In this paper, we study the problem of noisy, convex, zeroth order optimisation of a function $f$ over a bounded convex set $\bar{\mathcal X}\subset \mathbb{R}^d$. Given a budget $n$ of noisy queries to the function $f$ that can be allocated sequentially and adaptively, our aim is to construct an algorithm that returns a point $\hat x\in \bar{\mathcal X}$ such that $f(\hat x)$ is as small as possible. We provide a conceptually simple method inspired by the textbook center of gravity method, but adapted to the noisy and zeroth order setting. We prove that this method is such that the $f(\hat x) - \min_{x\in \bar{\mathcal X}} f(x)$ is of smaller order than $d^2/\sqrt{n}$ up to poly-logarithmic terms. We slightly improve upon existing literature, where to the best of our knowledge the best known rate is in [Lattimore, 2024] is of order $d^{2.5}/\sqrt{n}$, albeit for a more challenging problem. Our main contribution is however conceptual, as we believe that our algorithm and its analysis bring novel ideas and are significantly simpler than existing approaches.

翻译：本文研究有界凸集 $\bar{\mathcal X}\subset \mathbb{R}^d$ 上函数 $f$ 的噪声凸零阶优化问题。在给定可序贯自适应分配的 $n$ 次噪声函数查询预算条件下，我们的目标是构建一种算法，使其返回满足 $\hat x\in \bar{\mathcal X}$ 的点，并使 $f(\hat x)$ 尽可能小。我们提出了一种概念简洁的方法，其灵感来源于教科书中的重心法，但已适配噪声与零阶设置。我们证明该方法能使 $f(\hat x) - \min_{x\in \bar{\mathcal X}} f(x)$ 的阶低于 $d^2/\sqrt{n}$（忽略多对数项）。此结果较现有文献略有改进——据我们所知，[Lattimore, 2024] 中针对更具挑战性的问题所获得的最佳已知阶为 $d^{2.5}/\sqrt{n}$。然而，我们的主要贡献在于概念层面，我们认为所提出的算法及其分析引入了新颖的思路，且相较于现有方法显著简化。