# L2M: Mutual Information Scaling Law for Long-Context Language Modeling

arXiv: https://arxiv.org/abs/2503.04725 · HTML: https://mlsanity.com/paper/2503.04725/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 63 of its 114 references that are in the map.
- First posted on arXiv 6 Mar 2025.

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

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [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)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Longformer: The Long-Document Transformer](https://mlsanity.com/paper/2004.05150/) (2020)
- [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)
- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [HuggingFace's Transformers: State-of-the-art Natural Language Processing](https://mlsanity.com/paper/1910.03771/) (2019)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context](https://mlsanity.com/paper/1901.02860/) (2019)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Big Bird: Transformers for Longer Sequences](https://mlsanity.com/paper/2007.14062/) (2020)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [Learning deep representations by mutual information estimation and maximization](https://mlsanity.com/paper/1808.06670/) (2018)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)

## Built on by (most cited first, 9 of 9)

- [Demystifying Reasoning Dynamics with Mutual Information: Thinking Tokens are Information Peaks in LLM Reasoning](https://mlsanity.com/paper/2506.02867/) (2025)
- [Tensorization is a powerful but underexplored tool for compression and interpretability of neural networks](https://mlsanity.com/paper/2505.20132/) (2025)
- [Correlation Dimension of Auto-Regressive Large Language Models](https://mlsanity.com/paper/2510.21258/) (2025)
- [Advancing AI Challenges for the United States Department of the Air Force](https://mlsanity.com/paper/2511.00267/) (2025)
- [Semantic-aware Graph-guided Behavior Sequences Generation with Large Language Models for Smart Homes](https://mlsanity.com/paper/2508.03484/) (2025)
- [Semantic Rate Distortion and Posterior Design: Compute Constraints, Multimodality, and Strategic Inference](https://mlsanity.com/paper/2602.03949/) (2026)
- [Trading Complexity for Expressivity Through Structured Generalized Linear Token Mixing](https://mlsanity.com/paper/2605.31367/) (2026)
- [Do Quantum Models Scale Like LLMs?](https://mlsanity.com/paper/2609.20912/) (2026)
- [The Cost of Long Memory: State, Context, and Stability Complexity in Sequence Models](https://mlsanity.com/paper/2610.08816/) (2026)

## Abstract (arXiv)

We present a universal theoretical framework for understanding long-context language modeling based on a bipartite mutual information scaling law that we rigorously verify in natural language. We demonstrate that bipartite mutual information captures multi-token interactions distinct from and scaling independently of conventional two-point mutual information, and show that this provides a more complete characterization of the dependencies needed for accurately modeling long sequences. Leveraging this scaling law, we formulate the Long-context Language Modeling (L2M) condition, which lower bounds the necessary scaling of a model's history state -- the latent variables responsible for storing past information -- for effective long-context modeling. We validate the framework and its predictions on transformer and state-space models. Our work provides a principled foundation to understand long-context modeling and to design more efficient architectures with stronger long-context capabilities, with potential applications beyond natural language.