# The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow

arXiv: https://arxiv.org/abs/2609.01034 · HTML: https://mlsanity.com/paper/2609.01034/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 42 of its 45 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 30 of 42)

- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Gradient Descent Happens in a Tiny Subspace](https://mlsanity.com/paper/1812.04754/) (2018)
- [Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond](https://mlsanity.com/paper/1611.07476/) (2016)
- [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)
- [Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra](https://mlsanity.com/paper/2008.11865/) (2020)
- [On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length](https://mlsanity.com/paper/1807.05031/) (2018)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Understanding Warmup-Stable-Decay Learning Rates: A River Valley Loss Landscape Perspective](https://mlsanity.com/paper/2410.05192/) (2024)
- [A Walk with SGD](https://mlsanity.com/paper/1802.08770/) (2018)
- [Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians](https://mlsanity.com/paper/1901.08244/) (2019)
- [Understanding Optimization in Deep Learning with Central Flows](https://mlsanity.com/paper/2410.24206/) (2024)
- [Analyzing Sharpness along GD Trajectory: Progressive Sharpening and Edge of Stability](https://mlsanity.com/paper/2207.12678/) (2022)
- [Understanding Edge-of-Stability Training Dynamics with a Minimalist Example](https://mlsanity.com/paper/2210.03294/) (2022)
- [Beyond the Quadratic Approximation: the Multiscale Structure of Neural Network Loss Landscapes](https://mlsanity.com/paper/2204.11326/) (2022)
- [Implicit Bias of Gradient Descent for Logistic Regression at the Edge of Stability](https://mlsanity.com/paper/2305.11788/) (2023)
- [Second-order regression models exhibit progressive sharpening to the edge of stability](https://mlsanity.com/paper/2210.04860/) (2022)
- [Beyond the Edge of Stability via Two-step Gradient Updates](https://mlsanity.com/paper/2206.04172/) (2022)
- [Hessian based analysis of SGD for Deep Nets: Dynamics and Generalization](https://mlsanity.com/paper/1907.10732/) (2019)
- [Does SGD really happen in tiny subspaces?](https://mlsanity.com/paper/2405.16002/) (2024)
- [Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency](https://mlsanity.com/paper/2402.15926/) (2024)
- [Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory](https://mlsanity.com/paper/2307.04204/) (2023)
- [Good regularity creates large learning rate implicit biases: edge of stability, balancing, and catapult](https://mlsanity.com/paper/2310.17087/) (2023)
- [Universal Sharpness Dynamics in Neural Network Training: Fixed Point Analysis, Edge of Stability, and Route to Chaos](https://mlsanity.com/paper/2311.02076/) (2023)
- [Gradient Descent Monotonically Decreases the Sharpness of Gradient Flow Solutions in Scalar Networks and Beyond](https://mlsanity.com/paper/2305.13064/) (2023)
- [Large Stepsize Gradient Descent for Non-Homogeneous Two-Layer Networks: Margin Improvement and Fast Optimization](https://mlsanity.com/paper/2406.08654/) (2024)
- [Neural Thermodynamic Laws for Large Language Model Training](https://mlsanity.com/paper/2505.10559/) (2025)

## Abstract (arXiv)

The central flow of Cohen et al. (2025) is an empirically accurate continuous-time model of gradient descent at the edge of stability in deep learning, However, its derivation is heuristic. We propose a perturbative regime in which the central flow is the limit of gradient descent: we assume that the loss decomposes as f = g + varepsilon h; in the limit varepsilon → 0, the dynamics of gradient descent with learning rate η converge to the gradient flow of h constrained to the minimizers of g of sharpness at most 2/η. Our approach is formal rather than rigorous; it treats gradient descent as a singularly perturbed dynamical system in varepsilon. Three timescales emerge: a fast timescale of oscillations along the sharpest direction, an intermediate timescale of the self-stabilization mechanism, and a slow timescale of the dynamics along the minimizers of g-the central flow. Using the method of multiple scales, a classical formal method from singular perturbation theory, we derive the expansion of the dynamics in varepsilon: the central flow emerges as the leading-order term in the expansion, while the self-stabilization mechanism appears in the next-order term. We study this mechanism beyond previous analyses: with a single eigenvalue at the edge of stability, we compute the slow drift of the energy of the fluctuations; with several eigenvalues at the edge of stability, we derive the self-stabilization system and explain why fluctuations persist.