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

Excerpt

arXiv:2607.03998v4 Announce Type: replace 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 products. A single Armijo backtracking line search already carries this information at the cost of a few forward passes: the accepted step $\alpha$ brackets the directional curvature along the probed direction within the multiplicative band set by the backtracking fact