Title: On Gradient Descent Convergence beyond the Edge of Stability. Abstract: Gradient Descent (GD) is a powerful workhorse of modern machine learning thanks to its scalability and efficiency in high-dimensional spaces. Its ability to find local minimisers is only guaranteed for losses with Lipschitz gradients, where it can be seen as a `bona-fide' discretisation of an underlying gradient flow. Yet, many ML setups involving overparametrised models do not fall into this problem class, which has motivated research beyond the so-called ``Edge of Stability'' (EoS), where the step-size crosses the admissibility threshold inversely proportional to the Lipschitz constant above. Perhaps surprisingly, GD has been empirically observed to still converge regardless of local instability and oscillatory behavior.

The incipient theoretical analysis of this phenomena has mainly focused in the overparametrised regime, where the effect of choosing a large learning rate may be associated to a `Sharpness-Minimisation' implicit regularisation within the manifold of minimisers, under appropriate asymptotic limits. In contrast, in this work we directly examine the conditions for such unstable convergence, focusing on simple, yet representative, learning problems. Specifically, we characterize a local condition involving third-order derivatives that stabilizes oscillations of GD above the EoS, and leverage such property in a teacher-student setting, under population loss. Finally, focusing on Matrix Factorization, we establish a non-asymptotic `Local Implicit Bias' of GD above the EoS, whereby quasi-symmetric initializations converge to symmetric solutions --- where sharpness is minimum amongst all minimisers. 