Many expensive black-box optimisation problems are sensitive to their inputs. In these problems it makes more sense to locate a region of good designs, than a single, possible fragile, optimal design. Expensive black-box functions can be optimised effectively with Bayesian optimisation, where a Gaussian process is a popular choice as a prior over the expensive function. We propose a method for robust optimisation using Bayesian optimisation to find a region of design space in which the expensive function's performance is insensitive to the inputs whilst retaining a good quality. This is achieved by sampling realisations from a Gaussian process modelling the expensive function and evaluating the improvement for each realisation. The expectation of these improvements can be optimised cheaply with an evolutionary algorithm to determine the next location at which to evaluate the expensive function. We describe an efficient process to locate the optimum expected improvement. We show empirically that evaluating the expensive function at the location in the candidate sweet spot about which the model is most uncertain or at random yield the best convergence in contrast to exploitative schemes. We illustrate our method on six test functions in two, five, and ten dimensions, and demonstrate that it is able to outperform a state-of-the-art approach from the literature.

Bayesian optimization (BO) and its batch extensions are successful for optimizing expensive black-box functions. However, these traditional BO approaches are not yet ideal for optimizing less expensive functions when the computational cost of BO can dominate the cost of evaluating the blackbox function. Examples of these less expensive functions are cheap machine learning models, inexpensive physical experiment through simulators, and acquisition function optimization in Bayesian optimization. In this paper, we consider a batch BO setting for situations where function evaluations are less expensive. Our model is based on a new exploration strategy using geometric distance that provides an alternative way for exploration, selecting a point far from the observed locations. Using that intuition, we propose to use Sobol sequence to guide exploration that will get rid of running multiple global optimization steps as used in previous works. Based on the proposed distance exploration, we present an efficient batch BO approach. We demonstrate that our approach outperforms other baselines and global optimization methods when the function evaluations are less expensive.

When approximating a black-box function, sampling with active learning focussing on regions with non-linear responses tends to improve accuracy. We present the FLOLA-Voronoi method introduced previously for deterministic responses, and theoretically derive the impact of output uncertainty. The algorithm automatically puts more emphasis on exploration to provide more information to the models.