# Sparsing Law: Towards Large Language Models with Greater Activation Sparsity

arXiv: https://arxiv.org/abs/2411.02335 · HTML: https://mlsanity.com/paper/2411.02335/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 71 references that are in the map.
- First posted on arXiv 4 Nov 2024.

## Builds on (most cited first, 30 of 59)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [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)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [Challenging BIG-Bench Tasks and Whether Chain-of-Thought Can Solve Them](https://mlsanity.com/paper/2210.09261/) (2022)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [WizardLM: Empowering large pre-trained language models to follow complex instructions](https://mlsanity.com/paper/2304.12244/) (2023)
- [SparseGPT: Massive Language Models Can Be Accurately Pruned in One-Shot](https://mlsanity.com/paper/2301.00774/) (2023)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Enhancing Chat Language Models by Scaling High-quality Instructional Conversations](https://mlsanity.com/paper/2305.14233/) (2023)
- [StarCoder: may the source be with you!](https://mlsanity.com/paper/2305.06161/) (2023)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [AGIEval: A Human-Centric Benchmark for Evaluating Foundation Models](https://mlsanity.com/paper/2304.06364/) (2023)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [Knowledge Neurons in Pretrained Transformers](https://mlsanity.com/paper/2104.08696/) (2021)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [TyDi QA: A Benchmark for Information-Seeking Question Answering in Typologically Diverse Languages](https://mlsanity.com/paper/2003.05002/) (2020)
- [Dolma: an Open Corpus of Three Trillion Tokens for Language Model Pretraining Research](https://mlsanity.com/paper/2402.00159/) (2024)
- [Sheared LLaMA: Accelerating Language Model Pre-training via Structured Pruning](https://mlsanity.com/paper/2310.06694/) (2023)

## Built on by (most cited first, 22 of 22)

- [The Sparse Frontier: Sparse Attention Trade-offs in Transformer LLMs](https://mlsanity.com/paper/2504.17768/) (2025)
- [Meaningless Tokens, Meaningful Gains: How Activation Shifts Enhance LLM Reasoning](https://mlsanity.com/paper/2510.01032/) (2025)
- [Universal Properties of Activation Sparsity in Modern Large Language Models](https://mlsanity.com/paper/2509.00454/) (2025)
- [Polar Sparsity: High Throughput Batched LLM Inferencing with Scalable Contextual Sparsity](https://mlsanity.com/paper/2505.14884/) (2025)
- [BlockFFN: Towards End-Side Acceleration-Friendly Mixture-of-Experts with Chunk-Level Activation Sparsity](https://mlsanity.com/paper/2507.08771/) (2025)
- [Spark Transformer: Reactivating Sparsity in FFN and Attention](https://mlsanity.com/paper/2506.06644/) (2025)
- [SmallThinker: A Family of Efficient Large Language Models Natively Trained for Local Deployment](https://mlsanity.com/paper/2507.20984/) (2025)
- [Understanding and Harnessing Sparsity in Unified Multimodal Models](https://mlsanity.com/paper/2512.02351/) (2025)
- [Response Uncertainty and Probe Modeling: Two Sides of the Same Coin in LLM Interpretability?](https://mlsanity.com/paper/2505.18575/) (2025)
- [How Controlling the Variance can Improve Training Stability of Sparsely Activated DNNs and CNNs](https://mlsanity.com/paper/2602.05779/) (2026)
- [It Takes a MAESTRO To Prune Bad Experts](https://mlsanity.com/paper/2607.08601/) (2026)
- [What Should Feature Distillation Transfer in LLMs? A Task-Tangent Geometry View](https://mlsanity.com/paper/2507.10155/) (2025)
- [The Diminishing Returns of Early-Exit Decoding in Modern LLMs](https://mlsanity.com/paper/2603.23701/) (2026)
- [Finedeep: Mitigating Sparse Activation in Dense LLMs via Multi-Layer Fine-Grained Experts](https://mlsanity.com/paper/2502.12928/) (2025)
- [SAGE: Scale-Aware Gradual Evolution for Continual Knowledge Graph Embedding](https://mlsanity.com/paper/2508.11347/) (2025)
- [Resting Neurons, Active Insights: Robustifying Activation Sparsity in LLMs via Spontaneity](https://mlsanity.com/paper/2512.12744/) (2025)
- [Sparse or Dense? A Mechanistic Estimation of Computation Density in Transformer-based LLMs](https://mlsanity.com/paper/2601.22795/) (2026)
- [Uncovering Intra-expert Activation Sparsity for Efficient Mixture-of-Expert Model Execution](https://mlsanity.com/paper/2605.08575/) (2026)
- [DECO: Sparse Mixture-of-Experts with Dense-Comparable Performance on End-Side Devices](https://mlsanity.com/paper/2605.10933/) (2026)
- [Unifying Depth and Width Pruning for LLMs via Binary Knapsack Optimization](https://mlsanity.com/paper/2608.12953/) (2026)
- [Model Casting and Low-Parameter Gating: Towards More Sparsely Activated FFNs](https://mlsanity.com/paper/2609.31975/) (2026)
- [On the Necessity of Attention-FFN Split in Vision Transformers](https://mlsanity.com/paper/2610.10303/) (2026)

## Abstract (arXiv)

Activation sparsity denotes the existence of substantial weakly-contributed elements within activation outputs that can be eliminated, benefiting many important applications concerned with large language models (LLMs). Although promoting greater activation sparsity within LLMs deserves deep studies, existing works lack comprehensive and quantitative research on the correlation between activation sparsity and potentially influential factors. In this paper, we present a comprehensive study on the quantitative scaling properties and influential factors of the activation sparsity within decoder-only Transformer-based LLMs. Specifically, we propose PPL-p% sparsity, a precise and performance-aware activation sparsity metric that is applicable to any activation function. Through extensive experiments, we find several important phenomena. Firstly, different activation functions exhibit comparable performance but opposite training-time sparsity trends. The activation ratio (i.e., 1-sparsity ratio) evolves as a convergent increasing power-law and decreasing logspace power-law with the amount of training data for SiLU-activated and ReLU-activated LLMs, respectively. These demonstrate that ReLU is more efficient as the activation function than SiLU and can leverage more training data to improve activation sparsity. Secondly, the activation ratio linearly increases with the width-depth ratio below a certain bottleneck point, indicating the potential advantage of a deeper architecture at a fixed parameter scale. Finally, at similar width-depth ratios, we surprisingly find that the limit value of activation sparsity varies weakly with the parameter scale, i.e., the activation patterns within LLMs are insensitive to the parameter scale. These empirical laws towards LLMs with greater activation sparsity have important implications for making LLMs more efficient and interpretable.