Disentangling Geometry, Performance, and Training in Language Models

TL;DR

This paper systematically investigates how the geometry of Transformer weights, especially the unembedding matrix's effective rank, relates to model performance, revealing that geometry mainly reflects training choices rather than performance.

Contribution

The study provides a comprehensive analysis of the relationship between model performance and unembedding matrix geometry across various training conditions, challenging prior assumptions.

Findings

High effective rank often correlates with better performance, but not always.

Low effective rank does not necessarily cause performance degradation.

Pre-training hyperparameters significantly influence the model's geometry and performance.

Abstract

Geometric properties of Transformer weights, particularly the unembedding matrix, have been widely useful in language model interpretability research. Yet, their utility for estimating downstream performance remains unclear. In this work, we systematically investigate the relationship between model performance and the unembedding matrix geometry, particularly its effective rank. Our experiments, involving a suite of 108 OLMo-style language models trained under controlled variation, reveal several key findings. While the best-performing models often exhibit a high effective rank, this trend is not universal across tasks and training setups. Contrary to prior work, we find that low effective rank does not cause late-stage performance degradation in small models, but instead co-occurs with it; we find adversarial cases where low-rank models do not exhibit saturation. Moreover, we show that effective rank is strongly influenced by pre-training hyperparameters, such as batch size and weight decay, which in-turn affect the model's performance. Lastly, extending our analysis to other geometric metrics and final-layer representation, we find that these metrics are largely aligned, but none can reliably predict downstream performance. Overall, our findings suggest that the model's geometry, as captured by existing metrics, primarily reflects training choices rather than performance.