Mean--Fluctuation Dynamics at the Edge of Stability

תקציר מקורי באנגליתarXiv:2606.05326v2 Announce Type: replace-cross Abstract: We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the trajectory, which has been linked to better generalization performance. We introduce the mean--fluctuation dynamics, a tractable continuous-time model coupling the window-averaged trajectory to its fluctuation covariance. Among our contributions, we rigorously derive this model from gradient descent in a sharp-valley framework, characterize its stationary states and their linear stability, and establish precise connections with other effective dynamics. Numerical experiments illustrate these predictions and their finite-time limitations. We also study our model in the overparamet