  This work establishes new convergence guarantees for gradient descent in
smooth convex optimization via a computer-assisted analysis technique. Our
theory allows nonconstant stepsize policies with frequent long steps
potentially violating descent by analyzing the overall effect of many
iterations at once rather than the typical one-iteration inductions used in
most first-order method analyses. We show that long steps, which may increase
the objective value in the short term, lead to provably faster convergence in
the long term. A conjecture towards proving a faster $O(1/T\log T)$ rate for
gradient descent is also motivated along with simple numerical validation.
