Refining asymptotic complexity bounds for nonconvex optimization methods, including why steepest descent is o(eps^{-2}) rather than O(eps^{-2})

We revisit the standard ``telescoping sum'' argument ubiquitous in the final steps of analyzing evaluation complexity of algorithms for smooth nonconvex optimization, and obtain a refined formulation of the resulting bound as a function of the requested accuracy eps. While bounds obtained using the standard argument typically are of the form O(eps^{-\alpha}) for some positive alpha, the refined results are of the form o(eps^{-\alpha}). We then explore to which known algorithms our refined bounds are applicable and finally describe an example showing how close the standard and refined bounds can be.