# Variational Structure at the Edge of Stability

arXiv: https://arxiv.org/abs/2608.21660 · HTML: https://mlsanity.com/paper/2608.21660/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 25 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 16 of 16)

- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [The Break-Even Point on Optimization Trajectories of Deep Neural Networks](https://mlsanity.com/paper/2002.09572/) (2020)
- [Understanding Gradient Descent on Edge of Stability in Deep Learning](https://mlsanity.com/paper/2205.09745/) (2022)
- [Adaptive Gradient Methods at the Edge of Stability](https://mlsanity.com/paper/2207.14484/) (2022)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Understanding Optimization in Deep Learning with Central Flows](https://mlsanity.com/paper/2410.24206/) (2024)
- [Beyond the Edge of Stability via Two-step Gradient Updates](https://mlsanity.com/paper/2206.04172/) (2022)
- [Conformal Symplectic and Relativistic Optimization](https://mlsanity.com/paper/1903.04100/) (2019)
- [Optimization with Momentum: Dynamical, Control-Theoretic, and Symplectic Perspectives](https://mlsanity.com/paper/2002.12493/) (2020)
- [Momentum Further Constrains Sharpness at the Edge of Stochastic Stability](https://mlsanity.com/paper/2604.14108/) (2026)
- [Rod Flow: A Continuous-Time Model for Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2602.01480/) (2026)
- [The Origin of Edge of Stability](https://mlsanity.com/paper/2604.20446/) (2026)
- [Gradient Descent with Polyak's Momentum Finds Flatter Minima via Large Catapults](https://mlsanity.com/paper/2311.15051/) (2023)
- [A Rod Flow Model for Adam at the Edge of Stability](https://mlsanity.com/paper/2605.06821/) (2026)
- [Edge Flow: A Tractable and Predictive Continuous-Time Model for Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2606.18080/) (2026)

## Abstract (arXiv)

When discrete-time optimizers operate at the edge of stability, they exhibit near-two-periodic behavior. These oscillatory dynamics are reminiscent of conservative systems, such as the dynamics generated by symplectic integrators. However, a precise formulation of the connection between discrete-time optimizers at the edge of stability and discrete mechanics remains underexplored. Recently, Litman introduced the "edge coupling": a functional on consecutive gradient descent iterates whose critical points encode the fixed points and two-point orbits of the gradient descent dynamics. Here we extend the edge coupling to heavy-ball and Nesterov momentum. We show that its critical points characterize the fixed points and two-point orbits, with its Hessian characterizing their stability. We also show that the edge coupling can be identified with the symmetric Verlet action, formalizing the connection between the edge of stability and discrete mechanics.