Distributionally Robust Bayesian Quadrature Optimization

• Published in: arXiv.org 2020-01
• Main Authors: Thanh Tang Nguyen, Gupta, Sunil, Ha, Huong, Rana, Santu, Venkatesh, Svetha
• Format: Article
• Language: English
• Subjects: Algorithms; Bayesian analysis; Decision theory; Optimization; Probability distribution; Quadratures; Robustness
• ISSN: 2331-8422

Bayesian quadrature optimization (BQO) maximizes the expectation of an expensive black-box integrand taken over a known probability distribution. In this work, we study BQO under distributional uncertainty in which the underlying probability distribution is unknown except for a limited set of its i.i.d. samples. A standard BQO approach maximizes the Monte Carlo estimate of the true expected objective given the fixed sample set. Though Monte Carlo estimate is unbiased, it has high variance given a small set of samples; thus can result in a spurious objective function. We adopt the distributionally robust optimization perspective to this problem by maximizing the expected objective under the most adversarial distribution. In particular, we propose a novel posterior sampling based algorithm, namely distributionally robust BQO (DRBQO) for this purpose. We demonstrate the empirical effectiveness of our proposed framework in synthetic and real-world problems, and characterize its theoretical convergence via Bayesian regret.