Finite-Time Analysis of Gradient Descent for Shallow Transformers

2026-01-23

AI Summary

This work investigates the reasons behind the superior performance of Transformers in non-convex optimization, with a focus on the relationship between optimization error and sequence length. By analyzing a shallow multi-head Transformer trained via kernelized mechanisms and projected gradient descent, the authors theoretically establish that the model width need only grow logarithmically with the sample size to guarantee finite-time non-asymptotic convergence, and that the optimization error remains independent of sequence length. This result overcomes the well-known limitation in recurrent architectures, where optimization error typically grows exponentially with sequence length. Numerical experiments within a teacher-student framework corroborate the theoretically predicted scaling laws, highlighting the structural advantage of Transformers in optimizing long sequences.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Understanding why Transformers perform so well remains challenging due to their non-convex optimization landscape. In this work, we analyze a shallow Transformer with $m$ independent heads trained by projected gradient descent in the kernel regime. Our analysis reveals two main findings: (i) the width required for nonasymptotic guarantees scales only logarithmically with the sample size $n$, and (ii) the optimization error is independent of the sequence length $T$. This contrasts sharply with recurrent architectures, where the optimization error can grow exponentially with $T$. The trade-off is memory: to keep the full context, the Transformer's memory requirement grows with the sequence length. We validate our theoretical results numerically in a teacher-student setting and confirm the predicted scaling laws for Transformers.

Problem

• Transformers
• non-convex optimization
• gradient descent
• optimization error
• sequence length

Innovation

• finite-time analysis
• shallow Transformers
• kernel regime
• optimization error
• sequence length independence