Parallel Bayesian Global Optimization of Expensive Functions

Abstract

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 et al. (2007). At the heart of this algorithm is maximizing the information criterion called the "multi-points expected improvement", or the q-EI.To accomplish this, we use infinitessimal 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 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).

Bayesian Optimization of Composite Functions

“…Our method for optimizing the EI-CF acquisition function uses an unbiased estimator of the gradient of EI-CF within a multistart stochastic gradient ascent framework. This technique is structurally similar to methods developed for optimizing acquisition functions in other BO settings without composite objectives, including the parallel expected improvement Wang et al (2016) and the parallel knowledge-gradient Wu & Frazier (2016).…”

Predicting the utility of search spaces for black-box optimization: a simple, budget-aware approach

“…However, because we do not have access to f (x), we can instead estimate the expected utility E ŷ∼p(y|x,D) [u(y + , ŷ)] in place of u(y + , f (x)). If we adopt improvement over y + as our utility function, our expected utility expression becomes the multi-point expected improvement (b-EI) batch acquisition function (Wang et al, 2016a). Similarly, if we use binarized improvement as our utility, our expected utility will reproduce the multi-point probability of improvement (b-PI) batch acquisition function (Wang et al, 2016a).…”

“…If we adopt improvement over y + as our utility function, our expected utility expression becomes the multi-point expected improvement (b-EI) batch acquisition function (Wang et al, 2016a). Similarly, if we use binarized improvement as our utility, our expected utility will reproduce the multi-point probability of improvement (b-PI) batch acquisition function (Wang et al, 2016a). Of course unlike with the batch acquisition functions common in Bayesian optimization, we are not trying to find the location of the next batch of query points, but are instead predicting the score of a search space by averaging over batches of points within that search space.…”

aphBO-2GP-3B: A budgeted asynchronous parallel multi-acquisition functions for constrained Bayesian optimization on high-performing computing architecture

“…Marmin et al [44,45] proposed a analytical simplification for q-EI to avoid high-dimensional integration in large batch based on gradient-based ascent algorithms. Wang et al [46] also employed a gradient-based approach but relying on infinitesimal perturbation analysis to construct a stochastic gradient estimator to simplify the high-dimensional integration in the original q-EI framework. Wang et al [46] and Wu and…”

“…Wang et al [46] also employed a gradient-based approach but relying on infinitesimal perturbation analysis to construct a stochastic gradient estimator to simplify the high-dimensional integration in the original q-EI framework. Wang et al [46] and Wu and…”