# On the Nonlinearity of Learning Rate Scaling for LLM Training

arXiv: https://arxiv.org/abs/2606.29158 · HTML: https://mlsanity.com/paper/2606.29158/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 48 references that are in the map.
- First posted on arXiv 28 Jun 2026.

## Builds on (most cited first, 30 of 41)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [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)
- [DeepSeek LLM: Scaling Open-Source Language Models with Longtermism](https://mlsanity.com/paper/2401.02954/) (2024)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [An Empirical Model of Large-Batch Training](https://mlsanity.com/paper/1812.06162/) (2018)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [L2 Regularization versus Batch and Weight Normalization](https://mlsanity.com/paper/1706.05350/) (2017)
- [On the SDEs and Scaling Rules for Adaptive Gradient Algorithms](https://mlsanity.com/paper/2205.10287/) (2022)
- [Resolving Discrepancies in Compute-Optimal Scaling of Language Models](https://mlsanity.com/paper/2406.19146/) (2024)
- [Don't be lazy: CompleteP enables compute-efficient deep transformers](https://mlsanity.com/paper/2505.01618/) (2025)
- [Scaling Exponents Across Parameterizations and Optimizers](https://mlsanity.com/paper/2407.05872/) (2024)
- [How Does Critical Batch Size Scale in Pre-training?](https://mlsanity.com/paper/2410.21676/) (2024)
- [nGPT: Normalized Transformer with Representation Learning on the Hypersphere](https://mlsanity.com/paper/2410.01131/) (2024)
- [Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit](https://mlsanity.com/paper/2309.16620/) (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)
- [How to set AdamW's weight decay as you scale model and dataset size](https://mlsanity.com/paper/2405.13698/) (2024)
- [Power Lines: Scaling Laws for Weight Decay and Batch Size in LLM Pre-training](https://mlsanity.com/paper/2505.13738/) (2025)
- [Small Batch Size Training for Language Models: When Vanilla SGD Works, and Why Gradient Accumulation Is Wasteful](https://mlsanity.com/paper/2507.07101/) (2025)
- [Scaling Optimal LR Across Token Horizons](https://mlsanity.com/paper/2409.19913/) (2024)
- [Predictable Scale: Part I, Step Law -- Optimal Hyperparameter Scaling Law in Large Language Model Pretraining](https://mlsanity.com/paper/2503.04715/) (2025)
- [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)
- [u-μP: The Unit-Scaled Maximal Update Parametrization](https://mlsanity.com/paper/2407.17465/) (2024)
- [Scaling Law with Learning Rate Annealing](https://mlsanity.com/paper/2408.11029/) (2024)

## Built on by (most cited first, 2 of 2)

- [Effective Learning Rate Governs Loss Dynamics in Language Model Pretraining](https://mlsanity.com/paper/2608.24814/) (2026)
- [HyperTransfer: Understanding the Equivalence between Base Optimizer and Hyperball](https://mlsanity.com/paper/2609.07017/) (2026)

## Abstract (arXiv)

Learning-rate transfer can reduce the cost of training large language models: instead of sweeping learning rates at target scale, practitioners extrapolate from smaller runs. Existing approaches often assume that the optimal learning rate follows a log-linear scaling law in data scale and model size. We carefully examine and evaluate this scaling law. In our empirical study of GPT-2--style models from 22M to 707M parameters trained on 5B to 100B tokens, the optimal learning rate develops upward curvature at larger scales, leading to inaccurate extrapolation. We find that this curvature largely disappears when learning rates are replaced by effective learning rate (the step size in normalized weight space), and when data D extrapolation is used instead of model size N extrapolation. Next, we explain nonlinearity in scaling: weight-norm converges to equilibrium slower when optimal learning is small, requiring a larger step size to reduce the transient phase. Experiments with AdamH, which directly controls the effective learning rate, further support this explanation.