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

Abstract

Local sharpness, defined by the largest Hessian eigenvalue $\lambda_1$, 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 $\alpha$ determines the directional curvature along the search direction up to the multiplicative band set by the backtracking factor. The correlation between $\log\alpha$ and $\log\lambda_1$ 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 $-0.60$ to $-0.70$. This allows for a cheap online Edge-of-Stability estimate of the slow sharpness component. The employed probing mechanism searches along Adam's first-step update direction, at initialisation and nine times over the course of the first 50 optimiser steps. The learning-rate cap is set as twice the smallest observed step size, and in the studied learning-rate ranges ($10^{-3}$ to $3.0$) and GPT-2 pretraining experiments all capped runs avoid divergence; in the general architecture analysis, one MLP architecture is still sensitive to the initial batch order. This probing protocol incurs an approximately one percent overhead, and using a non-binding cap means that the optimiser's state and first update are bit-identical. There is no fine-tuning of any of the protocol parameters to any specific architecture; this is the sense in which this is a calibration-free safeguard. It is meant as a way of avoiding divergence, not achieving accuracy. At GPT-2 scale, multiple measurements also illustrate why an initialisation-only cap is insufficient: directional curvature rises substantially in the first five optimiser steps, which motivates the short probationary window.