# Distributionally Robust Bayesian Quadrature Optimization

arXiv: https://arxiv.org/abs/2001.06814 · HTML: https://mlsanity.com/paper/2001.06814/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jan 2020.

## Built on by (most cited first, 15 of 15)

- [Misspecified Gaussian Process Bandit Optimization](https://mlsanity.com/paper/2111.05008/) (2021)
- [Value-at-Risk Optimization with Gaussian Processes](https://mlsanity.com/paper/2105.06126/) (2021)
- [Risk-averse Heteroscedastic Bayesian Optimization](https://mlsanity.com/paper/2111.03637/) (2021)
- [Distributionally Robust Model-based Reinforcement Learning with Large State Spaces](https://mlsanity.com/paper/2309.02236/) (2023)
- [Distributionally Robust Bayesian Optimization with varphi-divergences](https://mlsanity.com/paper/2203.02128/) (2022)
- [Active learning for distributionally robust level-set estimation](https://mlsanity.com/paper/2102.04000/) (2021)
- [Robust Bandit Learning with Imperfect Context](https://mlsanity.com/paper/2102.05018/) (2021)
- [Bayesian Optimization for Distributionally Robust Chance-constrained Problem](https://mlsanity.com/paper/2201.13112/) (2022)
- [Adversarial Causal Bayesian Optimization](https://mlsanity.com/paper/2307.16625/) (2023)
- [Stochastic Bayesian Optimization with Unknown Continuous Context Distribution via Kernel Density Estimation](https://mlsanity.com/paper/2312.10423/) (2023)
- [Wasserstein Distributionally Robust Bayesian Optimization with Continuous Context](https://mlsanity.com/paper/2503.20341/) (2025)
- [Batched Kernelized Bandits: Refinements and Extensions](https://mlsanity.com/paper/2603.12627/) (2026)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Bayesian Optimization for General Reaction Conditions](https://mlsanity.com/paper/2502.18966/) (2025)
- [Ensemble Distributionally Robust Bayesian Optimisation with Continuous Context](https://mlsanity.com/paper/2605.07565/) (2026)

## Abstract (arXiv)

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.