# Stochastic Bayesian Optimization with Unknown Continuous Context Distribution via Kernel Density Estimation

arXiv: https://arxiv.org/abs/2312.10423 · HTML: https://mlsanity.com/paper/2312.10423/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 50 references that are in the map.
- First posted on arXiv 16 Dec 2023.

## Builds on (most cited first, 21 of 21)

- [Automatic chemical design using a data-driven continuous representation of molecules](https://mlsanity.com/paper/1610.02415/) (2016)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Learning to Optimize Via Posterior Sampling](https://mlsanity.com/paper/1301.2609/) (2013)
- [Neural Architecture Search with Bayesian Optimisation and Optimal Transport](https://mlsanity.com/paper/1802.07191/) (2018)
- [Variance-based regularization with convex objectives](https://mlsanity.com/paper/1610.02581/) (2016)
- [Fast Bayesian Optimization of Machine Learning Hyperparameters on Large Datasets](https://mlsanity.com/paper/1605.07079/) (2016)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [Adversarially Robust Optimization with Gaussian Processes](https://mlsanity.com/paper/1810.10775/) (2018)
- [Distributionally Robust Bayesian Optimization](https://mlsanity.com/paper/2002.09038/) (2020)
- [Bayesian Optimization in AlphaGo](https://mlsanity.com/paper/1812.06855/) (2018)
- [Monte Carlo Tree Search based Variable Selection for High Dimensional Bayesian Optimization](https://mlsanity.com/paper/2210.01628/) (2022)
- [Noisy-Input Entropy Search for Efficient Robust Bayesian Optimization](https://mlsanity.com/paper/2002.02820/) (2020)
- [Multi-Period Trading via Convex Optimization](https://mlsanity.com/paper/1705.00109/) (2017)
- [Value-at-Risk Optimization with Gaussian Processes](https://mlsanity.com/paper/2105.06126/) (2021)
- [Unscented Bayesian Optimization for Safe Robot Grasping](https://mlsanity.com/paper/1603.02038/) (2016)
- [Bayesian optimisation under uncertain inputs](https://mlsanity.com/paper/1902.07908/) (2019)
- [Stochastic Bandits with Context Distributions](https://mlsanity.com/paper/1906.02685/) (2019)
- [Mean-Variance Analysis in Bayesian Optimization under Uncertainty](https://mlsanity.com/paper/2009.08166/) (2020)
- [Distributionally Robust Bayesian Quadrature Optimization](https://mlsanity.com/paper/2001.06814/) (2020)
- [Bayesian Optimization using Pseudo-Points](https://mlsanity.com/paper/1910.05484/) (2019)
- [Bayesian Optimization for Distributionally Robust Chance-constrained Problem](https://mlsanity.com/paper/2201.13112/) (2022)

## Built on by (most cited first, 3 of 3)

- [Wasserstein Distributionally Robust Bayesian Optimization with Continuous Context](https://mlsanity.com/paper/2503.20341/) (2025)
- [Bayesian Optimization for Non-Convex Two-Stage Stochastic Optimization Problems](https://mlsanity.com/paper/2408.17387/) (2024)
- [Ensemble Distributionally Robust Bayesian Optimisation with Continuous Context](https://mlsanity.com/paper/2605.07565/) (2026)

## Abstract (arXiv)

Bayesian optimization (BO) is a sample-efficient method and has been widely used for optimizing expensive black-box functions. Recently, there has been a considerable interest in BO literature in optimizing functions that are affected by context variable in the environment, which is uncontrollable by decision makers. In this paper, we focus on the optimization of functions' expectations over continuous context variable, subject to an unknown distribution. To address this problem, we propose two algorithms that employ kernel density estimation to learn the probability density function (PDF) of continuous context variable online. The first algorithm is simpler, which directly optimizes the expectation under the estimated PDF. Considering that the estimated PDF may have high estimation error when the true distribution is complicated, we further propose the second algorithm that optimizes the distributionally robust objective. Theoretical results demonstrate that both algorithms have sub-linear Bayesian cumulative regret on the expectation objective. Furthermore, we conduct numerical experiments to empirically demonstrate the effectiveness of our algorithms.