Exact worst-case convergence rates of gradient descent: a complete analysis for all constant stepsizes over nonconvex and convex functions

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. Among the challenging parts of the analysis, we note the necessity to exploit dependencies between non-consecutive iterates. While this complicates the proofs to some extent, it enables us to achieve an exact full-range analysis of gradient descent for any constant stepsize (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 worst-case performance bounds, 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 in the worst-case over any constant stepsize schedule.