# Extending μP: Spectral Conditions for Feature Learning Across Optimizers

arXiv: https://arxiv.org/abs/2602.20937 · HTML: https://mlsanity.com/paper/2602.20937/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 29 references that are in the map.
- First posted on arXiv 24 Feb 2026.

## Builds on (most cited first, 19 of 19)

- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [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)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [An Empirical Model of Large-Batch Training](https://mlsanity.com/paper/1812.06162/) (2018)
- [Train longer, generalize better: closing the generalization gap in large batch training of neural networks](https://mlsanity.com/paper/1705.08741/) (2017)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [One weird trick for parallelizing convolutional neural networks](https://mlsanity.com/paper/1404.5997/) (2014)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [A Spectral Condition for Feature Learning](https://mlsanity.com/paper/2310.17813/) (2023)
- [Tensor Programs VI: Feature Learning in Infinite-Depth Neural Networks](https://mlsanity.com/paper/2310.02244/) (2023)
- [Don't be lazy: CompleteP enables compute-efficient deep transformers](https://mlsanity.com/paper/2505.01618/) (2025)
- [u-μP: The Unit-Scaled Maximal Update Parametrization](https://mlsanity.com/paper/2407.17465/) (2024)
- [ADOPT: Modified Adam Can Converge with Any β2 with the Optimal Rate](https://mlsanity.com/paper/2411.02853/) (2024)
- [On the Parameterization of Second-Order Optimization Effective Towards the Infinite Width](https://mlsanity.com/paper/2312.12226/) (2023)
- [μLO: Compute-Efficient Meta-Generalization of Learned Optimizers](https://mlsanity.com/paper/2406.00153/) (2024)

## Abstract (arXiv)

Several variations of adaptive first-order and second-order optimization methods have been proposed to accelerate and scale the training of large language models. The performance of these optimization routines is highly sensitive to the choice of hyperparameters (HPs), which are computationally expensive to tune for large-scale models. Maximal update parameterization (μP) is a set of scaling rules which aims to make the optimal HPs independent of the model size, thereby allowing the HPs tuned on a smaller (computationally cheaper) model to be transferred to train a larger, target model. Despite promising results for SGD and Adam, deriving μP for other optimizers is challenging because the underlying tensor programming approach is difficult to grasp. Building on recent work that introduced spectral conditions as an alternative to tensor programs, we propose a novel framework to derive μP for a broader class of optimizers, including AdamW, ADOPT, LAMB, Sophia, Shampoo and Muon. We implement our μP derivations on multiple benchmark models and demonstrate zero-shot learning rate transfer across increasing model width for the above optimizers. Further, we provide empirical insights into depth-scaling parameterization for these optimizers.