# Scaling Laws for Sparsely-Connected Foundation Models

arXiv: https://arxiv.org/abs/2309.08520 · HTML: https://mlsanity.com/paper/2309.08520/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 64 references that are in the map.
- First posted on arXiv 15 Sep 2023.

## Builds on (most cited first, 30 of 53)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [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)
- [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)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications](https://mlsanity.com/paper/1704.04861/) (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)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [Scaling Language Models: Methods, Analysis & Insights from Training Gopher](https://mlsanity.com/paper/2112.11446/) (2021)
- [SparseGPT: Massive Language Models Can Be Accurately Pruned in One-Shot](https://mlsanity.com/paper/2301.00774/) (2023)
- [Scaling Vision Transformers](https://mlsanity.com/paper/2106.04560/) (2021)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [GLaM: Efficient Scaling of Language Models with Mixture-of-Experts](https://mlsanity.com/paper/2112.06905/) (2021)
- [A Survey of Quantization Methods for Efficient Neural Network Inference](https://mlsanity.com/paper/2103.13630/) (2021)
- [Scaling Vision with Sparse Mixture of Experts](https://mlsanity.com/paper/2106.05974/) (2021)
- [Scaling Vision Transformers to 22 Billion Parameters](https://mlsanity.com/paper/2302.05442/) (2023)
- [Sparsity in Deep Learning: Pruning and growth for efficient inference and training in neural networks](https://mlsanity.com/paper/2102.00554/) (2021)
- [ZeroQuant: Efficient and Affordable Post-Training Quantization for Large-Scale Transformers](https://mlsanity.com/paper/2206.01861/) (2022)
- [Rigging the Lottery: Making All Tickets Winners](https://mlsanity.com/paper/1911.11134/) (2019)
- [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)
- [Movement Pruning: Adaptive Sparsity by Fine-Tuning](https://mlsanity.com/paper/2005.07683/) (2020)
- [Scaling Data-Constrained Language Models](https://mlsanity.com/paper/2305.16264/) (2023)

## Built on by (most cited first, 30 of 43)

- [Data Mixing Laws: Optimizing Data Mixtures by Predicting Language Modeling Performance](https://mlsanity.com/paper/2403.16952/) (2024)
- [RouterEval: A Comprehensive Benchmark for Routing LLMs to Explore Model-level Scaling Up in LLMs](https://mlsanity.com/paper/2503.10657/) (2025)
- [Parameters vs FLOPs: Scaling Laws for Optimal Sparsity for Mixture-of-Experts Language Models](https://mlsanity.com/paper/2501.12370/) (2025)
- [Scaling Laws for Predicting Downstream Performance in LLMs](https://mlsanity.com/paper/2410.08527/) (2024)
- [D-CPT Law: Domain-specific Continual Pre-Training Scaling Law for Large Language Models](https://mlsanity.com/paper/2406.01375/) (2024)
- [Scaling Optimal LR Across Token Horizons](https://mlsanity.com/paper/2409.19913/) (2024)
- [Quartet: Native FP4 Training Can Be Optimal for Large Language Models](https://mlsanity.com/paper/2505.14669/) (2025)
- [On the Scalability of GNNs for Molecular Graphs](https://mlsanity.com/paper/2404.11568/) (2024)
- [Selecting Large Language Model to Fine-tune via Rectified Scaling Law](https://mlsanity.com/paper/2402.02314/) (2024)
- [Parallel Scaling Law for Language Models](https://mlsanity.com/paper/2505.10475/) (2025)
- [The Sparse Frontier: Sparse Attention Trade-offs in Transformer LLMs](https://mlsanity.com/paper/2504.17768/) (2025)
- [(Mis)Fitting: A Survey of Scaling Laws](https://mlsanity.com/paper/2502.18969/) (2025)
- [Sparsing Law: Towards Large Language Models with Greater Activation Sparsity](https://mlsanity.com/paper/2411.02335/) (2024)
- [QuEST: Stable Training of LLMs with 1-Bit Weights and Activations](https://mlsanity.com/paper/2502.05003/) (2025)
- [A Multi-Power Law for Loss Curve Prediction Across Learning Rate Schedules](https://mlsanity.com/paper/2503.12811/) (2025)
- [ATLAS: Adaptive Transfer Scaling Laws for Multilingual Pretraining, Finetuning, and Decoding the Curse of Multilinguality](https://mlsanity.com/paper/2510.22037/) (2025)
- [Gemstones: A Model Suite for Multi-Faceted Scaling Laws](https://mlsanity.com/paper/2502.06857/) (2025)
- [Toward Inference-optimal Mixture-of-Expert Large Language Models](https://mlsanity.com/paper/2404.02852/) (2024)
- [Effective Interplay between Sparsity and Quantization: From Theory to Practice](https://mlsanity.com/paper/2405.20935/) (2024)
- [Sparse maximal update parameterization: A holistic approach to sparse training dynamics](https://mlsanity.com/paper/2405.15743/) (2024)
- [Cost-Aware Contrastive Routing for LLMs](https://mlsanity.com/paper/2508.12491/) (2025)
- [Breaking Language Barriers: Cross-Lingual Continual Pre-Training at Scale](https://mlsanity.com/paper/2407.02118/) (2024)
- [Sparse Fine-tuning for Inference Acceleration of Large Language Models](https://mlsanity.com/paper/2310.06927/) (2023)
- [Not-Just-Scaling Laws: Towards a Better Understanding of the Downstream Impact of Language Model Design Decisions](https://mlsanity.com/paper/2503.03862/) (2025)
- [Compression Scaling Laws:Unifying Sparsity and Quantization](https://mlsanity.com/paper/2502.16440/) (2025)
- [DAM: Towards A Foundation Model for Time Series Forecasting](https://mlsanity.com/paper/2407.17880/) (2024)
- [Physics of Skill Learning](https://mlsanity.com/paper/2501.12391/) (2025)
- [textttMoE-RBench: Towards Building Reliable Language Models with Sparse Mixture-of-Experts](https://mlsanity.com/paper/2406.11353/) (2024)
- [UMoE: Unifying Attention and FFN with Shared Experts](https://mlsanity.com/paper/2505.07260/) (2025)
- [How to Upscale Neural Networks with Scaling Law? A Survey and Practical Guidelines](https://mlsanity.com/paper/2502.12051/) (2025)

## Abstract (arXiv)

We explore the impact of parameter sparsity on the scaling behavior of Transformers trained on massive datasets (i.e., "foundation models"), in both vision and language domains. In this setting, we identify the first scaling law describing the relationship between weight sparsity, number of non-zero parameters, and amount of training data, which we validate empirically across model and data scales; on ViT/JFT-4B and T5/C4. These results allow us to characterize the "optimal sparsity", the sparsity level which yields the best performance for a given effective model size and training budget. For a fixed number of non-zero parameters, we identify that the optimal sparsity increases with the amount of data used for training. We also extend our study to different sparsity structures (such as the hardware-friendly n:m pattern) and strategies (such as starting from a pretrained dense model). Our findings shed light on the power and limitations of weight sparsity across various parameter and computational settings, offering both theoretical understanding and practical implications for leveraging sparsity towards computational efficiency improvements.