Asymptotic Analysis of Discrete-Time Self-Attention Dynamics in Transformers

Abstract:In machine learning, a self-attention dynamics describes the attention mechanism of transformers, and is obtained treating the layer index of a transformer as time index. Even though naturally discrete-time, this dynamical system is most often studied in continuous-time. Scope of this paper is to provide a thorough investigation of the discrete-time version of the self-attention dynamics: a complete classification of its equilibria is given, and their stability properties are studied, in both the single-head and the multihead case. We show that in both cases the main attractor of the dynamics is a consensus equilibrium aligned with the principal eigenvector of the value matrix (in the single-head case, a corresponding matrix of weights in the multihead case), even though other locally asymptotically stable bipartite equilibria can be show to exist at least in the single-head situation. The behavior is confirmed by looking at the learned weight matrices of a real transformer.

Dynamical Systems (math.DS); Machine Learning (cs.LG); Systems and Control (eess.SY)

arXiv:2610.13518 [math.DS]

(or arXiv:2610.13518v1 [math.DS] for this version)

https://doi.org/10.48550/arXiv.2610.13518 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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