Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models
Provides a rigorous mathematical foundation for understanding synchronization and noise effects in deep transformer models, relevant to theorists studying neural network dynamics.
The paper proves convergence of deep transformer models to a stochastic interacting particle system, identifies the limiting SPDE, and shows that common noise can synchronize tokens and exponentially dissipate interaction energy under certain conditions.
We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.