# Mean-Variance Analysis in Bayesian Optimization under Uncertainty

arXiv: https://arxiv.org/abs/2009.08166 · HTML: https://mlsanity.com/paper/2009.08166/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2020.

## Built on by (most cited first, 18 of 18)

- [Robust Multi-Objective Bayesian Optimization Under Input Noise](https://mlsanity.com/paper/2202.07549/) (2022)
- [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)
- [Bayesian Optimization for Cascade-type Multi-stage Processes](https://mlsanity.com/paper/2111.08330/) (2021)
- [Randomized Gaussian Process Upper Confidence Bound with Tighter Bayesian Regret Bounds](https://mlsanity.com/paper/2302.01511/) (2023)
- [Active learning for distributionally robust level-set estimation](https://mlsanity.com/paper/2102.04000/) (2021)
- [A Robust Multi-Objective Bayesian Optimization Framework Considering Input Uncertainty](https://mlsanity.com/paper/2202.12848/) (2022)
- [Batch Bayesian Optimization for Replicable Experimental Design](https://mlsanity.com/paper/2311.01195/) (2023)
- [Bounding Box-based Multi-objective Bayesian Optimization of Risk Measures under Input Uncertainty](https://mlsanity.com/paper/2301.11588/) (2023)
- [Adversarial Causal Bayesian Optimization](https://mlsanity.com/paper/2307.16625/) (2023)
- [Robust Entropy Search for Safe Efficient Bayesian Optimization](https://mlsanity.com/paper/2405.19059/) (2024)
- [Regret Analysis for Randomized Gaussian Process Upper Confidence Bound](https://mlsanity.com/paper/2409.00979/) (2024)
- [Robust Bayesian Target Value Optimization](https://mlsanity.com/paper/2301.04344/) (2023)
- [Stochastic Bayesian Optimization with Unknown Continuous Context Distribution via Kernel Density Estimation](https://mlsanity.com/paper/2312.10423/) (2023)
- [Uncertainty quantification and exploration-exploitation trade-off in humans](https://mlsanity.com/paper/2102.07647/) (2021)
- [Robust Satisficing Gaussian Process Bandits Under Adversarial Attacks](https://mlsanity.com/paper/2506.01625/) (2025)
- [Nearly-Optimal Algorithm for Adversarial Kernelized Bandits](https://mlsanity.com/paper/2605.10299/) (2026)
- [Gamifying optimization: a Wasserstein distance-based analysis of human search](https://mlsanity.com/paper/2112.06292/) (2021)

## Abstract (arXiv)

We consider active learning (AL) in an uncertain environment in which trade-off between multiple risk measures need to be considered. As an AL problem in such an uncertain environment, we study Mean-Variance Analysis in Bayesian Optimization (MVA-BO) setting. Mean-variance analysis was developed in the field of financial engineering and has been used to make decisions that take into account the trade-off between the average and variance of investment uncertainty. In this paper, we specifically focus on BO setting with an uncertain component and consider multi-task, multi-objective, and constrained optimization scenarios for the mean-variance trade-off of the uncertain component. When the target blackbox function is modeled by Gaussian Process (GP), we derive the bounds of the two risk measures and propose AL algorithm for each of the above three problems based on the risk measure bounds. We show the effectiveness of the proposed AL algorithms through theoretical analysis and numerical experiments.