As Language Models Scale, Low-order Linear Depth Dynamics Emerge

TL;DR

This paper demonstrates that as language models grow larger, their complex layerwise behaviors can be accurately approximated by simple low-dimensional linear models, revealing new insights into their internal dynamics and control.

Contribution

It introduces a low-order linear surrogate that accurately captures layerwise sensitivities in large language models and uncovers a scaling principle for their depth dynamics.

Findings

A 32-dimensional linear surrogate reproduces GPT-2-large sensitivities.

Agreement with the full model improves with size across GPT-2 models.

Linear surrogates enable energy-efficient multi-layer interventions.

Abstract

Large language models are often viewed as high-dimensional nonlinear systems and treated as black boxes. Here, we show that transformer depth dynamics admit accurate low-order linear surrogates within context. Across tasks including toxicity, irony, hate speech and sentiment, a 32-dimensional linear surrogate reproduces the layerwise sensitivity profile of GPT-2-large with near-perfect agreement, capturing how the final output shifts under additive injections at each layer. We then uncover a surprising scaling principle: for a fixed-order linear surrogate, agreement with the full model improves monotonically with model size across the GPT-2 family. This linear surrogate also enables principled multi-layer interventions that require less energy than standard heuristic schedules when applied to the full model. Together, our results reveal that as language models scale, low-order linear depth dynamics emerge within contexts, offering a systems-theoretic foundation for analyzing and controlling them.