Efficient Minimax Optimal Global Optimization of Lipschitz Continuous Multivariate Functions

TL;DR

This paper introduces a minimax optimal global optimization algorithm for multivariate Lipschitz functions, using average regret to address non-convex challenges and achieving computational efficiency and optimal bounds.

Contribution

It proposes a novel optimization method that employs a query creation rule, improving computational efficiency and providing minimax optimal average regret bounds for multivariate Lipschitz functions.

Findings

Achieves an average regret bound of O(L√n T^{-1/n})

Demonstrates minimax optimality of the algorithm

Outperforms traditional proxy-based methods in computational efficiency

Abstract

In this work, we propose an efficient minimax optimal global optimization algorithm for multivariate Lipschitz continuous functions. To evaluate the performance of our approach, we utilize the average regret instead of the traditional simple regret, which, as we show, is not suitable for use in the multivariate non-convex optimization because of the inherent hardness of the problem itself. Since we study the average regret of the algorithm, our results directly imply a bound for the simple regret as well. Instead of constructing lower bounding proxy functions, our method utilizes a predetermined query creation rule, which makes it computationally superior to the Piyavskii-Shubert variants. We show that our algorithm achieves an average regret bound of $O(L\sqrt{n}T^{-\frac{1}{n}})$ for the optimization of an $n$-dimensional $L$-Lipschitz continuous objective in a time horizon $T$, which we show to be minimax optimal.