A simple and improved algorithm for noisy, convex, zeroth-order optimisation

• URL: http://arxiv.org/abs/2406.18672v1
• Date: Wed, 26 Jun 2024 18:19:10 GMT
• Title: A simple and improved algorithm for noisy, convex, zeroth-order optimisation
• Authors: Alexandra Carpentier,
• Abstract summary: We construct an algorithm that returns a point $hat xin barmathcal X$ such that $f(hat x)$ is as small as possible. We prove that this method is such that the $f(hat x) - min_xin barmathcal X f(x)$ is of smaller order than $d2/sqrtn$ up to poly-logarithmic terms.
• Score: 59.51990161522328
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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