The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow
Berthier gives a formal perturbative route from discrete gradient descent to the central flow model.
The paper assumes a loss split as `f = g + εh` and studies the `ε → 0` limit at a fixed learning rate. In that regime, gradient descent is described as moving along minimizers of `g` whose sharpness stays at most `2/η`. The derivation separates three timescales: fast oscillations, intermediate self-stabilization, and slow central-flow motion. The author says the argument is formal, not rigorous, and extends the self-stabilization analysis to both single and multiple edge eigenvalues. ArXiv · AI/CL/LG's note
The paper assumes a loss split as `f = g + εh` and studies the `ε → 0` limit at a fixed learning rate. In that regime, gradient descent is described as moving along minimizers of `g` whose sharpness stays at most `2/η`. The derivation separates three timescales: fast oscillations, intermediate self-stabilization, and slow central-flow motion. The author says the argument is formal, not rigorous, and extends the self-stabilization analysis to both single and multiple edge eigenvalues. ArXiv · AI/CL/LG's note
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