Distributionally Robust Bayesian Quadrature Optimization
This addresses the challenge of high variance in Monte Carlo estimates for expensive black-box integrands in optimization, which is incremental as it builds on existing Bayesian quadrature methods by incorporating robustness to distributional uncertainty.
The paper tackles the problem of Bayesian quadrature optimization under distributional uncertainty, where the underlying probability distribution is unknown except for a limited set of samples, by proposing a distributionally robust approach that maximizes the expected objective under the most adversarial distribution, and demonstrates empirical effectiveness in synthetic and real-world problems with theoretical convergence guarantees via Bayesian regret.
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.