A Theoretical Framework for LLM Fine-tuning Using Early Stopping for Non-random Initialization

TL;DR

This paper develops a theoretical framework combining early stopping and Neural Tangent Kernel theory to explain why few epochs of fine-tuning pretrained large language models are effective, supported by empirical results.

Contribution

It extends NTK theory to non-random initializations and provides convergence guarantees for attention-based fine-tuning of LLMs.

Findings

Convergence rate linked to eigenvalue decay of the NTK

Framework explains task vectors in LLMs

Empirical results support theoretical insights

Abstract

In the era of large language models (LLMs), fine-tuning pretrained models has become ubiquitous. Yet the theoretical underpinning remains an open question. A central question is why only a few epochs of fine-tuning are typically sufficient to achieve strong performance on many different tasks. In this work, we approach this question by developing a statistical framework, combining rigorous early stopping theory with the attention-based Neural Tangent Kernel (NTK) for LLMs, offering new theoretical insights on fine-tuning practices. Specifically, we formally extend classical NTK theory [Jacot et al., 2018] to non-random (i.e., pretrained) initializations and provide a convergence guarantee for attention-based fine-tuning. One key insight provided by the theory is that the convergence rate with respect to sample size is closely linked to the eigenvalue decay rate of the empirical kernel matrix induced by the NTK. We also demonstrate how the framework can be used to explain task vectors for multiple tasks in LLMs. Finally, experiments with modern language models on real-world datasets provide empirical evidence supporting our theoretical insights.