Directional Curvature from Armijo Backtracking: A Low-Cost Sharpness Probe and a Calibration-Free Learning-Rate Safeguard for Adam

arXiv:2607.03998v1 Announce Type: new Abstract: The local sharpness of the loss, the top Hessian eigenvalue $\lambda_1$, determines the largest stable gradient step, but measuring it normally requires Lanczos or Hessian-vector iterations. We observe that a single Armijo backtracking line search already carries this information at the cost of a few forward passes: the accepted step $\alpha$ brackets the \emph{directional} curvature $q = g^\top H g/\|g\|^2$ within the multiplicative band set by the backtracking factor. Across CIFAR-10, Fashion-MNIST and Imagenette, $\log\alpha$ tracks $\log\lamb