$1D$ to $nD$: A Meta Algorithm for Multivariate Global Optimization via Univariate Optimizers

TL;DR

This paper introduces a meta algorithm that leverages univariate global optimizers to efficiently solve multivariate global optimization problems, providing theoretical regret bounds and demonstrating practical relevance.

Contribution

It presents a novel meta algorithm that reduces multivariate optimization to univariate optimization, with theoretical analysis and regret guarantees.

Findings

The meta algorithm effectively solves multivariate problems using univariate optimizers.

Regret bounds are established in terms of time horizon and univariate optimizer performance.

The approach demonstrates practical relevance despite limited prior focus on univariate global optimization.

Abstract

In this work, we propose a meta algorithm that can solve a multivariate global optimization problem using univariate global optimizers. Although the univariate global optimization does not receive much attention compared to the multivariate case, which is more emphasized in academia and industry; we show that it is still relevant and can be directly used to solve problems of multivariate optimization. We also provide the corresponding regret bounds in terms of the time horizon $T$ and the average regret of the univariate optimizer, when it is robust against nonnegative noises with robust regret guarantees.