The paper investigates gradient descent dynamics in the Edge of Stability regime, where a large learning rate causes persistent oscillations linked to improved generalization. It introduces a tractable continuous‑time mean–fluctuation model that couples the window‑averaged trajectory with its fluctuation covariance, derives this model rigorously from a sharp‑valley framework, and analyzes its stationary states and linear stability. The authors also extend the model to wide two‑layer networks, deriving a Wasserstein‑2 gradient flow for weights and fluctuations, proving well‑posedness, a mean‑field limit, and conditional convergence results, with numerical experiments illustrating the predictions and finite‑time limitations.
By Antonin Chodron de Courcel
arXiv:2606. 18080v1 Announce Type: new Abstract: Gradient descent in deep learning may operate at the edge of stability (EoS), a regime in which the largest eigenvalue of the loss Hessian hovers near the stability threshold $2/\eta$, where $\eta$ is the learning rate.
By Pierre Marion
arXiv:2505. 13196v3 Announce Type: replace-cross Abstract: We introduce Velocity-Regularized Adam (VRAdam), a physics-inspired optimizer for training deep neural networks that draws on ideas from quartic terms for kinetic energy with its stabilizing effects on various system dynamics.
By Pranav Vaidhyanathan, Lucas Schorling, Natalia Ares, Maike Osborne
arXiv:2606. 30384v1 Announce Type: new Abstract: Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape.
By Pedro Jim\'enez-Gonz\'alez, Miguel C. Soriano, Lucas Lacasa
Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape. However, the large number of trainable parameters makes the direct analysis of these dynamics challenging.
arXiv:2607. 04993v1 Announce Type: cross Abstract: Many phenomena of deep learning are dynamical: they concern not only which minima exist, but how gradient descent reaches, avoids, or selects among them.
By Thomas Hofmann