# Fundamental limits of learning in sequence multi-index models and deep attention networks: High-dimensional asymptotics and sharp thresholds

arXiv: https://arxiv.org/abs/2502.00901 · HTML: https://mlsanity.com/paper/2502.00901/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2025.

## Built on by (most cited first, 12 of 12)

- [The Nuclear Route: Sharp Asymptotics of ERM in Overparameterized Quadratic Networks](https://mlsanity.com/paper/2505.17958/) (2025)
- [Asymptotics of SGD in Sequence-Single Index Models and Single-Layer Attention Networks](https://mlsanity.com/paper/2506.02651/) (2025)
- [Preventing Model Collapse Under Overparametrization: Optimal Mixing Ratios for Interpolation Learning and Ridge Regression](https://mlsanity.com/paper/2509.22341/) (2025)
- [Statistical Advantage of Softmax Attention: Insights from Single-Location Regression](https://mlsanity.com/paper/2509.21936/) (2025)
- [Single-Head Attention in High Dimensions: A Theory of Generalization, Weights Spectra, and Scaling Laws](https://mlsanity.com/paper/2509.24914/) (2025)
- [Bayes optimal learning of attention-indexed models](https://mlsanity.com/paper/2506.01582/) (2025)
- [How Do Transformers Learn to Associate Tokens: Gradient Leading Terms Bring Mechanistic Interpretability](https://mlsanity.com/paper/2601.19208/) (2026)
- [High-Dimensional Theory of LoRA Fine-Tuning in a Solvable Attention Model](https://mlsanity.com/paper/2606.05899/) (2026)
- [Multi-layer Cross-attention is Provably Optimal for Multi-modal In-context Learning](https://mlsanity.com/paper/2602.04872/) (2026)
- [Incremental Learning of Sparse Attention Patterns in Transformers](https://mlsanity.com/paper/2602.19143/) (2026)
- [Specialization of softmax attention heads: insights from the high-dimensional single-location model](https://mlsanity.com/paper/2603.03993/) (2026)
- [Minimax Rates for Learning Pairwise Interactions in Attention-Style Models](https://mlsanity.com/paper/2510.11789/) (2025)

## Abstract (arXiv)

In this manuscript, we study the learning of deep attention neural networks, defined as the composition of multiple self-attention layers, with tied and low-rank weights. We first establish a mapping of such models to sequence multi-index models, a generalization of the widely studied multi-index model to sequential covariates, for which we establish a number of general results. In the context of Bayesian-optimal learning, in the limit of large dimension D and commensurably large number of samples N, we derive a sharp asymptotic characterization of the optimal performance as well as the performance of the best-known polynomial-time algorithm for this setting --namely approximate message-passing--, and characterize sharp thresholds on the minimal sample complexity required for better-than-random prediction performance. Our analysis uncovers, in particular, how the different layers are learned sequentially. Finally, we discuss how this sequential learning can also be observed in a realistic setup.