Clustering-Guided Gp-Ucb for Bayesian Optimization

Bayesian optimization is a powerful technique for finding extrema of an objective function, a closed-form expression of which is not given but expensive evaluations at query points are available. Gaussian Process (GP) regression is often used to estimate the objective function and uncertainty estimates that guide GP-Upper Confidence Bound (GP-UCB) to determine where next to sample from the objective function, balancing exploration and exploitation. In general, it requires an auxiliary optimization to tune the hyperparameter in GP-UCB, which is sometimes not easy to carry out in practice. In this paper we present a simple practical method which improves GP-UCB, especially in cases where the objective function is not smooth with sharp peaks and valleys. We first present a geometric interpretation of GP-UCB on which we base our development of the clustering-guided method to select the next observation. Clustering is applied to two-dimensional vectors whose entries correspond to the posterior mean and standard deviation computed by GP regression, which is followed by utility maximization with GP-UCB, in order to determine where next to sample from the objective function. Experiments on various functions demonstrate our method alleviates the chance of being trapped in local extrema, making small efforts for auxiliary optimization.