Dynamic metastability in the self-attention model

• URL: http://arxiv.org/abs/2410.06833v1
• Date: Wed, 9 Oct 2024 12:50:50 GMT
• Title: Dynamic metastability in the self-attention model
• Authors: Borjan Geshkovski, Hugo Koubbi, Yury Polyanskiy, Philippe Rigollet,
• Abstract summary: We consider the self-attention model - an interacting particle system on the unit sphere - which serves as a toy model for Transformers. We prove the appearance of dynamic metastability conjectured in [GLPR23]. We show that under an appropriate time-rescaling, the energy reaches its global maximum in finite time and has a staircase profile.
• Score: 22.689695473655906
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the self-attention model - an interacting particle system on the unit sphere, which serves as a toy model for Transformers, the deep neural network architecture behind the recent successes of large language models. We prove the appearance of dynamic metastability conjectured in [GLPR23] - although particles collapse to a single cluster in infinite time, they remain trapped near a configuration of several clusters for an exponentially long period of time. By leveraging a gradient flow interpretation of the system, we also connect our result to an overarching framework of slow motion of gradient flows proposed by Otto and Reznikoff [OR07] in the context of coarsening and the Allen-Cahn equation. We finally probe the dynamics beyond the exponentially long period of metastability, and illustrate that, under an appropriate time-rescaling, the energy reaches its global maximum in finite time and has a staircase profile, with trajectories manifesting saddle-to-saddle-like behavior, reminiscent of recent works in the analysis of training dynamics via gradient descent for two-layer neural networks.

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