# Hyperparameter Scaling Laws Across MoE Sparsity

arXiv: https://arxiv.org/abs/2609.08690 · HTML: https://mlsanity.com/paper/2609.08690/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in MoE theory and gating functions, one of 12 directions in Mixture-of-experts.
- Builds on 29 of its 35 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Placed in

- Mixture-of-experts / [MoE theory and gating functions](https://mlsanity.com/moe/moe-theory-and-gating-functions/)

## Builds on (most cited first, 29 of 29)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence](https://mlsanity.com/paper/2606.19348/) (2026)
- [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)
- [Unified Scaling Laws for Routed Language Models](https://mlsanity.com/paper/2202.01169/) (2022)
- [Kimi K3: Open Frontier Intelligence](https://mlsanity.com/paper/2607.24653/) (2026)
- [Language models scale reliably with over-training and on downstream tasks](https://mlsanity.com/paper/2403.08540/) (2024)
- [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)
- [How Does Critical Batch Size Scale in Pre-training?](https://mlsanity.com/paper/2410.21676/) (2024)
- [Towards Greater Leverage: Scaling Laws for Efficient Mixture-of-Experts Language Models](https://mlsanity.com/paper/2507.17702/) (2025)
- [Power Lines: Scaling Laws for Weight Decay and Batch Size in LLM Pre-training](https://mlsanity.com/paper/2505.13738/) (2025)
- [Scaling Optimal LR Across Token Horizons](https://mlsanity.com/paper/2409.19913/) (2024)
- [TransNormerLLM: A Faster and Better Large Language Model with Improved TransNormer](https://mlsanity.com/paper/2307.14995/) (2023)
- [Predictable Scale: Part I, Step Law -- Optimal Hyperparameter Scaling Law in Large Language Model Pretraining](https://mlsanity.com/paper/2503.04715/) (2025)
- [Every Activation Boosted: Scaling General Reasoner to 1 Trillion Open Language Foundation](https://mlsanity.com/paper/2510.22115/) (2025)
- [Joint MoE Scaling Laws: Mixture of Experts Can Be Memory Efficient](https://mlsanity.com/paper/2502.05172/) (2025)
- [Scaling Laws Across Model Architectures: A Comparative Analysis of Dense and MoE Models in Large Language Models](https://mlsanity.com/paper/2410.05661/) (2024)
- [Hyperparameter Transfer with Mixture-of-Expert Layers](https://mlsanity.com/paper/2601.20205/) (2026)
- [Ling and Ring 2.6 Technical Report: Efficient and Instant Agentic Intelligence at Trillion-Parameter Scale](https://mlsanity.com/paper/2606.15079/) (2026)
- [How to Set the Learning Rate for Large-Scale Pre-training?](https://mlsanity.com/paper/2601.05049/) (2026)
- [Complete-muE: Optimal Hyperparameter Transfer and Scaling for MoE Models](https://mlsanity.com/paper/2605.23893/) (2026)
- [SuperValid: Capability-Aligned OOD Validation for Generalizable Downstream Scaling](https://mlsanity.com/paper/2605.28179/) (2026)

## Abstract (arXiv)

Mixture-of-Experts (MoE) models expand model capacity without a proportional increase in training compute, but increasing sparsity makes reliable hyperparameter transfer challenging. In this work, we show that conventional hyperparameter scaling laws are insufficient for ultra-sparse MoEs: the optimal learning rate and batch size vary with activation ratio, and these shifts cannot be explained by either total or activated parameter count alone. To characterize this dependence, we conduct 1,800 pre-training runs spanning six activated-parameter scales and models with up to 6B total non-embedding parameters, processing approximately 20 trillion tokens at a cost of 200,000 equivalent H800 GPU-hours. Our results reconcile conflicting findings in prior work by revealing two scaling regimes. At fixed sparsity, the optimal batch size follows a power-law relationship with training tokens D, whereas the optimal learning rate scales with training compute C and remains robust to the allocation between model size and data. Across sparsity levels, the activation ratio A enters both relationships as an additional multiplicative power-law factor. These observations lead to unified hyperparameter scaling laws that transfer across MoE sparsity levels. Large-scale evaluation shows that the scaling form outperforms alternative functional forms. On a held-out ultra-sparse MoE with 12B total parameters and only 1/64 of its experts activated, the predicted hyperparameters remain close to the observed optima, supporting joint extrapolation across model scale and sparsity. Further experiments demonstrate transfer across expert granularities and isolate the effect of activation ratio from that of total expert count.