Training Dynamics of Softmax Self-Attention: Fast Global Convergence via Preconditioning

2026-03-02

AI Summary

This work addresses the challenge of achieving global convergence in Softmax-based self-attention layers for regression tasks by modeling the problem as a non-convex matrix factorization in the infinite data limit. The authors propose a structure-aware optimization framework that integrates spectral initialization—tailored to the underlying data distribution—with preconditioned gradient descent and regularization techniques. This approach effectively avoids spurious stationary points and ensures, with high probability, that the initial parameters lie in a neighborhood of the global optimum manifold. Theoretical analysis demonstrates that the proposed algorithm achieves global convergence at a geometric rate, substantially improving both training efficiency and stability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

We study the training dynamics of gradient descent in a softmax self-attention layer trained to perform linear regression and show that a simple first-order optimization algorithm can converge to the globally optimal self-attention parameters at a geometric rate. Our analysis proceeds in two steps. First, we show that in the infinite-data limit the regression problem solved by the self-attention layer is equivalent to a nonconvex matrix factorization problem. Second, we exploit this connection to design a novel "structure-aware" variant of gradient descent which efficiently optimizes the original finite-data regression objective. Our optimization algorithm features several innovations over standard gradient descent, including a preconditioner and regularizer which help avoid spurious stationary points, and a data-dependent spectral initialization of parameters which lie near the manifold of global minima with high probability.

Problem

• softmax self-attention
• training dynamics
• global convergence
• gradient descent
• nonconvex optimization

Innovation

• softmax self-attention
• gradient descent
• preconditioning
• matrix factorization
• spectral initialization