A short proof of near-linear convergence of adaptive gradient descent under fourth-order growth and convexity

ArXi:2604.13393v1 Announce Type: cross Davis, Drusvyatskiy, and Jiang showed that gradient descent with an adaptive stepsize converges locally at a nearly-linear rate for smooth functions that grow at least quartically away from their minimizers. The argument is intricate, relying on monitoring the performance of the algorithm relative to a certain manifold of slow growth -- called the ravine. In this work, we provide a direct Lyapuno-based argument that bypasses these difficulties when the objective is in addition convex and a has a unique minimizer.