Propagation of Chaos in Contextual Flow Maps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910225678729216 |
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| author | Chen, Shi Lin, Zhengjiang Liu, Kaizhao Rigollet, Philippe |
| author_facet | Chen, Shi Lin, Zhengjiang Liu, Kaizhao Rigollet, Philippe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16747 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Propagation of Chaos in Contextual Flow Maps Chen, Shi Lin, Zhengjiang Liu, Kaizhao Rigollet, Philippe Machine Learning Analysis of PDEs Optimization and Control Probability Statistics Theory 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. |
| title | Propagation of Chaos in Contextual Flow Maps |
| topic | Machine Learning Analysis of PDEs Optimization and Control Probability Statistics Theory |
| url | https://arxiv.org/abs/2605.16747 |