  Recently Grimmer [1] showed for smooth convex optimization by utilizing
longer steps periodically, gradient descent's textbook $LD^2/2T$ convergence
guarantees can be improved by constant factors, conjecturing an accelerated
rate strictly faster than $O(1/T)$ could be possible. Here we prove such a
big-O gain, establishing gradient descent's first accelerated convergence rate
in this setting. Namely, we prove a $O(1/T^{1.0564})$ rate for smooth convex
minimization by utilizing a nonconstant nonperiodic sequence of increasingly
large stepsizes. It remains open if one can achieve the $O(1/T^{1.178})$ rate
conjectured by Das Gupta et. al. [2] or the optimal gradient method rate of
$O(1/T^2)$. Big-O convergence rate accelerations from long steps follow from
our theory for strongly convex optimization, similar to but somewhat weaker
than those concurrently developed by Altschuler and Parrilo [3].
