Explicit and Non-asymptotic Query Complexities of Rank-Based Zeroth-order Algorithms on Smooth Functions

TL;DR

This work tackles the theoretical gap for rank-based zeroth-order optimization by establishing explicit non-asymptotic query complexities for a simple rank-based algorithm that uses only order information from Gaussian probes. The method selectively aggregates information from top and bottom ranked samples with positive and negative weights, including a log-weights strategy inspired by CMA-ES. The main results show that for $L$-smooth and $\mu$-strongly convex functions, the algorithm achieves $\tilde{O}\left(\frac{dL}{\mu}\log\frac{dL}{\mu\delta}\log\frac{1}{\varepsilon}\right)$ queries to reach $\varepsilon$-suboptimality, and for $L$-smooth nonconvex objectives, $O\left(\frac{dL}{\varepsilon}\log\frac{1}{\varepsilon}\right)$ queries, with high-probability guarantees. Crucially, the analysis avoids drift and information-geometric techniques, offering new insights into why rank-based heuristics can be as efficient as their value-based counterparts and informing future study of evolution strategies.

Abstract

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.

Theorems & Definitions (52)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof