Rank-based zeroth-order (ZO) optimization -- which relies only on the ordering of function evaluations -- offers strong robustness to noise and monotone transformations, and underlies many successful algorithms such as CMA-ES, natural evolution strategies, and rank-based genetic algorithms. Despite its widespread use, the theoretical understanding of rank-based ZO methods remains limited: existing analyses provide only asymptotic insights and do not yield explicit convergence rates for algorithms selecting the top-$k$ directions. This work closes this gap by analyzing a simple rank-based ZO algorithm and establishing the first \emph{explicit}, and \emph{non-asymptotic} query complexities. For a $d$-dimension problem, if the function is $L$-smooth and $μ$-strongly convex, the algorithm achieves $\widetilde{\mathcal O}\!\left(\frac{dL}μ\log\!\frac{dL}{μδ}\log\!\frac{1}{\varepsilon}\right)$ to find an $\varepsilon$-suboptimal solution, and for smooth nonconvex objectives it reaches $\mathcal O\!\left(\frac{dL}{\varepsilon}\log\!\frac{1}{\varepsilon}\right)$. Notation $\cO(\cdot)$ hides constant terms and $\widetilde{\mathcal O}(\cdot)$ hides extra $\log\log\frac{1}{\varepsilon}$ term. These query complexities hold with a probability at least $1-δ$ with $0<δ<1$. The analysis in this paper is novel and avoids classical drift and information-geometric techniques. Our analysis offers new insight into why rank-based heuristics lead to efficient ZO optimization.

翻译：基于排序的零阶优化——仅依赖于函数评估值的排序关系——对噪声和单调变换具有强鲁棒性，是CMA-ES、自然进化策略及基于排序的遗传算法等众多成功算法的理论基础。尽管该方法应用广泛，但其理论理解仍显不足：现有分析仅提供渐近性结论，未能给出选取前$k$个方向算法的显式收敛速率。本文通过分析一种简单的基于排序的零阶算法，首次建立了显式且非渐近的查询复杂度界。对于$d$维问题，若函数满足$L$-光滑且$μ$-强凸性，该算法以$\widetilde{\mathcal O}\!\left(\frac{dL}μ\log\!\frac{dL}{μδ}\log\!\frac{1}{\varepsilon}\right)$的查询复杂度找到$\varepsilon$-次优解；对于光滑非凸目标函数，则达到$\mathcal O\!\left(\frac{dL}{\varepsilon}\log\!\frac{1}{\varepsilon}\right)$的复杂度。符号$\mathcal O(\cdot)$隐藏常数项，$\widetilde{\mathcal O}(\cdot)$额外隐藏$\log\log\frac{1}{\varepsilon}$项。上述查询复杂度以至少$1-δ$的概率成立（其中$0<δ<1$）。本文分析避免了经典的漂移分析与信息几何技术，提出了新颖的理论框架，为基于排序的启发式方法能实现高效零阶优化的内在机理提供了新的理论见解。