Classification at the Edge of Stability: Unifying Self-Stabilization and Convergence Rates

Neural networks are often trained with gradient descent (GD) at the edge of stability (EoS), where step sizes exceed classical stability thresholds and standard convergence theory no longer applies. Despite exhibiting non-monotonic loss and oscillatory dynamics, this regime frequently leads to faster convergence and better generalization. Existing theoretical perspectives offer complementary but incomplete insights: one identifies a self-stabilization mechanism that drives GD toward flat regions but does not yield convergence rates; another derives rigorous rates for specific losses but does not explain the underlying stabilization mechanism; and a third unifies these views for overparameterized least squares, but relies on the existence of a minimizer manifold, which is absent in classification settings. In this work, we extend this unified geometric perspective to exponential-tail losses, including logistic and cross-entropy losses, where no finite minimizer exists. We identify a curvature-defined reference subspace that replaces the role of the minimizer manifold, yielding a coupled dynamical system in which orthogonal oscillations are damped by parallel progress along directions of decreasing sharpness. Under regularity conditions, which we verify for logistic and multiclass cross-entropy losses in symmetric data models, we derive explicit bounds for the iterates, not just the loss, covering both the EoS and stable regimes, and revealing a three-phase damping structure in the latter. Our analysis further uncovers a transient implicit bias induced by large step sizes: although GD converges asymptotically to the max-margin direction, large steps can leave persistent orthogonal residuals and, in certain geometries, amplify them.