Efficient Minimax Optimal Global Optimization of Lipschitz Continuous Multivariate Functions

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.