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arXiv · 2509.15958

Localmax dynamics for attention in transformers and its asymptotic behavior

Henri Cimetière·Maria Teresa Chiri·Bahman Gharesifard
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Abstract

We introduce a new discrete-time attention model, termed the localmax dynamics, which interpolates between the classic softmax dynamics and the hardmax dynamics, where only the tokens that maximize the influence toward a given token have a positive weight. As in hardmax, uniform weights are determined by a parameter controlling neighbor influence, but the key extension lies in relaxing neighborhood interactions through an alignment-sensitivity parameter, which allows controlled deviations from pure hardmax behavior. As we prove, while the convex hull of the token states still converges to a convex polytope, its structure can no longer be fully described by a maximal alignment set, prompting the introduction of quiescent sets to capture the invariant behavior of tokens near vertices. We show that these sets play a key role in understanding the asymptotic behavior of the system, even under time-varying alignment sensitivity parameters. We further show that localmax dynamics does not exhibit finite-time convergence and provide results for vanishing, nonzero, time-varying alignment-sensitivity parameters, recovering the limiting behavior of hardmax as a by-product. Finally, we adapt Lyapunov-based methods from classical opinion dynamics, highlighting their limitations in the asymmetric setting of localmax interactions and outlining directions for future research.

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Henri Cimetière, Maria Teresa Chiri, Bahman Gharesifard. 2025-09-19. Localmax dynamics for attention in transformers and its asymptotic behavior. https://arxiv.org/abs/2509.15958

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