Parallel Bayesian Global Optimization of Expensive Functions

We consider parallel global optimization of derivative-free expensive-to-evaluate functions, and propose an efficient method based on stochastic approximation for implementing a conceptual Bayesian optimization algorithm proposed by Ginsbourger in 2008. At the heart of this algorithm is maximizing the information criterion called the “multipoints expected improvement,” or the q - EI . To accomplish this, we use infinitesimal perturbation analysis (IPA) to construct a stochastic gradient estimator and show that this estimator is unbiased. We also show that the stochastic gradient ascent algorithm using the constructed gradient estimator converges to a stationary point of the q - EI surface, and therefore, as the number of multiple starts of the gradient ascent algorithm and the number of steps for each start grow large, the one-step Bayes-optimal set of points is recovered. We show in numerical experiments using up to 128 parallel evaluations that our method for maximizing the q - EI is faster than methods based on closed-form evaluation using high-dimensional integration, when considering many parallel function evaluations, and is comparable in speed when considering few. We also show that the resulting one-step Bayes-optimal algorithm for parallel global optimization finds high-quality solutions with fewer evaluations than a heuristic based on approximately maximizing the q - EI . A high-quality open source implementation of this algorithm is available in the open source Metrics Optimization Engine (MOE).