Asymptotics of SGD in Sequence-Single Index Models and Single-Layer Attention Networks

• URL: http://arxiv.org/abs/2506.02651v1
• Date: Tue, 03 Jun 2025 09:03:27 GMT
• Title: Asymptotics of SGD in Sequence-Single Index Models and Single-Layer Attention Networks
• Authors: Luca Arnaboldi, Bruno Loureiro, Ludovic Stephan, Florent Krzakala, Lenka Zdeborova,
• Abstract summary: We study the dynamics of gradient descent (SGD) for a class of sequence models termed Sequence Single-Index (SSI) models.<n>This setting generalizes classical single-index models to the sequential domain.
• Score: 24.882327415229295
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the dynamics of stochastic gradient descent (SGD) for a class of sequence models termed Sequence Single-Index (SSI) models, where the target depends on a single direction in input space applied to a sequence of tokens. This setting generalizes classical single-index models to the sequential domain, encompassing simplified one-layer attention architectures. We derive a closed-form expression for the population loss in terms of a pair of sufficient statistics capturing semantic and positional alignment, and characterize the induced high-dimensional SGD dynamics for these coordinates. Our analysis reveals two distinct training phases: escape from uninformative initialization and alignment with the target subspace, and demonstrates how the sequence length and positional encoding influence convergence speed and learning trajectories. These results provide a rigorous and interpretable foundation for understanding how sequential structure in data can be beneficial for learning with attention-based models.

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