The Origin of Edge of Stability

Summary

Introduces the edge coupling A_η(x,y)=L(x)+L(y)-(1/(2η))||x-y||^2, a discrete generating function whose x-criticality is the GD update. Differencing consecutive edges yields a propagator with stability boundary 2/η; a second-order expansion telescopes so weighted curvature is forced toward 2/η. Mean-value localization transfers the forcing to the true Hessian eigenvalue on each step segment with no residual gap. Full criticality classifies fixed points and period-two orbits; a center reduction reduces branching to a quartic on the half-amplitude. For two-layer linear nets the transverse theory is width-invariant and period-doubling appears continuously for η>1/σ_1. No official code on the abs page.

Keywords

edge-of-stability · sharpness · hessian · period-two · stanford · catapult

Topics

optimization, Edge of Stability, gradient descent

Research notes

• Primary: arxiv abs (cs.LG / stat.ML). Stanford; correspondence elonlit@stanford.edu. Discord posted abs. HF has no paper page (API 404). No official code on abs. ArXiv license widget not visible in converted abs HTML, so license left blank. Experiments use CIFAR-10 and synthetic linear-net targets; no new corpus, so no datasets_local row.