Information-Theoretic Analysis of Next-Token Prediction under Markovian Data

Abstract:We develop an information-theoretic framework for generalization in next-token prediction under temporally dependent data. We consider independent trajectories generated by finite-memory Markov processes and distinguish algorithmic dependence, quantified by mutual information, from temporal dependence, characterized by mixing. For cross-entropy loss, we derive an expected generalization bound using the Donsker--Varadhan variational representation and a McDiarmid-type concentration inequality for Markov chains. A refinement captures the joint effect of context length and temporal mixing through the mixing properties of the history-state process. We then extend the bound through a rate--distortion formulation, replacing mutual information with the minimum information rate required to represent the learned model within a prescribed distortion in the generalization gap, yielding informative guarantees for deterministic algorithms over continuous hypothesis spaces. For margin-based prediction, we derive explicit bounds for linear and self-attention next-token predictors via noisy low-dimensional compression, revealing the roles of context length, model complexity, sample size, margin, and temporal mixing. Experiments on TinyStories and ETTh2 show that longer contexts can reduce both training and test losses, but typically reduce training loss more, enlarging the generalization gap. A complementary ETTh2 analysis identifies an effective predictive-memory scale near 24 hours, with no statistically supported improvement beyond this scale, offering a plausible explanation for test-performance saturation at larger contexts.

Machine Learning (stat.ML); Information Theory (cs.IT); Machine Learning (cs.LG)

arXiv:2609.34731 [stat.ML]

(or arXiv:2609.34731v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.34731 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.