Variational Structure at the Edge of Stability

Full-batch gradient descent on neural networks drives the largest Hessian eigenvalue to the threshold $2/η$, where $η$ is the learning rate. This phenomenon, the Edge of Stability, has resisted a unified explanation: existing accounts establish self-regulation near the edge but do not explain why the trajectory is forced toward $2/η$ from arbitrary initialization. We introduce the edge coupling, a functional on consecutive iterate pairs whose coefficient is uniquely fixed by the gradient-descent update. Differencing its criticality condition yields a step recurrence with stability boundary $2/η$, and a second-order expansion yields a loss-change formula whose telescoping sum forces curvature toward $2/η$. The two formulas involve different Hessian averages, but the mean value theorem localizes each to the true Hessian at an interior point of the step segment, yielding exact forcing of the Hessian eigenvalue with no gap. Setting both gradients of the edge coupling to zero classifies fixed points and period-two orbits; near a fixed point, the problem reduces to a function of the half-amplitude alone, which determines which directions support period-two orbits and on which side of the critical learning rate they appear.

The edge of stability refers to a phenomenon in deep learning with gradient-based optimizers where the Hessian eigenvalues of the loss remain stable above a threshold that the classical descent lemma predicts to be unstable. Previous works formulate the edge of stability with respect to the maximum Hessian eigenvalue and the learning rate. However, we observe that many first-order methods, including gradient descent, significantly violate the stability bound predicted by these theories by a factor as large as $\times 21.1$. Moreover, this deviation turns out to be systematic and highly dependent on the underlying optimizer, which is not captured by previous formulations. This calls for a new formulation of the stability threshold, which we derive from the directional Hessian and the gradient-alignment score with respect to the actual update taken by the optimizer, rather than the maximum curvature mode. Our new formulation of the realized edge of stability not only removes optimizer-dependent offsets and provides more consistent predictions of the stability threshold, but also introduces new diagnostic tools that reveal the unique role of the optimizer in actively balancing between the temporal and spatial budgets in first-order optimization.

This work establishes a rigorous connection between stability properties of discrete-time algorithms (DTAs) and corresponding continuous-time dynamical systems derived through $O(s^r)$-resolution ordinary differential equations (ODEs). We show that for discrete- and continuous-time dynamical systems satisfying a mild error assumption, exponential stability of a common equilibrium with respect to the continuous time dynamics implies exponential stability of the corresponding equilibrium for the discrete-time dynamics, provided that the step size is chosen sufficiently small. We extend this result to common compact invariant sets. We prove that if an equilibrium is exponentially stable for the $O(s^r)$-resolution ODE, then it is also exponentially stable for the associated DTA. We apply this framework to analyse the limit point properties of several prominent optimisation algorithms, including Two-Timescale Gradient Descent--Ascent (TT-GDA), Generalised Extragradient (GEG), Two-Timescale Proximal Point (TT-PPM), Damped Newton (DN), Regularised Damped Newton (RDN), and the Jacobian method (JM), by studying their $O(1)$- and $O(s)$-resolution ODEs. We show that under a proper choice of hyperparameters, the set of saddle points of the objective function is a subset of the set of exponentially stable equilibria of GEG, TT-PPM, DN, and RDN. We relax the common Hessian invariance assumption through direct analysis of the resolution ODEs, broadening the applicability of our results. Numerical examples illustrate the theoretical findings.