A Simple and Efficient Measure of Loss Landscape Curvature

The Edge of Stability characterizes training dynamics through a sharpness measure — the Hessian eigenvalue for GD, a preconditioned variant for adaptive methods — reaching an optimizer-dependent threshold, yet measuring this quantity is prohibitive at scale and incompatible with modern training kernels. We revisit directional curvature along the optimizer's update direction as a scalable alternative, and show that its one-step descent boundary stays at $2/\eta$ across optimizers — including momentum and adaptive methods — by absorbing the optimizer state into the update direction rather than the threshold. To make this measure practical, we introduce two forward-pass estimators that avoid Hessian-vector products: a symmetric finite-difference estimator and a KL-divergence estimator requiring only one or two extra forward passes per step. On full-batch CIFAR-10 training under GD and AdamW, both estimators reproduce progressive sharpening and oscillate around the predicted boundary, with the KL-based estimator tracking the reference curvature cleanly at lower cost.