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
The paper investigates how continuous symmetries in neural networks, which give rise to conserved quantities under gradient flow, behave when training with finite‑step gradient descent at the edge of stability (EOS). By applying central flow dynamics, the authors show that curvature corrections cause a finite‑dimensional drift driven by oscillation covariance along critical parameter directions. They analyze this effect in a scalar two‑layer model, two‑layer matrix factorisation, and a two‑layer ReLU network, demonstrating which conservation laws persist or break under EOS conditions.
By Naima Elosegui Borras, Georgios Arvanitidis
The paper investigates how stochastic gradient descent (SGD) selects specific functional decompositions when training a deep linear residual network to learn the identity function. Although many weight configurations minimize the population loss, SGD consistently prefers particular solutions, especially under anisotropic label noise or different parametrizations. The authors explain this bias using an entropic loss term that penalizes the expected squared norm of the minibatch gradient, analytically characterizing its minimizers and showing that trained networks align with these predictions.
By Andy Arditi, Weian Xie, David Bau, Liu Ziyin
arXiv:2607. 21005v1 Announce Type: new Abstract: Most explanations of training instability focus on \emph{learning-rate criticality}, typically characterized by the Edge of Stability, beyond which optimization becomes unstable.
By Xiaolong Li, Zhangchen Zhou, Zhi-Qin John Xu
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:2607. 07845v1 Announce Type: new Abstract: The Hessian of the training loss governs the local geometry of the loss landscape, yet despite existing explanations for its largest eigenvalues, the origin of the vast multitude of vanishingly small eigenvalues remains elusive.
By Marcel K\"uhn, Bernd Rosenow