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.

通过噪声、凸性和零阶优化等概念，研究在有界凸集合中的一类函数的优化问题，我们提出了一个基于质心方法的改进算法，证明其相对于最小值的差距的阶数小于 d^2/√n，且比现有文献中已知的 d^{2.5}/√n 的速度更快，并且更简化。