Distributionally Robust Bayesian Quadrature Optimization
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Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:19211931, 2020.
Abstract
Bayesian quadrature optimization (BQO) maximizes the expectation of an expensive blackbox 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 realworld problems, and characterize its theoretical convergence via Bayesian regret.
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