Directional Curvature from Armijo Backtracking: A Low-Cost Sharpness Probe and a Calibration-Free Learning-Rate Safeguard for Adam
Local sharpness, defined by the largest Hessian eigenvalue $λ_1$, sets the maximum stable gradient update size, but its computation would usually require running Lanczos or Hessian-vector products. However, we notice that even a single Armijo backtracking line search already contains this information with just a few forward passes, as the accepted step $α$ determines the directional curvature along the search direction up to the multiplicative band set by the backtracking factor. The correlation between $\logα$ and $\logλ_1$ on CIFAR-10, Fashion-MNIST and Imagenette reaches $-0.91$ to $-0.95$ in Pearson correlation, and even after removing the trend per run the correlation remains at $-0....
| Authors | Ashmitha R, Jörg Frochte |
|---|---|
| Primary category | cs.LG |
| Published | 4 July 2026 |
| Last updated at source | 7 October 2026 |
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