  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 $O\left(N^{-1}\right)$.
