# Densing Law of LLMs

arXiv: https://arxiv.org/abs/2412.04315 · HTML: https://mlsanity.com/paper/2412.04315/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 80 references that are in the map.
- First posted on arXiv 5 Dec 2024.

## Builds on (most cited first, 30 of 54)

- [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)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [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)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [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)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [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)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone](https://mlsanity.com/paper/2404.14219/) (2024)
- [Challenging BIG-Bench Tasks and Whether Chain-of-Thought Can Solve Them](https://mlsanity.com/paper/2210.09261/) (2022)
- [Is Your Code Generated by ChatGPT Really Correct? Rigorous Evaluation of Large Language Models for Code Generation](https://mlsanity.com/paper/2305.01210/) (2023)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [GQA: Training Generalized Multi-Query Transformer Models from Multi-Head Checkpoints](https://mlsanity.com/paper/2305.13245/) (2023)
- [Scaling Language Models: Methods, Analysis & Insights from Training Gopher](https://mlsanity.com/paper/2112.11446/) (2021)
- [DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model](https://mlsanity.com/paper/2405.04434/) (2024)

## Built on by (most cited first, 30 of 40)

- [TIP: Token Importance in On-Policy Distillation](https://mlsanity.com/paper/2604.14084/) (2026)
- [MiniCPM4: Ultra-Efficient LLMs on End Devices](https://mlsanity.com/paper/2506.07900/) (2025)
- [Ultra-FineWeb: Efficient Data Filtering and Verification for High-Quality LLM Training Data](https://mlsanity.com/paper/2505.05427/) (2025)
- [Scaling LLM Test-Time Compute with Mobile NPU on Smartphones](https://mlsanity.com/paper/2509.23324/) (2025)
- [Unveiling Downstream Performance Scaling of LLMs: A Clustering-Based Perspective](https://mlsanity.com/paper/2502.17262/) (2025)
- [Information Capacity: Evaluating the Efficiency of Large Language Models via Text Compression](https://mlsanity.com/paper/2511.08066/) (2025)
- [Think Before You Accept: Semantic Reflective Verification for Faster Speculative Decoding](https://mlsanity.com/paper/2505.18629/) (2025)
- [Can AI Truly Represent Your Voice in Deliberations? A Comprehensive Study of Large-Scale Opinion Aggregation with LLMs](https://mlsanity.com/paper/2510.05154/) (2025)
- [Reasoning emerges from constrained inference manifolds in large language models](https://mlsanity.com/paper/2605.08142/) (2026)
- [A Theory of Inference Compute Scaling: Reasoning through Directed Stochastic Skill Search](https://mlsanity.com/paper/2507.00004/) (2025)
- [Towards Alignment-Centric Paradigm: A Survey of Instruction Tuning in Large Language Models](https://mlsanity.com/paper/2508.17184/) (2025)
- [Predicting LLM Reasoning Performance with Small Proxy Model](https://mlsanity.com/paper/2509.21013/) (2025)
- [RooflineBench: A Benchmarking Framework for On-Device LLMs via Roofline Analysis](https://mlsanity.com/paper/2602.11506/) (2026)
- [Do Code Semantics Help? A Comprehensive Study on Execution Trace-Based Information for Code Large Language Models](https://mlsanity.com/paper/2509.11686/) (2025)
- [Semantic Retention and Extreme Compression in LLMs: Can We Have Both?](https://mlsanity.com/paper/2505.07289/) (2025)
- [MedCoG: Maximizing LLM Inference Density in Medical Reasoning via Meta-Cognitive Regulation](https://mlsanity.com/paper/2602.07905/) (2026)
- [SemRF: A Semantic Reference Frame for Residual-Stream Dynamics in Language Models](https://mlsanity.com/paper/2606.32022/) (2026)
- [Research Superalignment Should Advance Now with Alternating Competence and Conformity Optimization](https://mlsanity.com/paper/2503.07660/) (2025)
- [Logical Structure as Knowledge: Enhancing LLM Reasoning via Structured Logical Knowledge Density Estimation](https://mlsanity.com/paper/2509.24836/) (2025)
- [Mixture of Lookup Key-Value Experts](https://mlsanity.com/paper/2512.09723/) (2025)
- [Compute-Accuracy Pareto Frontiers for Open-Source Reasoning Large Language Models](https://mlsanity.com/paper/2512.24776/) (2025)
- [Benchmarking the Safety of Large Language Models for Robotic Health Attendant Control](https://mlsanity.com/paper/2604.26577/) (2026)
- [MiniOpt: Reasoning to Model and Solve General Optimization Problems with Limited Resources](https://mlsanity.com/paper/2606.25832/) (2026)
- [Do LLMs Triage Like Clinicians? A Dynamic Study of Outpatient Referral](https://mlsanity.com/paper/2503.08292/) (2025)
- [Invoke Interfaces Only When Needed: Adaptive Invocation for Large Language Models in Question Answering](https://mlsanity.com/paper/2505.02311/) (2025)
- [Renormalizable Spectral-Shell Dynamics as the Origin of Neural Scaling Laws](https://mlsanity.com/paper/2512.10427/) (2025)
- [LLMs Can Assist with Proposal Selection at Large User Facilities](https://mlsanity.com/paper/2512.10895/) (2025)
- [Constrained Language Model Policy Optimization via Risk-aware Stepwise Alignment](https://mlsanity.com/paper/2512.24263/) (2025)
- [Research Paradigm of Materials Science Tetrahedra with Artificial Intelligence](https://mlsanity.com/paper/2603.13744/) (2026)
- [A collaborative agent with two lightweight synergistic models for autonomous crystal materials research](https://mlsanity.com/paper/2604.11540/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) have emerged as a milestone in artificial intelligence, and their performance can improve as the model size increases. However, this scaling brings great challenges to training and inference efficiency, particularly for deploying LLMs in resource-constrained environments, and the scaling trend is becoming increasingly unsustainable. This paper introduces the concept of ``textitcapacity density'' as a new metric to evaluate the quality of the LLMs across different scales and describes the trend of LLMs in terms of both effectiveness and efficiency. To calculate the capacity density of a given target LLM, we first introduce a set of reference models and develop a scaling law to predict the downstream performance of these reference models based on their parameter sizes. We then define the textiteffective parameter size of the target LLM as the parameter size required by a reference model to achieve equivalent performance, and formalize the capacity density as the ratio of the effective parameter size to the actual parameter size of the target LLM. Capacity density provides a unified framework for assessing both model effectiveness and efficiency. Our further analysis of recent open-source base LLMs reveals an empirical law (the densing law)that the capacity density of LLMs grows exponentially over time. More specifically, using some widely used benchmarks for evaluation, the capacity density of LLMs doubles approximately every three months. The law provides new perspectives to guide future LLM development, emphasizing the importance of improving capacity density to achieve optimal results with minimal computational overhead.