Extending Expected Improvement for High-dimensional Stochastic Optimization of Expensive Black-Box Functions

TL;DR

This paper extends the expected improvement acquisition function within Bayesian optimization to handle high-dimensional stochastic problems with expensive black-box functions, incorporating uncertainty filtering and Bayesian hyperparameter inference.

Contribution

It introduces a robust EI-based Bayesian optimization method for uncertain, high-dimensional, expensive black-box functions, using a particle approximation of Gaussian process hyperparameters.

Findings

Successfully applied to synthetic problems with uncertainty.

Demonstrated effectiveness on oil-well placement under uncertainty.

Improved sample efficiency in high-dimensional stochastic optimization.

Abstract

Design optimization under uncertainty is notoriously difficult when the objective function is expensive to evaluate. State-of-the-art techniques, e.g, stochastic optimization or sampling average approximation, fail to learn exploitable patterns from collected data and require an excessive number of objective function evaluations. There is a need for techniques that alleviate the high cost of information acquisition and select sequential simulations optimally. In the field of deterministic single-objective unconstrained global optimization, the Bayesian global optimization (BGO) approach has been relatively successful in addressing the information acquisition problem. BGO builds a probabilistic surrogate of the expensive objective function and uses it to define an information acquisition function (IAF) whose role is to quantify the merit of making new objective evaluations. Specifically, BGO iterates between making the observations with the largest expected IAF and rebuilding the probabilistic surrogate, until a convergence criterion is met. In this work, we extend the expected improvement (EI) IAF to the case of design optimization under uncertainty wherein the EI policy is reformulated to filter out parametric and measurement uncertainties. To increase the robustness of our approach in the low sample regime, we employ a fully Bayesian interpretation of Gaussian processes by constructing a particle approximation of the posterior of its hyperparameters using adaptive Markov chain Monte Carlo. We verify and validate our approach by solving two synthetic optimization problems under uncertainty and demonstrate it by solving the oil-well-placement problem with uncertainties in the permeability field and the oil price time series.