  We consider gradient descent with constant stepsizes and derive exact
worst-case convergence rates on the minimum gradient norm of the iterates. Our
analysis covers all possible stepsizes and arbitrary upper/lower bounds on the
curvature of the objective function, thus including convex, strongly convex and
weakly convex (hypoconvex) objective functions. Unlike prior approaches, which
rely solely on inequalities connecting consecutive iterations, our analysis
employs inequalities involving an iterate and its two predecessors. While this
complicates the proofs to some extent, it enables us to achieve, for the first
time, an exact full-range analysis of gradient descent for any constant
stepsizes (covering, in particular, normalized stepsizes greater than one),
whereas the literature contained only conjectured rates of this type. In the
nonconvex case, allowing arbitrary bounds on upper and lower curvatures extends
existing partial results that are valid only for gradient Lipschitz functions
(i.e., where lower and upper bounds on curvature are equal), leading to
improved rates for weakly convex functions. From our exact rates, we deduce the
optimal constant stepsize for gradient descent. Leveraging our analysis, we
also introduce a new variant of gradient descent based on a unique, fixed
sequence of variable stepsizes, demonstrating its superiority over any constant
stepsize schedule.
