做 LLM 训练调参的团队会关心——嵌入层学习率是 μP 优势的关键，直接放大 SP 的嵌入层学习率就能获得类似效果，值得在实验中验证。
本文提出一个框架，通过三个指标量化超参数迁移效果：缩放律拟合质量、外推鲁棒性、参数化导致的渐近损失惩罚。研究发现，μP 相比标准参数化（SP）在 AdamW 训练中的优势，主要源于嵌入层学习率的最大化。SP 中嵌入层学习率是瓶颈，导致训练不稳定；将其按宽度因子放大以匹配 μP 可显著平滑训练并改善迁移。此外，权重衰减改善缩放律拟合，但在固定 token-per-parameter 设置下会损害外推鲁棒性。
Quantifying Hyperparameter Transfer and the Importance of Embedding Layer Learning Rate
Hyperparameter transfer allows extrapolating optimal optimization hyperparameters from small to large scales, making it critical for training large language models (LLMs). This is done either by fitting a scaling law to the hyperparameters or by a judicious choice of parameterization, such as Maximal Update ($μ$P), that renders optimal hyperparameters approximately scale invariant. In this paper, we first develop a framework to quantify hyperparameter transfer through three metrics: (1) the quality of the scaling law fit, (2) the robustness to extrapolation errors, and (3) the asymptotic loss penalty due to choice of parameterization. Next, we investigate through a comprehensive series of ablations why $μ$P appears to offer high-quality learning rate transfer relative to standard parameterization (SP), as existing theory is inadequate. We find that the overwhelming benefit of $μ$P relative to SP when training with AdamW arises simply from maximizing the learning rate of the embedding layer. In SP, the embedding layer learning rate acts as a bottleneck that induces training instabilities; increasing it by a factor of width to match $μ$P dramatically smooths out training while improving hyperparameter transfer. We also find that weight decay improves the scaling law fits, while, in the fixed token-per-parameter setting, it hurts the robustness of the extrapolation.