Global Convergence in Training Large-Scale Transformers

Cheng Gao, Yuan Cao, Zihao Li, Yihan He, Mengdi Wang, Han Liu, Jason Matthew Klusowski, Jianqing Fan

Keywords: Transformer, gradient flow, mean-field analysis, global convergence

TL;DR: We prove that large-scale Transformers closely approximate their mean-field limit and that gradient flow converges to the global optimization minimum.

Abstract: Despite the widespread success of Transformers across various domains, their optimization guarantees in large-scale model settings are not well-understood. This paper rigorously analyzes the convergence properties of gradient flow in training Transformers with weight decay regularization. First, we construct the mean-field limit of large-scale Transformers, showing that as the model width and depth go to infinity, gradient flow converges to the Wasserstein gradient flow, which is represented by a partial differential equation. Then, we demonstrate that the gradient flow reaches a global minimum consistent with the PDE solution when the weight decay regularization parameter is sufficiently small. Our analysis is based on a series of novel mean-field techniques that adapt to Transformers. Compared with existing tools for deep networks (Lu et al., 2020) that demand homogeneity and global Lipschitz smoothness, we utilize a refined analysis assuming only $\textit{partial homogeneity}$ and $\textit{local Lipschitz smoothness}$. These new techniques may be of independent interest.

Primary Area: Optimization for deep networks

Submission Number: 8415