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

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 $\epsilon$. While bounds obtained using the standard argument typically are of the form $\mathcal{O}({\epsilon }^{-\alpha })$ for some positive $\alpha$, the refined results are of the form $o({\epsilon }^{-\alpha })$. 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.