Bayesian Experimental Design for Finding Reliable Level Set under Input Uncertainty

Abstract

In the manufacturing industry, it is often necessary to repeat expensive operational testing of machine in order to identify the range of input conditions under which the machine operates properly. Since it is often difficult to accurately control the input conditions during the actual usage of the machine, there is a need to guarantee the performance of the machine after properly incorporating the possible variation in input conditions. In this paper, we formulate this practical manufacturing scenario as an Input Uncertain Reliable Level Set Estimation (IU-rLSE) problem, and provide an efficient algorithm for solving it. The goal of IU-rLSE is to identify the input range in which the outputs smaller/greater than a desired threshold can be obtained with high probability when the input uncertainty is properly taken into consideration. We propose an active learning method to solve the IU-rLSE problem efficiently, theoretically analyze its accuracy and convergence, and illustrate its empirical performance through numerical experiments on artificial and real data.

Bayesian Quadrature Optimization for Probability Threshold Robustness Measure

“…Proof. From Chebyshev's inequality and LemmaA.2 in [12], noting that |p t−1;η (x) − µ (p) t−1;η (x)| ≤ 1 the inequality holds for any τ > 0, t ≥ 1 and x ∈ X :…”

“…Other closely related works are [6] and [7] where a distributional robust optimization framework was introduced in the context of BQO. Our work is also related to robust BO/LSE methods under input uncertainty [8,9,10,11,12] in which one can only obtain the function values evaluated at noisy inputs. In addition to these related studies, various forms of robustness of GP modeling have been considered previously [13,14,15]; however, to our knowledge, none of these previous works studied AL problems for the PTR measure in the form of p upper (x), for which it is necessary to solve non-trivial and technically challenging problems.…”