Transformers converge pathwise to a stochastic particle system and SPDE in the scaling limit, exhibiting synchronization by noise and exponential energy dissipation when common noise is coercive relative to self-attention drift.
Using Cauchy-Schwarz and the Lipschitz assumption (A.4): A2 t ≤T K 2 Z t 0 1 N NX i=1 E h |X i u −X i tℓu |2 i du
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Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models
Transformers converge pathwise to a stochastic particle system and SPDE in the scaling limit, exhibiting synchronization by noise and exponential energy dissipation when common noise is coercive relative to self-attention drift.