细化非凸优化方法的渐近复杂度界，包括为何最速下降是$o(ε^{-2})$而非$\boldsymbol{\text{O}}(ε^{-2})$
Refining asymptotic complexity bounds for nonconvex optimization methods, including why steepest descent is $o(ε^{-2})$ rather than $\mathcal{O}(ε^{-2})$
AI总结：
本研究重新分析光滑非凸优化的望远镜求和论证，将复杂度界从$\boldsymbol{\text{O}}(ε^{-α})$细化为$o(ε^{-α})$，探究适用算法并给出两类界的接近度示例。
摘要：
我们重新审视了光滑非凸优化算法评估复杂度分析收尾阶段普遍使用的标准“望远镜求和”论证，得到了作为所求精度$ε$的函数的细化界表述。标准论证得到的界通常形如$\boldsymbol{\text{O}}(ε^{-α})$（$α$为某个正数），而细化结果形如$o(ε^{-α})$。我们随后探究了细化界适用于哪些已知算法，最后给出了一个示例，展示标准界与细化界的接近程度。
英文摘要：
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 $ε$. While bounds obtained using the standard argument typically are of the form $\mathcal{O}(ε^{-α})$ for some positive $α$, the refined results are of the form $o(ε^{-α})$. 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.
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