Single-Head Attention in High Dimensions: A Theory of Generalization, Weights Spectra, and Scaling Laws

Fabrizio Boncoraglio, Vittorio Erba, Emanuele Troiani, Yizhou Xu, Florent Krzakala, Lenka Zdeborová

Keywords: single-head attention, power law, spectrum, generalization

Abstract: Trained attention layers exhibit striking and reproducible spectral structure of the weights, including low-rank collapse, bulk deformation, and isolated spectral outliers, yet the origin of these phenomena and their implications for generalization remain poorly understood. We study empirical risk minimization in a single-head tied-attention layer trained on synthetic high-dimensional sequence tasks generated from the attention-indexed model. Using tools from random matrix theory, spin-glass theory, and approximate message passing, we obtain an exact high-dimensional characterization of training and test error, interpolation and recovery thresholds, and the spectrum of the key and query matrices. Our theory predicts the full singular-value distribution of the trained query–key map—including low-rank structure and isolated spectral outliers—in qualitative agreement with observations in more realistic transformers. Finally, for targets with power-law spectra, we show that learning proceeds through sequential spectral recovery, leading to the emergence of power-law scaling laws.

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Submission Number: 28