Black-box optimization has gained great attention for its success in recent applications. However, scaling up to high-dimensional problems with good query efficiency remains challenging. This paper proposes a novel Rank-1 Lattice Targeted Sampling (RLTS) technique to address this issue. Our RLTS benefits from random rank-1 lattice Quasi-Monte Carlo, which enables us to perform fast local exact Gaussian processes (GP) training and inference with O(nlogn)complexity w.r.t.

Müller et al. [2021], Nguyen et al. [2022] builds a GP model for both the function value and the

Quasi-Monte Carlo (QMC) is an essential tool for integral approximation, Bayesian inference, and sampling for simulation in science, etc.

In this work, we investigate black-box optimization from the perspective of frequentist kernel methods. We propose a novel batch optimization algorithm to jointly maximize the acquisition function and select points from a whole batch in a holistic way. Theoretically, we derive regret bounds for both the noise-free and perturbation settings. Moreover, we analyze the property of the adversarial regret that is required by robust initialization for Bayesian Optimization (BO), and prove that the adversarial regret bounds decrease with the decrease of covering radius, which provides a criterion for generating (initialization point set) to minimize the bound. We then propose fast searching algorithms to generate a point set with a small covering radius for the robust initialization. Experimental results on both synthetic benchmark problems and real-world problems show the effectiveness of the proposed algorithms.