L$^2$M: Mutual Information Scaling Law for Long-Context Language Modeling

Zhuo Chen (ByteDance Inc.) · Oriol Comas (Massachusetts Institute of Technology) · Zhuotao Jin (Massachusetts Institute of Technology) · Di Luo (University of California, Los Angeles) · Marin Soljacic (MIT)

architectural designbipartite mutual informationdependencieseffective modelinghistory statelatent variableslong-context capabilitieslong-context language modelingmulti-token interactionsnatural language applicationsprincipled foundation scaling law state-space models transformer modelstwo-point mutual information

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.