Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Steps

This work considers gradient descent for L-smooth convex optimization with stepsizes larger than the classic regime where descent can be ensured. The stepsize schedules considered are similar to but differ slightly from the recent silver stepsizes of Altschuler and Parrilo. For one of our stepsize sequences, we prove a $O\left(N^{- 1.2716\dots}\right)$ convergence rate in terms of objective gap decrease and for the other, we show the same rate of decrease for squared-gradient-norm decrease. This first result improves on the recent result of Altschuler and Parrilo by a constant factor, while the second results improve on the exponent of the prior best squared-gradient-norm convergence guarantee of $O\left(N^{-1}\right)$.