Bayes optimal learning of attention-indexed models
Fabrizio Boncoraglio, Emanuele Troiani, Vittorio Erba, Lenka Zdeborová
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-rank 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 modern attention architectures.