# Effective Learning Rate Governs Loss Dynamics in Language Model Pretraining

arXiv: https://arxiv.org/abs/2608.24814 · HTML: https://mlsanity.com/paper/2608.24814/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 44 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 30 of 37)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Query-Key Normalization for Transformers](https://mlsanity.com/paper/2010.04245/) (2020)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [Kimi Linear: An Expressive, Efficient Attention Architecture](https://mlsanity.com/paper/2510.26692/) (2025)
- [Small-scale proxies for large-scale Transformer training instabilities](https://mlsanity.com/paper/2309.14322/) (2023)
- [L2 Regularization versus Batch and Weight Normalization](https://mlsanity.com/paper/1706.05350/) (2017)
- [An Exponential Learning Rate Schedule for Deep Learning](https://mlsanity.com/paper/1910.07454/) (2019)
- [Three Mechanisms of Weight Decay Regularization](https://mlsanity.com/paper/1810.12281/) (2018)
- [Why Do We Need Weight Decay in Modern Deep Learning?](https://mlsanity.com/paper/2310.04415/) (2023)
- [Norm matters: efficient and accurate normalization schemes in deep networks](https://mlsanity.com/paper/1803.01814/) (2018)
- [Reconciling Modern Deep Learning with Traditional Optimization Analyses: The Intrinsic Learning Rate](https://mlsanity.com/paper/2010.02916/) (2020)
- [Rotational Equilibrium: How Weight Decay Balances Learning Across Neural Networks](https://mlsanity.com/paper/2305.17212/) (2023)
- [Controlled LLM Training on Spectral Sphere](https://mlsanity.com/paper/2601.08393/) (2026)
- [The Sharpness Disparity Principle in Transformers for Accelerating Language Model Pre-Training](https://mlsanity.com/paper/2502.19002/) (2025)
- [Functional Scaling Laws in Kernel Regression: Loss Dynamics and Learning Rate Schedules](https://mlsanity.com/paper/2509.19189/) (2025)
- [Weight Decay may matter more than muP for Learning Rate Transfer in Practice](https://mlsanity.com/paper/2510.19093/) (2025)
- [Training Scale-Invariant Neural Networks on the Sphere Can Happen in Three Regimes](https://mlsanity.com/paper/2209.03695/) (2022)
- [Fantastic Pretraining Optimizers and Where to Find Them II: Hyperball Optimization](https://mlsanity.com/paper/2606.16899/) (2026)
- [Cautious Weight Decay](https://mlsanity.com/paper/2510.12402/) (2025)
- [GradPower: Powering Gradients for Faster Language Model Pre-Training](https://mlsanity.com/paper/2505.24275/) (2025)

## Built on by (most cited first, 3 of 3)

- [Curvature-Conditioned Multiscale Momentum with Sphere Constraints for LLM Pretraining](https://mlsanity.com/paper/2608.28442/) (2026)
- [HyperTransfer: Understanding the Equivalence between Base Optimizer and Hyperball](https://mlsanity.com/paper/2609.07017/) (2026)
- [Understanding the Weight Averaging Mechanism in LLM Training for Post-Training Quantization](https://mlsanity.com/paper/2610.05329/) (2026)

## Abstract (arXiv)

We uncover ELR collapse in language model pretraining: learning rate (LR) and parameter norm govern loss dynamics primarily through their ratio, the effective learning rate (ELR). When ELR is matched across runs, their loss trajectories collapse throughout training despite substantially different LRs and parameter norms. Across optimizers, architectures, datasets, and model scales, mean collapse errors are typically a few x 10^-3, below the seed-to-seed variation measured in a representative configuration. Systematic ablations identify normalization design and the timescale of LR-norm variation as key determinants of collapse precision. Controlled interventions further show that weight decay and Hyperball shape loss dynamics primarily through the ELR schedules they induce. Replacing LR with ELR enables a fitted functional scaling law (FSL) to transfer across norm-control methods. The resulting ELR-based FSL also explains delayed acceleration, a recurring effect of norm control. Together, these results establish ELR as a common coordinate linking LR scheduling, norm control, and loss dynamics.