A Markov Categorical Framework for Language Modeling
This provides a foundational mathematical framework for understanding language models, which could impact all of ML/AI by unifying disparate aspects of model analysis.
The paper tackles the lack of a unified theory for autoregressive language models by introducing a Markov categorical framework that connects training objectives, representation geometry, and model capabilities, showing that minimizing negative log-likelihood induces spectral alignment in the representation space.
Autoregressive language models achieve remarkable performance, yet a unified theory explaining their internal mechanisms, how training shapes their representations, and enables complex behaviors, remains elusive. We introduce a new analytical framework that models the single-step generation process as a composition of information-processing stages using the language of Markov categories. This compositional perspective provides a unified mathematical language to connect three critical aspects of language modeling that are typically studied in isolation: the training objective, the geometry of the learned representation space, and practical model capabilities. First, our framework provides a precise information-theoretic rationale for the success of multi-token prediction methods like speculative decoding, quantifying the information surplus a model's hidden state contains about tokens beyond the immediate next one. Second, we clarify how the standard negative log-likelihood (NLL) objective compels the model to learn not just the next word, but also the data's intrinsic conditional uncertainty, a process we formalize using categorical entropy. Our central result shows that, under a linear-softmax head with bounded features, minimizing NLL induces spectral alignment: the learned representation space aligns with the eigenspectrum of a predictive similarity operator. This work presents a powerful new lens for understanding how information flows through a model and how the training objective shapes its internal geometry.