Provable Edge-of-Stability for Adam on a One-Dimensional Quadratic

What the math gives to ML

This paper isolates an optimizer-induced stability boundary for uncorrected Adam without relying on changing loss curvature or high-dimensional geometry. On a one-dimensional quadratic, freezing the second moment reduces Adam to a linear two-state recurrence whose spectral radius crosses one exactly at a normalized effective sharpness of 2, with momentum dependence absorbed by c=(1-beta_1)/(1+beta_1). The transferable asset is a computable stability diagnostic for Adam: estimate local curvature and compare c times eta times curvature divided by sqrt(v)+epsilon with the universal boundary 2. A practical adaptation is a layerwise controller that reduces Adam's learning rate before this effective sharpness becomes supercritical.