifically, we explore LDHD generalization in Boolean functions. We verify that different architectures trained with (S)GD converge to \emph{min-degree interpolators w.r.t. different independent sets}. LDHD generalization is achievable if and only if the target function coincides with this inductive bias. Applying the insights from LDHD generalization to length generalization, we explain the effectiveness of CoT as changing the structure latent space to enable better LDHD generalization. We also propose a principle for position embedding design to handle both the inherent LDHD generalization and the nuisances such as the data format. Following the principle, we propose a novel position embedding called RPE-Square that remedies the RPE for dealing with the data format nuisance.

Paper Structure

This paper contains 31 sections, 9 theorems, 78 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Our Main Works
  3. Notations
  4. Related Work
  5. LDHD Generalization
  6. Main Results
  7. Random Feature Model with Projection
  8. Further Implications
  9. Chain-of-Thought for Length Generalization
  10. Position Embeddings for Length Generalization
  11. Conclusion and Discussion
  12. Background on Boolean Analysis
  13. Detailed Explanation of PLAA
  14. Construction of PLAA
  15. Construction of PLAA with APE
  16. ...and 16 more sections

Key Result

Theorem 1

Suppose that the two sets ${\mathcal{X}}$ and ${\mathcal{Y}}$ are finite. For some $N > N_0$, consider two subsets ${\mathcal{X}}_{N_0}$, ${\mathcal{X}}_N$ of ${\mathcal{X}}$ such that ${\mathcal{X}}_{N_0}\subsetneq{\mathcal{X}}_N\subseteq{\mathcal{X}}$ and $\dim({\mathcal{X}}_{N_0})=N_0 < N=\dim({\

Figures (8)

  • Figure 1: Illustrative comparison of in-distribution generalization, typical OOD generalization, and LDHD generalization. (a) In-distribution generalization assumes identical training and testing distributions. (b) Typical OOD generalization involves a shift between training and testing distributions, which remain relatively "close" (e.g., sharing support or having small distributional distances). (c) LDHD generalization features a training distribution restricted to a low-dimensional subspace and a testing distribution on a high-dimensional space, often vastly different. While LDHD is a type of OOD generalization, its structured shift poses unique challenges, as training data provide no clues about the additional dimensions' contribution to the label.
  • Figure 2: Length generalization of Transformer with RPE and RPE-Square in the unaligned copy and the URF addition tasks. Unaligned Copy: Transformers with RPE (a) and RPE-Square (b) are trained on lengths 1–5 for 1000 steps and tested on lengths 1–10. While both models generalize in-distribution, only RPE-Square achieves out-of-distribution generalization for lengths 6–10. URF Addition: Models are trained on URF 4-addition and tested on URF 5-addition. (c) Both the models are trained for $20000$ steps. The comparison result shows that the RPE fails while the RPE-Square succeeds in achieving length generalization. (d) RPE-Square trained for 200,000 steps achieves nearly perfect accuracy, with digitwise accuracy shown for each $z_k$.
  • Figure 3: Addition. The models are trained on URF 4-addition and tested on URF 5-addition
  • Figure 4: Copy. The models are trained on scales 1-5 and tested on scales 1-10.
  • Figure 5: ParityCoT. The models are trained on scales 1-5 and tested on scales 1-10. An instance is like "$\mathtt{[BOS] x_0 \dots x_{n-1} = y_0 \dots y_{n-1} [EOS]}$", where $y_0=x_0$ and $y_k = x_k \oplus y_{k-1}$ for $k=1,\dots, n-1$.
  • ...and 3 more figures

Theorems & Definitions (25)

  • Example 1
  • Example 2: Addition
  • Definition 1: Low-Dimension-to-High-Dimension Generalization
  • Theorem 1: No-Free-Lunch Theorem of LDHD Generalization
  • Remark 1
  • Definition 2: Degree Profile w.r.t. Linearly Independent Set ${\mathcal{B}}$
  • Definition 3: Min-Degree Interpolator w.r.t. Linearly Independent Set ${\mathcal{B}}$
  • Definition 4: Random Feature Model with Projection
  • Theorem 2
  • Corollary 1
  • ...and 15 more