Propagation of Chaos in Contextual Flow Maps

Abstract:We develop a quantitative statistical theory of transformers in the large-context regime by adopting the abstraction of contextual flow maps (CFMs): dynamical systems that evolve a distinguished token in the presence of a contextual measure across a stack of attention blocks. Within this framework, the finite-context model approximates an idealized infinite-context system in which the contextual measure is replaced by its underlying population, so that the context length $n$ becomes a statistical resource. Exploiting the McKean--Vlasov structure of the dynamics and the classical machinery of propagation of chaos, we establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate $n^{-1/d}$ for general CFMs and parametric rate $n^{-1/2}$ for a restricted class of CFMs that includes transformers as a special case. The analysis rests on a new Eulerian adjoint formulation of the loss gradient and stability estimates for the resulting forward--adjoint system, both of which may be of independent interest.

Machine Learning (cs.LG); Analysis of PDEs (math.AP); Optimization and Control (math.OC); Probability (math.PR); Statistics Theory (math.ST)

arXiv:2605.16747 [cs.LG]

(or arXiv:2605.16747v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2605.16747 Focus to learn more arXiv-issued DOI via DataCite

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