Fundamental limits of learning in sequence multi-index models and deep attention networks: High-dimensional asymptotics and sharp thresholds (2502.00901v1)

Abstract: 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.

• The paper establishes a formal equivalence between deep attention networks and sequence multi-index models, enabling precise asymptotic performance analysis.
• It derives sharp phase transitions and optimal estimation thresholds in high-dimensional regimes using Bayesian inference and AMP algorithms.
• It uncovers sequential layer-wise learning dynamics, indicating different sample complexities for each layer in deep attention networks.

Analyzing the Fundamental Limits of Learning in Deep Attention Networks

Introduction

This paper provides a rigorous analysis of the fundamental limits of learning in sequence multi-index models and deep attention networks, which have become critical in modern AI applications, particularly in handling sequential data like language. By mapping deep attention networks with tied and low-rank weights to sequence multi-index models, the authors extend the theoretical framework of multi-index models to sequence models and explore the learning dynamics in high-dimensional regimes. This paper involves an asymptotic characterization of optimal performance and computational limits using Bayesian inference and approximate message-passing (AMP) algorithms.

Theoretical Contributions

The paper's major contributions can be summarized as follows:

1. Connection to Sequence Multi-Index Models: The authors reveal a formal equivalence between deep attention networks and sequence multi-index (SMI) models. This connection allows insights and techniques from traditional multi-index models to be applied to more complex attention networks. By expanding on the SMI function class, they navigate the complexities of sequential data and model the dependencies within.
2. Asymptotic Analysis and Sharp Thresholds: Using both rigorous theoretical methods and non-rigorous replica techniques, the paper derives sharp asymptotic expressions for statistically and computationally optimal estimation errors. This includes revealing phase transitions—critical sample complexities beyond which better-than-random prediction performance is achievable.
3. Algorithmic Investigation via GAMP: The use of generalized approximate message passing (GAMP) provides a computational lens to evaluate the model's performance, delivering insights about the sample complexity required for successful learning, with the state evolution equations explicitly detailing AMP's performance.
4. Sequential Layer-Wise Learning: A particularly insightful finding is the characterization of sequential layer-wise learning, especially in deep attention networks. The paper identifies different sample complexities for learning different layers, implying that attention layers are learned one after another, rather than simultaneously.

Implications and Future Directions

The implications of this research are profound, establishing a bridge between shallow multi-index models and deeper models necessary for sequence learning, like attention networks. By providing a statistically grounded foundation, this work opens the door for further theoretical exploration and practical enhancement of deep learning models for sequential data tasks.

For practitioners and researchers, this formulation of learning limits serves as both a theoretical baseline and a diagnostic tool for evaluating model performance relative to the fundamental statistical limits. It also points to dynamic model tuning strategies where certain layers could be prioritized based on sample availability and task complexity.

Future work could extend these insights to architectures involving more layers or complex sequence interactions, like those found in the full transformer models with multiple heads and trainable value layers. Another exciting direction could involve exploring how these theoretical findings translate in the context of non-Gaussian inputs, or how they integrate with recent trends in data augmentation and synthetic data regimes.

Conclusion

In summary, this paper rigorously delineates the statistical and computational constraints of learning in deep attention networks utilizing a sequence multi-index model formulation. The work stands as a significant contribution to the theoretical understanding of deep sequential learning, equipping researchers with potent analytical tools that complement empirical approaches in modern AI development.