{"ID":5938035,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-07T19:36:44.467391193Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03998","arxiv_id":"2607.03998","title":"Directional Curvature from Armijo Backtracking: A Low-Cost Sharpness Probe and a Calibration-Free Learning-Rate Safeguard for Adam","abstract":"The local sharpness of the loss, the top Hessian eigenvalue $λ_1$, 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 $α$ 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, $\\logα$ tracks $\\logλ_1$ at Pearson $-0.91$ to $-0.95$, giving a low-cost online Edge-of-Stability reading. Used once at initialisation, this measurement yields a learning-rate cap (a safeguard, not a faster optimiser) that makes Adam robust to a too-large initial learning rate across more than three orders of magnitude ($10^{-3}$ to $3.0$), at about one percent overhead, and it is a no-op when the chosen rate is already safe. One probe is enough: periodic in-training probing adds no robust benefit. The raw-gradient probe exposes the mechanism but needs a safety factor calibrated to the architecture by a one-minute divergence sweep. Probing along Adam's own update direction removes this calibration: a single fixed safety factor $κ= 2$ avoids divergence on all nine architectures we test and across the full learning-rate grids of all four benchmarks, and the recipe transfers to AdamW unchanged.","short_abstract":"The local sharpness of the loss, the top Hessian eigenvalue $λ_1$, 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 acce...","url_abs":"https://arxiv.org/abs/2607.03998","url_pdf":"https://arxiv.org/pdf/2607.03998v1","authors":"[\"Ashmitha R\",\"Jörg Frochte\"]","published":"2026-07-04T19:49:01Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}