A Bifurcation Theory Framework for Gradient Descent on the Edge of Stability
Mirrored from arXiv — Machine Learning for archival readability. Support the source by reading on the original site.
Computer Science > Machine Learning
Title:A Bifurcation Theory Framework for Gradient Descent on the Edge of Stability
Abstract: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.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2606.15551 [cs.LG] |
| (or arXiv:2606.15551v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.15551
arXiv-issued DOI via DataCite (pending registration)
|
Access Paper:
- View PDF
- HTML (experimental)
- TeX Source
References & Citations
Bookmark
Bibliographic and Citation Tools
Code, Data and Media Associated with this Article
Demos
Recommenders and Search Tools
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
More from arXiv — Machine Learning
-
Can AI Draw Science? A Benchmark for Evaluating Scientific Figure Generation by Text-to-Image and Multimodal Models
Jun 30
-
On the Necessity of a Liquid Substrate for Mesh Intelligence
Jun 30
-
Position: RL Researchers Need to Distinguish Between Solving Simulators and Using Simulators as a Proxy
Jun 30
-
Learning to Distributedly Estimate under Partially Known Dynamics: A Covariance-Agnostic Neural Kalman Consensus Filter
Jun 30
Discussion (0)
Sign in to join the discussion. Free account, 30 seconds — email code or GitHub.
Sign in →No comments yet. Sign in and be the first to say something.