L$^2$M: Mutual Information Scaling Law for Long-Context Language Modeling
Zhuo Chen, Oriol Mayné i Comas, Zhuotao Jin, Di Luo, Marin Soljacic
Abstract
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 **L**ong-context **L**anguage **M**odeling (**L**$^2$**M**) 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.
BibTeX
@inproceedings{
chen2025lm,
title={L\${\textasciicircum}2\$M: Mutual Information Scaling Law for Long-Context Language Modeling},
author={Zhuo Chen and Oriol Mayn{\'e} i Comas and Zhuotao Jin and Di Luo and Marin Soljacic},
booktitle={The Thirty-ninth Annual Conference on Neural Information Processing Systems},
year={2025},
url={https://openreview.net/forum?id=s3maemwE5M}
}