Bayesian Quadrature Optimization for Probability Threshold Robustness Measure

Abstract

In many product development problems, the performance of the product is governed by two types of parameters called design parameter and environmental parameter. While the former is fully controllable, the latter varies depending on the environment in which the product is used. The challenge of such a problem is to find the design parameter that maximizes the probability that the performance of the product will meet the desired requisite level given the variation of the environmental parameter. In this paper, we formulate this practical problem as active learning (AL) problems and propose efficient algorithms with theoretically guaranteed performance. Our basic idea is to use Gaussian Process (GP) model as the surrogate model of the product development process, and then to formulate our AL problems as Bayesian Quadrature Optimization problems for probabilistic threshold robustness (PTR) measure. We derive credible intervals for the PTR measure and propose AL algorithms for the optimization and level set estimation of the PTR measure. We clarify the theoretical properties of the proposed algorithms and demonstrate their efficiency in both synthetic and real-world product development problems.Preprint. Under review.

Active learning for distributionally robust level-set estimation

“…In addition, we regarded economic risk as a black-box function f (x, w). Note that similar numerical experiments were performed in [13] under the setting where the distribution of w, p † (w), is known. Furthermore, we rescaled the ranges of x and w in the interval [−1, 1].…”

“…We used the following economic risk function f (x, w): f (x, w) = n infected (x, w) − 150x, where n infected (x, w) is the maximum number of infected people in a given period of time, calculated using the SIR model. Note that this risk function was also used by [13], and in this experiment, we used the same function they used in their experiment. Under this setup, we took one initial point at random and ran the algorithms until the number of iterations reached 100.…”

Mean-Variance Analysis in Bayesian Optimization under Uncertainty

“…When the blackbox function follows a GP, the mean function (1a) also follows a GP, suggesting that one can efficiently solve BQO problems by properly modifying the AFs in conventional BO. By replacing the integrand in (1a) with different uncertainty measures, one can consider various types of AL problems under uncertainty [14,15]. Another line of research dealing with uncontrollable and uncertain factors in BO is known as robust BO.…”