Geometric Signatures of Compositionality Across a Language Model's Lifetime (2410.01444v5)

Abstract: By virtue of linguistic compositionality, few syntactic rules and a finite lexicon can generate an unbounded number of sentences. That is, language, though seemingly high-dimensional, can be explained using relatively few degrees of freedom. An open question is whether contemporary LLMs (LMs) reflect the intrinsic simplicity of language that is enabled by compositionality. We take a geometric view of this problem by relating the degree of compositionality in a dataset to the intrinsic dimension (ID) of its representations under an LM, a measure of feature complexity. We find not only that the degree of dataset compositionality is reflected in representations' ID, but that the relationship between compositionality and geometric complexity arises due to learned linguistic features over training. Finally, our analyses reveal a striking contrast between nonlinear and linear dimensionality, showing they respectively encode semantic and superficial aspects of linguistic composition.

• The paper reveals that language models encode linguistic compositionality by representing information on low-dimensional nonlinear manifolds while linear dimensionality scales with model size.
• The paper identifies a pivotal phase transition in training where adjustments in representational geometry lead to a sharp increase in task performance.
• The paper demonstrates that nonlinear intrinsic dimensionality aligns with semantic compositionality, whereas linear measures capture the combinatorial complexity of grammatical forms.

Geometric Signatures of Compositionality Across a LLM's Lifetime

The paper "Geometric Signatures of Compositionality Across a LLM's Lifetime" explores the complex relationship between linguistic compositionality and the geometric features of representations within neural LLMs (LMs). Compositionality is a fundamental property that allows human language to derive infinite meanings from a finite set of elements and syntactic rules. This paper investigates how LLMs, particularly those based on transformer architectures, encode such compositional structures over the course of their training.

Core Contributions and Findings

The authors adopt a geometric perspective on compositionality, exploring how the intrinsic dimensionality (ID) of representation manifolds reflects the compositional complexity of input datasets. They focus on the correlation between dataset compositionality and two measures of feature complexity: nonlinear intrinsic dimensionality and linear effective dimensionality. The key contributions of the paper can be summarized as follows:

1. Dimensionality and Model Scale:
   • The paper reveals that LLMs represent linguistic information on low-dimensional nonlinear manifolds, even as linear dimensionality scales proportionately to the model size. This dichotomy underscores distinct roles in encoding superficial input complexity versus latent semantic features.
2. Dynamics of Feature Complexity:
   • The paper longitudinally tracks the evolution of LMs' representational geometry, identifying a phase transition marking the development of compositional understanding. Around a pivotal training point, a sharp increase in task performance corresponds to adjustments in representational dimensionality, illustrating a non-trivial transition in linguistic competency.
3. Form and Meaning Compositionality:
   • Evaluation of grammatical (sane) and agrammatical (shuffled) sequences clarifies how linear and nonlinear dimensionalities correspond to different aspects of compositionality. Nonlinear ID appears to correlate with semantic compositionality, whereas linear dimensionality captures form or combinatorial complexity. This dual encoding suggests nonlinear features are more aligned with semantic understanding.

Methodology

The research leverages a synthetic dataset composed using a controlled grammar with tunable compositionality. Through dynamic experiments on several Pythia model sizes, the authors quantify both formal and semantic compositionality. Intrinsic dimensionality is estimated using the TwoNN method, while linear dimensionality is assessed via Principal Component Analysis (PCA).

Significantly, the paper isolates various aspects of compositionality through the design of the synthetic dataset, allowing for precise control over the grammatical integrity and lexical distribution. This control enables the extraction of rigorous insights into how LMs process these compositions over different training phases.

Implications

The results from this research carry substantial implications for both theoretical understanding and practical application:

• Theoretical Implications:
  • The identification of intrinsic manifolds in language representations aligns with the manifold hypothesis, supporting the notion that high-dimensional linguistic data can be efficiently represented within low-dimensional structures.
  • The phase transition in feature complexity echoes findings in cognitive neuroscience, where representational geometry may indicate emergent cognition in artificial systems.
• Practical Applications:
  • The form-meaning dichotomy and the distinct roles of linear and nonlinear dimensional measures may guide the development of more interpretable and efficient LLMs.
  • The geometric approach could enhance model diagnostics, offering new avenues to evaluate and potentially improve their compositional generalization capabilities.

Future Directions

Further exploration is required to deepen the understanding of the causal contributions of nonlinear and linear dimensions in predictive coding. New research might explore the observed dual role of dimensional spaces in more sophisticated linguistic settings, potentially leading to innovations in model design and training strategies. The intersection of this work with cognitive modeling could also illuminate how artificial systems mirror human compositional reasoning.

In summary, the paper provides a detailed inquiry into the geometric intricacies underpinning linguistic compositionality in LLMs, presenting significant findings that bridge neural representation theory and practical AI enhancement.