# Sparse maximal update parameterization: A holistic approach to sparse training dynamics

arXiv: https://arxiv.org/abs/2405.15743 · HTML: https://mlsanity.com/paper/2405.15743/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 63 of its 76 references that are in the map.
- First posted on arXiv 24 May 2024.

## Builds on (most cited first, 30 of 63)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [Jailbroken: How Does LLM Safety Training Fail?](https://mlsanity.com/paper/2307.02483/) (2023)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [SparseGPT: Massive Language Models Can Be Accurately Pruned in One-Shot](https://mlsanity.com/paper/2301.00774/) (2023)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [Model Cards for Model Reporting](https://mlsanity.com/paper/1810.03993/) (2018)
- [StereoSet: Measuring stereotypical bias in pretrained language models](https://mlsanity.com/paper/2004.09456/) (2020)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [SNIP: Single-shot Network Pruning based on Connection Sensitivity](https://mlsanity.com/paper/1810.02340/) (2018)
- [To prune, or not to prune: exploring the efficacy of pruning for model compression](https://mlsanity.com/paper/1710.01878/) (2017)
- [Rethinking the Value of Network Pruning](https://mlsanity.com/paper/1810.05270/) (2018)
- [Carbon Emissions and Large Neural Network Training](https://mlsanity.com/paper/2104.10350/) (2021)
- [The Woman Worked as a Babysitter: On Biases in Language Generation](https://mlsanity.com/paper/1909.01326/) (2019)
- [Sparsity in Deep Learning: Pruning and growth for efficient inference and training in neural networks](https://mlsanity.com/paper/2102.00554/) (2021)
- [Rigging the Lottery: Making All Tickets Winners](https://mlsanity.com/paper/1911.11134/) (2019)
- [Pruning neural networks without any data by iteratively conserving synaptic flow](https://mlsanity.com/paper/2006.05467/) (2020)
- [The State of Sparsity in Deep Neural Networks](https://mlsanity.com/paper/1902.09574/) (2019)
- [Picking Winning Tickets Before Training by Preserving Gradient Flow](https://mlsanity.com/paper/2002.07376/) (2020)
- [Scalable Training of Artificial Neural Networks with Adaptive Sparse Connectivity inspired by Network Science](https://mlsanity.com/paper/1707.04780/) (2017)
- [Lipstick on a Pig: Debiasing Methods Cover up Systematic Gender Biases in Word Embeddings But do not Remove Them](https://mlsanity.com/paper/1903.03862/) (2019)
- [Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time](https://mlsanity.com/paper/2310.17157/) (2023)

## Built on by (most cited first, 16 of 16)

- [Practical Efficiency of Muon for Pretraining](https://mlsanity.com/paper/2505.02222/) (2025)
- [Don't be lazy: CompleteP enables compute-efficient deep transformers](https://mlsanity.com/paper/2505.01618/) (2025)
- [Straight to Zero: Why Linearly Decaying the Learning Rate to Zero Works Best for LLMs](https://mlsanity.com/paper/2502.15938/) (2025)
- [u-μP: The Unit-Scaled Maximal Update Parametrization](https://mlsanity.com/paper/2407.17465/) (2024)
- [Dynamic Sparse Training versus Dense Training: The Unexpected Winner in Image Corruption Robustness](https://mlsanity.com/paper/2410.03030/) (2024)
- [Robust Layerwise Scaling Rules by Proper Weight Decay Tuning](https://mlsanity.com/paper/2510.15262/) (2025)
- [μLO: Compute-Efficient Meta-Generalization of Learned Optimizers](https://mlsanity.com/paper/2406.00153/) (2024)
- [When Does Sparsity Mitigate the Curse of Depth in LLMs](https://mlsanity.com/paper/2603.15389/) (2026)
- [Quantifying Hyperparameter Transfer and the Importance of Embedding Layer Learning Rate](https://mlsanity.com/paper/2605.21486/) (2026)
- [MeTA-LoRA: Data-Efficient Multi-Task Fine-Tuning for Large Language Models](https://mlsanity.com/paper/2510.11598/) (2025)
- [One Step Closer to Ground Truth: A Multi-Scale Residual-Aware Representation Learning Pipeline for Predicting Time Series Data](https://mlsanity.com/paper/2606.10678/) (2026)
- [Towards a Principled Muon under μmathsfP: Ensuring Spectral Conditions throughout Training](https://mlsanity.com/paper/2601.01306/) (2026)
- [When is Warmstarting Effective for Scaling Language Models?](https://mlsanity.com/paper/2605.13405/) (2026)
- [Memory-Efficient LLM Training with Dynamic Sparsity: From Stability to Practical Scaling](https://mlsanity.com/paper/2606.00888/) (2026)
- [Unlocking Feature Learning in Gated Delta Networks at Scale](https://mlsanity.com/paper/2606.04048/) (2026)
- [Learning Rate Transfer for Hybrid Transformer-SSM Architectures](https://mlsanity.com/paper/2610.01172/) (2026)

## Abstract (arXiv)

Several challenges make it difficult for sparse neural networks to compete with dense models. First, setting a large fraction of weights to zero impairs forward and gradient signal propagation. Second, sparse studies often need to test multiple sparsity levels, while also introducing new hyperparameters (HPs), leading to prohibitive tuning costs. Indeed, the standard practice is to re-use the learning HPs originally crafted for dense models. Unfortunately, we show sparse and dense networks do not share the same optimal HPs. Without stable dynamics and effective training recipes, it is costly to test sparsity at scale, which is key to surpassing dense networks and making the business case for sparsity acceleration in hardware. A holistic approach is needed to tackle these challenges and we propose SμPar as one such approach. For random unstructured static sparsity, SμPar ensures activations, gradients, and weight updates all scale independently of sparsity level. Further, by reparameterizing the HPs, SμPar enables the same HP values to be optimal as we vary both sparsity level and model width. HPs can be tuned on small dense networks and transferred to large sparse models, greatly reducing tuning costs. On large-scale language modeling, SμPar shows increasing improvements over standard parameterization as sparsity increases, leading up to 11.9% relative loss improvement at 99.2% sparsity. A minimal implementation of SμPar is available at https://github.com/EleutherAI/nanoGPT-mup/tree/supar.