Knowing The What But Not The Where in Bayesian Optimization

Bayesian optimization has demonstrated impressive success in finding the optimum input x∗ and output f∗ = f(x∗) = max f(x) of a black-box function f. In some applications, however, the optimum output is known in advance and the goal is to find the corresponding optimum input. Existing work in Bayesian optimization (BO) has not effectively exploited the knowledge of f∗ for optimization. In this paper, we consider a new setting in BO in which the knowledge of the optimum output is available. Our goal is to exploit the knowledge about f∗ to search for the input x∗ efficiently. To achieve this goal, we first transform the Gaussian process surrogate using the information about the optimum output. Then, we propose two acquisition functions, called confidence bound minimization and expected regret minimization, which exploit the knowledge about the optimum output to identify the optimum input more efficient. We show that our approaches work intuitively and quantitatively better performance against standard BO methods. We demonstrate real applications in tuning a deep reinforcement learning algorithm on the CartPole problem and XGBoost on Skin Segmentation dataset in which the optimum values are publicly available.