Negligible in Size, Significant in Effect: On Scale Vectors in Large Language Models

2026-05-26

AI Summary

This work investigates the mechanism of lightweight yet critical learnable scaling vectors in large language models. Through theoretical analysis and large-scale pretraining experiments, it reveals for the first time that, within Pre-Norm architectures, these vectors primarily improve optimization dynamics via a self-amplifying preconditioning effect rather than enhancing model expressivity. The study further identifies divergent responses to weight decay between Input-Norm and Output-Norm layers. Building on these insights, three efficient improvements are proposed: heterogeneous branch design, optimized placement of linear mappings, and a magnitude-direction decoupled reparameterization. Evaluated across dense and mixture-of-experts models ranging from 0.12B to 2B parameters, the unified scaling strategy consistently reduces final training loss, enhances stability, and incurs negligible additional parameters or computational cost.

Abstract

Normalization layers in modern large language models (LLMs) consist of a deterministic normalization operation and a learnable scale vector. While the normalization operation has been extensively studied, the scale vector remains poorly understood despite its ubiquitous use. In this work, we present a systematic study of scale vectors in LLMs from the perspectives of expressivity, optimization, and architectural structure. First, we show empirically that although scale vectors constitute only a negligible fraction of model parameters, removing them substantially degrades LLM pre-training. Our theory further shows that, in Pre-Norm architectures, scale vectors do not increase expressivity; instead, they improve optimization through a self-amplifying preconditioning effect on subsequent linear mappings. Second, we investigate the role of weight decay for scale vectors. By distinguishing Input-Norm and Output-Norm layers, we theoretically show that weight decay is beneficial for the former but harmful for the latter, due to their distinct roles in optimization and expressivity. Third, motivated by this understanding, we propose three lightweight and complementary improvements to scale vectors: branch-specific heterogeneity, improved placement around linear mappings, and magnitude-direction reparameterization. Both theory and experiments show that each improvement yields consistent gains. Finally, we combine these improvements into a unified scale-vector strategy and evaluate it through extensive LLM pre-training experiments on dense and mixture-of-experts models ranging from 0.12B to 2B parameters, across multiple optimizers and learning rate schedules, under industrial-scale token budgets. The unified strategy consistently achieves lower terminal loss than well-tuned baselines and exhibits more favorable scaling behavior, while adding negligible parameter and computational overhead.

Problem

• scale vectors
• large language models
• normalization layers
• optimization
• expressivity

Innovation

• scale vectors
• preconditioning effect
• weight decay
• reparameterization
• large language models