Refining asymptotic complexity bounds for nonconvex optimization methods, including why steepest descent is o(ɛ^-2) rather than O(ɛ^-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 ɛ. While bounds obtained using the standard argument typically are of the form O(ɛ^-α) for some positive α, the refined results are of the form o(ɛ^-α). 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.