Bayes optimal learning of attention-indexed models

Fabrizio Boncoraglio, Emanuele Troiani, Vittorio Erba, Lenka Zdeborova

Keywords: Bayes optimal analysis, Transformers, Self-attention mechanism, Random matrix theory, Matrix models, Extensive-rank analysis

TL;DR: We introduce and analyze the Attention-Indexed Model (AIM), a theoretical framework for analyzing learning in deep attention layers.

Abstract: We introduce the attention-indexed model (AIM), a theoretical framework for analyzing learning in deep attention layers. Inspired by multi-index models, AIM captures how token-level outputs emerge from layered bilinear interactions over high-dimensional embeddings. Unlike prior tractable attention models, AIM allows full-width key and query matrices, aligning more closely with practical transformers. Using tools from statistical mechanics and random matrix theory, we derive closed-form predictions for Bayes-optimal generalization error and identify sharp phase transitions as a function of sample complexity, model width, and sequence length. We propose a matching approximate message passing algorithm and show that gradient descent can reach optimal performance. AIM offers a solvable playground for understanding learning in self-attention layers, that are key components of modern architectures.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 12680