---
title: "A Bifurcation Theory Framework for Gradient Descent on the Edge of Stability"
created: 2026-06-23
type: paper-raw
arxiv: "2606.15551v1"
category: cs.LG
author: "Eric Gan"
date: 2026-06-14
venue: Preprint
---

# A Bifurcation Theory Framework for Gradient Descent on the Edge of Stability

- **作者**: Eric Gan (Independent Researcher, egan8@ucla.edu)
- **arXiv**: 2606.15551v1
- **领域**: cs.LG (Machine Learning)
- **日期**: 2026-06-14
- **来源**: https://arxiv.org/abs/2606.15551

## 摘要

The Edge of Stability (EoS) phenomenon, where gradient descent operates with sharpness exceeding the classical convergence threshold yet the loss decreases over long timescales, is ubiquitous in modern deep learning but remains poorly understood in realistic settings. Prior rigorous analyses have been largely confined to scalar or low-dimensional losses with specific structural forms. In this work, we develop a bifurcation theory framework for gradient descent on the edge of stability that applies directly to overparameterized neural networks. By decomposing the training dynamics into components normal and tangent to the manifold of minimizers, we show that stable EoS training arises from a flip bifurcation in the normal direction, governed by the sign of the first Lyapunov coefficient, while the tangent dynamics drift toward regions of decreasing sharpness. Under mild spectral and geometric assumptions on the loss landscape, we prove convergence to the minimizing manifold when training at the EoS threshold. As a corollary, we recover and unify prior results: we show that the product-stability condition of Gan (2026) is an instance of our framework.

## 核心贡献

1. 发展了一个适用于过参数化网络的分岔理论 EoS 框架
2. 将 EoS 动力学分解为法向 flip 分岔 + 切向 sharpness 递减漂移
3. 证明了在 EoS 阈值处（η = 2/λ_max）收敛到极小值流形 (Theorem 4.4)
4. 统一了乘积稳定性 (Gan 2026) 为框架特例

## 关键技术工具

- 中心流形定理 (Center Manifold Theorem)
- 投影法 (Projection Method)
- 第一 Lyapunov 系数 (c₁)
- Morse-Bott 条件 + 谱间隙假设
