# Scale Weight Decay and Train Better

arXiv: https://arxiv.org/abs/2607.23777 · HTML: https://mlsanity.com/paper/2607.23777/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 58 of its 70 references that are in the map.
- First posted on arXiv 26 Jul 2026.

## Builds on (most cited first, 30 of 58)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence](https://mlsanity.com/paper/2606.19348/) (2026)
- [Scaling Vision Transformers](https://mlsanity.com/paper/2106.04560/) (2021)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [GLM-4.5: Agentic, Reasoning, and Coding (ARC) Foundation Models](https://mlsanity.com/paper/2508.06471/) (2025)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer](https://mlsanity.com/paper/2203.03466/) (2022)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Samba: Simple Hybrid State Space Models for Efficient Unlimited Context Language Modeling](https://mlsanity.com/paper/2406.07522/) (2024)

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

- [Convergence of Practical Muon](https://mlsanity.com/paper/2609.33152/) (2026)
- [Second-Moment Stochastic Approximation Methods](https://mlsanity.com/paper/2609.36600/) (2026)

## Abstract (arXiv)

The discovery of scaling laws has motivated training neural networks on ever increasing quantities of data. This is typically done with a constant decoupled weight decay which causes the network weights to shrink steadily over the course of training. Taking inspiration from the Robbins--Monro conditions, we propose to scale weight decay by the fraction of the peak learning rate η/ηmax. We prove that this scaled weight decay preserves the asymptotic stationarity guarantees of the corresponding unregularized methods for both stochastic gradient descent and the non-Euclidean spectral optimizer Muon, thereby avoiding the additional asymptotic bias introduced by constant decoupled weight decay. This retains the stability benefits of weight decay without changing the asymptotic optimization target. Using a steady-state analysis, we explain why under standard weight decay the weight norm shrinks steadily as training proceeds, whereas under scaled weight decay it settles to a roughly constant value. When applied to the training of mixture-of-experts models, Muon with scaled weight decay (Muon-SW) consistently outpaces Muon with identical hyperparameters, reaching the same validation loss 30% faster at our largest scale across models from 72 - 930 million parameters trained at ~ 600 tokens per active parameter. If this trend continues to hold, the method promises to substantially accelerate the pre-training of frontier models while requiring only a few lines of code to implement.