@article{ang2002,
title = {{Asymmetric Correlations of Equity Portfolios}},
journal = {Journal of Financial Economics},
volume = {63},
number = {3},
pages = {443-494},
year = {2002},
COMMENTED_issn = {0304-405X},
COMMENTED_doi = {https://doi.org/10.1016/S0304-405X(02)00068-5},
COMMENTED_url = {https://www.sciencedirect.com/science/article/pii/S0304405X02000685},
author = {Andrew Ang and Joseph Chen},
}

@inproceedings{beland2017,
  title={Bayesian Optimization under Uncertainty},
  author={Beland, Justin J and Nair, Prasanth B},
  booktitle={{NIPS BayesOpt 2017 Workshop}},
  year={2017}
}

@article{berkenkamp2023bayesian,
  title={{Bayesian Optimization with Safety Constraints: Safe and Automatic Parameter Tuning in Robotics}},
  author={Berkenkamp, Felix and Krause, Andreas and Schoellig, Angela P},
  journal={Machine Learning},
  volume={112},
  number={10},
  pages={3713--3747},
  year={2023},
  publisher={Springer}
}

@article{Bertsimas2010NonconvexRO,
  title={Nonconvex Robust Optimization for Problems with Constraints},
  author={Dimitris Bertsimas and Omid Nohadani and Kwong Meng Teo},
  journal={INFORMS J. Comput.},
  year={2010},
  volume={22},
  pages={44-58},
  COMMENTED_url={https://api.semanticscholar.org/CorpusID:352002}
}

@inproceedings{Bogunovic,
 author = {Bogunovic, Ilija and Scarlett, Jonathan and Jegelka, Stefanie and Cevher, Volkan},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {S. Bengio and H. Wallach and H. Larochelle and K. Grauman and N. Cesa-Bianchi and R. Garnett},
 pages = {},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Adversarially Robust Optimization with Gaussian Processes},
 COMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2018/file/60243f9b1ac2dba11ff8131c8f4431e0-Paper.pdf},
 volume = {31},
 year = {2018}
}

@inproceedings{Bottero2022,
 author = {Bottero, Alessandro and Luis, Carlos and Vinogradska, Julia and Berkenkamp, Felix and Peters, Jan R},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {S. Koyejo and S. Mohamed and A. Agarwal and D. Belgrave and K. Cho and A. Oh},
 pages = {30707--30719},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Information-Theoretic Safe Exploration with Gaussian Processes},
 COMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2022/file/c628644624c1be9c8cfb1541fa6421fd-Paper-Conference.pdf},
 volume = {35},
 year = {2022}
}

@inproceedings{Cakmak2020,
 author = {Cakmak, Sait and Astudillo Marban, Raul and Frazier, Peter and Zhou, Enlu},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {H. Larochelle and M. Ranzato and R. Hadsell and M.F. Balcan and H. Lin},
 pages = {20130--20141},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Bayesian Optimization of Risk Measures},
 COMMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2020/file/e8f2779682fd11fa2067beffc27a9192-Paper.pdf},
 volume = {33},
 year = {2020}
}

@article{Christianson2023,
abstract = {Bayesian Optimization (BO) links Gaussian Process (GP) surrogates with sequential design toward optimizing expensive-to-evaluate black-box functions. Example design heuristics, or so-called acquisition functions, like expected improvement (EI), balance exploration and exploitation to furnish global solutions under stringent evaluation budgets. However, they fall short when solving for robust optima, meaning a preference for solutions in a wider domain of attraction. Robust solutions are useful when inputs are imprecisely specified, or where a series of solutions is desired. A common mathematical programming technique in such settings involves an adversarial objective, biasing a local solver away from ``sharp'' troughs. Here we propose a surrogate modeling and active learning technique called robust expected improvement (REI) that ports adversarial methodology into the BO/GP framework. After describing the methods, we illustrate and draw comparisons to several competitors on benchmark synthetic exercises and real problems of varying complexity.},
archivePrefix = {arXiv},
arxivId = {2302.08612},
author = {Christianson, Ryan B. and Gramacy, Robert B.},
COMMENTED_doi = {10.1080/24725854.2023.2275166},
eprint = {2302.08612},
file = {:C\:/Users/dweichert/Downloads/Literatur_BO/robust_EI.pdf:pdf},
COMMENTED_issn = {2472-5854},
journal = {IISE Transactions},
keywords = {Expected Improvement,active learning,adversarially robust,gaussian process,robust optimization,sequential design},
mendeley-groups = {RobustBO_UAI},
mendeley-tags = {adversarially robust,Expected Improvement},
pages = {1--22},
title = {{Robust expected improvement for Bayesian optimization}},
year = {2023}
}

@article{Daulton2022,
abstract = {Bayesian optimization (BO) is a sample-efficient approach for tuning design parameters to optimize expensive-to-evaluate, black-box performance metrics. In many manufacturing processes, the design parameters are subject to random input noise, resulting in a product that is often less performant than expected. Although BO methods have been proposed for optimizing a single objective under input noise, no existing method addresses the practical scenario where there are multiple objectives that are sensitive to input perturbations. In this work, we propose the first multi-objective BO method that is robust to input noise. We formalize our goal as optimizing the multivariate value-at-risk (MVAR), a risk measure of the uncertain objectives. Since directly optimizing MVAR is computationally infeasible in many settings, we propose a scalable, theoretically-grounded approach for optimizing MVAR using random scalarizations. Empirically, we find that our approach significantly outperforms alternative methods and efficiently identifies optimal robust designs that will satisfy specifications across multiple metrics with high probability.},
archivePrefix = {arXiv},
arxivId = {2202.07549},
author = {Daulton, Samuel and Cakmak, Sait and Balandat, Maximilian and Osborne, Michael A. and Zhou, Enlu and Bakshy, Eytan},
eprint = {2202.07549},
file = {:C\:/Users/dweichert/Downloads/Literatur_BO/daulton22a.pdf:pdf},
COMMENTED_issn = {26403498},
journal = {Proceedings of Machine Learning Research},
keywords = {multi-objective,risk measures},
mendeley-groups = {RobustBO_UAI},
mendeley-tags = {risk measures,multi-objective},
pages = {4831--4866},
title = {{Robust Multi-Objective Bayesian Optimization Under Input Noise}},
volume = {162},
year = {2022}
}

@article{Dorschel2021,
abstract = {As control systems become more and more complex, the optimal tuning of control parameters using Bayesian Optimisation gained an increased interest of research in recent years. Safe Bayesian Optimisation, tries to prevent sampling of unsafe parametrizations and therefore allow parameter tuning in real world experiments. Usually this is achieved by approximating a safe set using probabilistic Gaussian Process Regression predictions. In contrast in this work, analytical knowledge about robustly stable parameter configurations is gained by the parameter space approach and then incorporated within the optimisation as constraint. Simulation results on linear and non-linear systems with uncertain parameters show a significant performance gain compared to standard approaches.},
author = {D{\"{o}}rschel, Lorenz and Stenger, David and Abel, Dirk},
file = {:C\:/Users/dweichert/Downloads/Literatur_BO/dorschel21a.pdf:pdf},
COMMENTED_issn = {26403498},
journal = {Proceedings of Machine Learning Research},
keywords = {Bayesian Optimisation,Optimisation of controller parameters,Safe Bayesian Optimization,parameter space approach,performance index ITAE},
mendeley-groups = {RobustBO_UAI},
mendeley-tags = {Safe Bayesian Optimization},
number = {2020},
pages = {299--311},
title = {{Safe Bayesian Optimisation for Controller Design by Utilising the Parameter Space Approach}},
volume = {144},
year = {2021}
}

@book{fisher1930genetical,
  title={The genetical theory of natural selection: a complete variorum edition},
  author={Fisher, Ronald Aylmer},
  year={1930},
  publisher={Oxford University Press}
}

@article{frazier2008knowledge,
  title={A knowledge-gradient policy for sequential information collection},
  author={Frazier, Peter I and Powell, Warren B and Dayanik, Savas},
  journal={SIAM Journal on Control and Optimization},
  volume={47},
  number={5},
  pages={2410--2439},
  year={2008},
  publisher={SIAM}
}

@InProceedings{Frohlich2020,
  title = 	 {Noisy-Input Entropy Search for Efficient Robust Bayesian Optimization},
  author =       {Fr\"ohlich, Lukas and Klenske, Edgar and Vinogradska, Julia and Daniel, Christian and Zeilinger, Melanie},
  booktitle = 	 {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
  pages = 	 {2262--2272},
  year = 	 {2020},
  COMMENTED_editor = 	 {Chiappa, Silvia and Calandra, Roberto},
  volume = 	 {108},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {26--28 Aug},
  COMMENTED_publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v108/frohlich20a/frohlich20a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v108/frohlich20a.html},
  abstract = 	 {We consider the problem of robust optimization within the well-established Bayesian Optimization (BO) framework.While BO is intrinsically robust to noisy evaluations of the objective function, standard approaches do not consider the case of uncertainty about the input parameters.In this paper, we propose Noisy-Input Entropy Search (NES), a novel information-theoretic acquisition function that is designed to find robust optima for problems with both input and measurement noise.NES is based on the key insight that the robust objective in many cases can be modeled as a Gaussian process, however, it cannot be observed directly.We evaluate NES on several benchmark problems from the optimization literature and from engineering.The results show that NES reliably finds robust optima, outperforming existing methods from the literature on all benchmarks.}
}

@book{garnett2023bayesian,
  title={Bayesian optimization},
  author={Garnett, Roman},
  year={2023},
  publisher={Cambridge University Press}
}

@article{Genz1992,
author = {Alan Genz},
title = {Numerical Computation of Multivariate Normal Probabilities},
journal = {Journal of Computational and Graphical Statistics},
volume = {1},
number = {2},
pages = {141-149},
year  = {1992},
publisher = {Taylor & Francis},
COMMENTED_doi = {10.1080/10618600.1992.10477010},
}

@InProceedings{Gessner2020,
  title = 	 {Integrals over Gaussians under Linear Domain Constraints},
  author =       {Gessner, Alexandra and Kanjilal, Oindrila and Hennig, Philipp},
  booktitle = 	 {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
  pages = 	 {2764--2774},
  year = 	 {2020},
  COMMENTED_editor = 	 {Chiappa, Silvia and Calandra, Roberto},
  volume = 	 {108},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {26--28 Aug},
  COMMENTED_publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v108/gessner20a/gessner20a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v108/gessner20a.html},
  abstract = 	 {Integrals of linearly constrained multivariate Gaussian densities are a frequent problem in machine learning and statistics, arising in tasks like generalized linear models and Bayesian optimization. Yet they are notoriously hard to compute, and to further complicate matters, the numerical values of such integrals may be very small. We present an efficient black-box algorithm that exploits geometry for the estimation of integrals over a small, truncated Gaussian volume, and to simulate therefrom. Our algorithm uses the Holmes-Diaconis-Ross (HDR) method combined with an analytic version of elliptical slice sampling (ESS). Adapted to the linear setting, ESS allows for rejection-free sampling, because intersections of ellipses and domain boundaries have closed-form solutions. The key idea of HDR is to decompose the integral into easier-to-compute conditional probabilities by using a sequence of nested domains. Remarkably, it allows for direct computation of the logarithm of the integral value and thus enables the computation of extremely small probability masses. We demonstrate the effectiveness of our tailored combination of HDR and ESS on high-dimensional integrals and on entropy search for Bayesian optimization.}
}

@Misc{gpy2014,
  author =   {{GPy}},
  title =    {{GPy}: A Gaussian process framework in python},
  howpublished = {\url{http://github.com/SheffieldML/GPy}},
  year = {since 2012}
}

@inproceedings{he2022solve,
  title={GDA-AM: On the Effectiveness of Solving Min-Imax Optimization via Anderson Mixing},
  author={Huan He and Shifan Zhao and Yuanzhe Xi and Joyce C. Ho and Yousef Saad},
  booktitle={International Conference on Learning Representations},
  year={2022},
  COMMENTED_url={https://api.semanticscholar.org/CorpusID:251648111}
}

@article{Hennig2012,
  author    = {Philipp Hennig and
               Christian J. Schuler},
  title     = {Entropy Search for Information-Efficient Global Optimization},
  journal   = {Journal of Machine Learning Research},
  volume    = {13},
  pages     = {1809--1837},
  year      = {2012},
  timestamp = {Wed, 10 Jul 2019 15:28:31 +0200},
  biburl    = {https://dblp.org/rec/journals/jmlr/HennigS12.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@techreport{Herbrich,
 author={Ralf Herbrich},
 title={On Gaussian expectation propagation},
 year={2005},
 institution={Microsoft Research Cambridge},
}

@inproceedings{Hernandez-Lobato2014,
 author = {Hern\'{a}ndez-Lobato, Jos\'{e} Miguel and Hoffman, Matthew W and Ghahramani, Zoubin},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {Z. Ghahramani and M. Welling and C. Cortes and N. Lawrence and K. Q. Weinberger},
 pages = {},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Predictive Entropy Search for Efficient Global Optimization of Black-box Functions},
 COMMENTED_url = {https://proceedings.neurips.cc/paper/2014/file/069d3bb002acd8d7dd095917f9efe4cb-Paper.pdf},
 volume = {27},
 year = {2014}
}

@inproceedings{hoffman2015output,
  title={Output-space predictive entropy search for flexible global optimization},
  author={Hoffman, Matthew W and Ghahramani, Zoubin},
  booktitle={NIPS workshop on Bayesian Optimization},
  pages={1--5},
  year={2015}
}

@inproceedings{Husain2022,
abstract = {The study of robustness has received much attention due to its inevitability in data-driven settings where many systems face uncertainty. One such example of concern is Bayesian Optimization (BO), where uncertainty is multi-faceted, yet there only exists a limited number of works dedicated to this direction. In particular, there is the work of Kirschner et al. (2020), which bridges the existing literature of Distributionally Robust Optimization (DRO) by casting the BO problem from the lens of DRO. While this work is pioneering, it admittedly suffers from various practical shortcomings such as finite contexts assumptions, leaving behind the main question Can one devise a computationally tractable algorithm for solving this DRO-BO problem? In this work, we tackle this question to a large degree of generality by considering robustness against data-shift in $\phi$-divergences, which subsumes many popular choices, such as the $\chi^2$-divergence, Total Variation, and the extant Kullback-Leibler (KL) divergence. We show that the DRO-BO problem in this setting is equivalent to a finite-dimensional optimization problem which, even in the continuous context setting, can be easily implemented with provable sublinear regret bounds. We then show experimentally that our method surpasses existing methods, attesting to the theoretical results},
archivePrefix = {arXiv},
arxivId = {2203.02128},
author = {Husain, Hisham and Nguyen, Vu and van den Hengel, Anton},
eprint = {2203.02128},
COMMENTED_file = {:C\:/Users/dweichert/Downloads/Literatur_BO/2203.02128.pdf:pdf},
keywords = {Distributionally Robust,GP-UCB},
mendeley-groups = {RobustBO_UAI},
mendeley-tags = {Distributionally Robust,GP-UCB},
title = {{Distributionally Robust Bayesian Optimization with $\phi$-divergences}},
COMMENTED_url = {http://arxiv.org/abs/2203.02128},
booktitle={{Gaussian Processes, Spatiotemporal Modeling, and Decision-making Systems, NeurIPS Workshop}},
year = {2022}
}

@inproceedings{Hvarfner2022,
 author = {Hvarfner, Carl and Hutter, Frank and Nardi, Luigi},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {S. Koyejo and S. Mohamed and A. Agarwal and D. Belgrave and K. Cho and A. Oh},
 pages = {11494--11506},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Joint Entropy Search For Maximally-Informed Bayesian Optimization},
 COMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2022/file/4b03821747e89ce803b2dac590f6a39b-Paper-Conference.pdf},
 volume = {35},
 year = {2022}
}

@misc{figdeepdrawing,
    author={Baran Ivo},
    title={{CC BY 3.0, https://creativecommons.org/licenses/by/3.0, via Wikimedia Commons, adapted to include labels}},
    year={2007},
}

@InProceedings{iwazaki2021,
  title = 	 { Mean-Variance Analysis in Bayesian Optimization under Uncertainty },
  author =       {Iwazaki, Shogo and Inatsu, Yu and Takeuchi, Ichiro},
  booktitle = 	 {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics},
  pages = 	 {973--981},
  year = 	 {2021},
  COMMENTED_editor = 	 {Banerjee, Arindam and Fukumizu, Kenji},
  volume = 	 {130},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {13--15 Apr},
  COMMENTED_publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v130/iwazaki21a/iwazaki21a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v130/iwazaki21a.html},
  abstract = 	 { 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 scenarios based on the risk measure bounds. We show the effectiveness of the proposed AL algorithms through theoretical analysis and numerical experiments. }
}

@article{Jones1998,
  author    = {Donald R. Jones and
               Matthias Schonlau and
               William J. Welch},
  title     = {Efficient Global Optimization of Expensive Black-Box Functions},
  journal   = {Journal of Global Optimization},
  volume    = {13},
  number    = {4},
  pages     = {455--492},
  year      = {1998},
  timestamp = {Wed, 17 May 2017 14:25:32 +0200},
  biburl    = {https://dblp.org/rec/journals/jgo/JonesSW98.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{Kirschner2020,
  author    = {Johannes Kirschner and
               Ilija Bogunovic and
               Stefanie Jegelka and
               Andreas Krause},
  COMMENTED_editor    = {Silvia Chiappa and
               Roberto Calandra},
  title     = {Distributionally Robust Bayesian Optimization},
  booktitle = {The 23rd International Conference on Artificial Intelligence and Statistics},
  series    = {Proceedings of Machine Learning Research},
  volume    = {108},
  pages     = {2174--2184},
  COMMENTED_publisher = {{PMLR}},
  year      = {2020},
  COMMENTED_url       = {http://proceedings.mlr.press/v108/kirschner20a.html},
  timestamp = {Mon, 29 Jun 2020 18:03:58 +0200},
  biburl    = {https://dblp.org/rec/conf/aistats/KirschnerBJ020.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{lam2018advances,
  title={Advances in Bayesian optimization with applications in aerospace engineering},
  author={Lam, R{\'e}mi and Poloczek, Matthias and Frazier, Peter and Willcox, Karen E},
  booktitle={2018 AIAA Non-Deterministic Approaches Conference},
  pages={},
  year={2018}
}

@article{lazaro-gredilla10a,
  author    = {Miguel L{\'{a}}zaro{-}Gredilla and
               Joaquin Qui{\~{n}}onero Candela and
               Carl Edward Rasmussen and
               An{\'{\i}}bal R. Figueiras{-}Vidal},
  title     = {Sparse Spectrum Gaussian Process Regression},
  journal   = {Journal of Machine Learning Research},
  volume    = {11},
  pages     = {1865--1881},
  year      = {2010},
  timestamp = {Wed, 10 Jul 2019 15:28:39 +0200},
  biburl    = {https://dblp.org/rec/journals/jmlr/Lazaro-GredillaCRF10.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{Minka2001,
author = {Minka, Thomas P.},
title = {Expectation Propagation for Approximate Bayesian Inference},
year = {2001},
COMMENTED_isbn = {1558608001},
COMMENTED_publisher = {Morgan Kaufmann Publishers Inc.},
COMMENTED_address = {San Francisco, CA, USA},
booktitle = {Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence},
pages = {362–369},
numpages = {8},
location = {Seattle, Washington},
COMMENTED_series = {UAI'01}
}

@inproceedings{Neiswanger2022,
 author = {Neiswanger, Willie and Yu, Lantao and Zhao, Shengjia and Meng, Chenlin and Ermon, Stefano},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {S. Koyejo and S. Mohamed and A. Agarwal and D. Belgrave and K. Cho and A. Oh},
 pages = {21016--21029},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Generalizing Bayesian Optimization with Decision-theoretic Entropies},
 COMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2022/file/8451a20c5a7e0ee5671dda28f7daf7f3-Paper-Conference.pdf},
 volume = {35},
 year = {2022}
}

@InProceedings{Nguyen2021,
  title = 	 {Value-at-Risk Optimization with Gaussian Processes},
  author =       {Nguyen, Quoc Phong and Dai, Zhongxiang and Low, Bryan Kian Hsiang and Jaillet, Patrick},
  booktitle = 	 {Proceedings of the 38th International Conference on Machine Learning},
  pages = 	 {8063--8072},
  year = 	 {2021},
  COMMENTED_editor = 	 {Meila, Marina and Zhang, Tong},
  volume = 	 {139},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {18--24 Jul},
  COMMENTED_publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v139/nguyen21b/nguyen21b.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v139/nguyen21b.html},
  abstract = 	 {Value-at-risk (VaR) is an established measure to assess risks in critical real-world applications with random environmental factors. This paper presents a novel VaR upper confidence bound (V-UCB) algorithm for maximizing the VaR of a black-box objective function with the first no-regret guarantee. To realize this, we first derive a confidence bound of VaR and then prove the existence of values of the environmental random variable (to be selected to achieve no regret) such that the confidence bound of VaR lies within that of the objective function evaluated at such values. Our V-UCB algorithm empirically demonstrates state-of-the-art performance in optimizing synthetic benchmark functions, a portfolio optimization problem, and a simulated robot task.}
}

@article{Nguyen2021a,
abstract = {This paper presents two Bayesian optimization (BO) algorithms with theoretical performance guarantee to maximize the conditional value-at-risk (CVaR) of a black-box function: CV-UCB and CV-TS which are based on the well-established principle of optimism in the face of uncertainty and Thompson sampling, respectively. To achieve this, we develop an upper confidence bound of CVaR and prove the no-regret guarantee of CV-UCB by utilizing an interesting connection between CVaR and value-at-risk (VaR). For CV-TS, though it is straightforwardly performed with Thompson sampling, bounding its Bayesian regret is non-trivial because it requires a tail expectation bound for the distribution of CVaR of a black-box function, which has not been shown in the literature. The performances of both CV-UCB and CV-TS are empirically evaluated in optimizing CVaR of synthetic benchmark functions and simulated real-world optimization problems.},
author = {Nguyen, Quoc Phong and Dai, Zhongxiang and Low, Bryan Kian Hsiang and Jaillet, Patrick},
file = {:C\:/Users/dweichert/Downloads/Literatur_BO/NeurIPS-2021-optimizing-conditional-value-at-risk-of-black-box-functions-Paper.pdf:pdf},
isbn = {9781713845393},
COMMENTED_issn = {10495258},
journal = {Advances in Neural Information Processing Systems},
keywords = {GP-UCB,TS,risk measures},
mendeley-groups = {RobustBO_UAI},
mendeley-tags = {risk measures,GP-UCB,TS},
COMMENTED_number = {NeurIPS},
pages = {4170--4180},
title = {{Optimizing Conditional Value-At-Risk of Black-Box Functions}},
volume = {6},
year = {2021}
}

@inproceedings{Nogueira2016,
  author    = {Jos{\'{e}} Nogueira and
               Ruben Martinez{-}Cantin and
               Alexandre Bernardino and
               Lorenzo Jamone},
  title     = {Unscented Bayesian optimization for safe robot grasping},
  booktitle = {2016 {IEEE/RSJ} International Conference on Intelligent Robots and
               Systems},
  pages     = {1967--1972},
  COMMENTED_publisher = {{IEEE}},
  year      = {2016},
  COMMENTED_url       = {https://doi.org/10.1109/IROS.2016.7759310},
  COMMENTED_doi       = {10.1109/IROS.2016.7759310},
  timestamp = {Fri, 27 Mar 2020 08:49:57 +0100},
  biburl    = {https://dblp.org/rec/conf/iros/NogueiraMBJ16.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{Oliveira2019,
  author    = {Rafael Oliveira and
               Lionel Ott and
               Fabio Ramos},
  COMMENTED_editor    = {Kamalika Chaudhuri and
               Masashi Sugiyama},
  title     = {Bayesian optimisation under uncertain inputs},
  booktitle = {The 22nd International Conference on Artificial Intelligence and Statistics},
  series    = {Proceedings of Machine Learning Research},
  volume    = {89},
  pages     = {1177--1184},
  COMMENTED_publisher = {{PMLR}},
  year      = {2019},
  COMMENTED_url       = {http://proceedings.mlr.press/v89/oliveira19a.html},
  timestamp = {Wed, 10 Jun 2020 11:25:35 +0200},
  biburl    = {https://dblp.org/rec/conf/aistats/OliveiraOR19.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@article{emukit2023,
  title={Emukit: A {P}ython toolkit for decision making under uncertainty},
  author={Andrei Paleyes and Maren Mahsereci and Neil D. Lawrence},
  journal={Proceedings of the Python in Science Conference},
  year={2023}
}

@InProceedings{Picheny2022,
  title = 	 {Bayesian quantile and expectile optimisation},
  author =       {Picheny, Victor and Moss, Henry and Torossian, L\'eonard and Durrande, Nicolas},
  booktitle = 	 {Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence},
  pages = 	 {1623--1633},
  year = 	 {2022},
  COMMENTED_editor = 	 {Cussens, James and Zhang, Kun},
  volume = 	 {180},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {01--05 Aug},
  COMMENTED_publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v180/picheny22a/picheny22a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v180/picheny22a.html},
  abstract = 	 {Bayesian optimisation (BO) is widely used  to optimise stochastic black box functions. While most BO approaches focus on optimising conditional expectations, many applications require risk-averse strategies and alternative criteria accounting for the distribution tails need to be considered. In this paper, we propose new variational models for Bayesian quantile and expectile regression that are well-suited for heteroscedastic noise settings. Our models consist of two latent Gaussian processes accounting respectively for the conditional quantile (or expectile) and the scale parameter of an asymmetric likelihood functions. Furthermore, we propose two BO strategies based on max-value entropy search and Thompson sampling, that are tailored to such models and that can accommodate large batches of points. Contrary to existing BO approaches for risk-averse optimisation, our strategies can directly optimise for the quantile and expectile, without requiring replicating observations or assuming a parametric form for the noise. As illustrated in the experimental section, the proposed approach clearly outperforms the state of the art in the heteroscedastic, non-Gaussian case.}
}

@InProceedings{Qing2022,
  title = 	 {Spectral Representation of Robustness Measures for Optimization Under Input Uncertainty},
  author =       {Qing, Jixiang and Dhaene, Tom and Couckuyt, Ivo},
  booktitle = 	 {Proceedings of the 39th International Conference on Machine Learning},
  pages = 	 {18096--18121},
  year = 	 {2022},
  COMMENTED_editor = 	 {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
  volume = 	 {162},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {17--23 Jul},
  COMMENTED_publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v162/qing22a/qing22a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v162/qing22a.html},
  abstract = 	 {We study the inference of mean-variance robustness measures to quantify input uncertainty under the Gaussian Process (GP) framework. These measures are widely used in applications where the robustness of the solution is of interest, for example, in engineering design. While the variance is commonly used to characterize the robustness, Bayesian inference of the variance using GPs is known to be challenging. In this paper, we propose a Spectral Representation of Robustness Measures based on the GP’s spectral representation, i.e., an analytical approach to approximately infer both robustness measures for normal and uniform input uncertainty distributions. We present two approximations based on different Fourier features and compare their accuracy numerically. To demonstrate their utility and efficacy in robust Bayesian Optimization, we integrate the analytical robustness measures in three standard acquisition functions for various robust optimization formulations. We show their competitive performance on numerical benchmarks and real-life applications.}
}

@book{Rasmussen2006,
  author    = {Carl Edward Rasmussen and
               Christopher K. I. Williams},
  title     = {Gaussian processes for machine learning},
  publisher = {{MIT} Press},
  year      = {2006},
  timestamp = {Wed, 26 Apr 2017 17:48:08 +0200},
  biburl    = {https://dblp.org/rec/books/lib/RasmussenW06.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@article{Rosenbaum1961,
 COMMENTED_ISSN = {00359246},
 COMMENTED_URL = {http://www.jstor.org/stable/2984029},
 author = {Rosenbaum, Sidney},
 journal = {Journal of the Royal Statistical Society. Series B (Methodological)},
 number = {2},
 pages = {405--408},
 publisher = {[Royal Statistical Society, Wiley]},
 title = {Moments of a Truncated Bivariate Normal Distribution},
 urldate = {2023-05-03},
 volume = {23},
 year = {1961}
}

@article{Shahriari2016,
  author    = {Bobak Shahriari and
               Kevin Swersky and
               Ziyu Wang and
               Ryan P. Adams and
               Nando de Freitas},
  title     = {Taking the Human Out of the Loop: {A} Review of Bayesian Optimization},
  journal   = {Proceedings of the {IEEE}},
  volume    = {104},
  number    = {1},
  pages     = {148--175},
  year      = {2016},
  biburl    = {https://dblp.org/rec/journals/pieee/ShahriariSWAF16.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@article{shields2021bayesian,
  title={Bayesian reaction optimization as a tool for chemical synthesis},
  author={Shields, Benjamin J and Stevens, Jason and Li, Jun and Parasram, Marvin and Damani, Farhan and Alvarado, Jesus I Martinez and Janey, Jacob M and Adams, Ryan P and Doyle, Abigail G},
  journal={Nature},
  volume={590},
  number={7844},
  pages={89--96},
  year={2021},
  publisher={Nature Publishing Group UK London}
}

@inproceedings{Srinivas2010,
author = {Srinivas, Niranjan and Krause, Andreas and Kakade, Sham and Seeger, Matthias},
title = {Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design},
year = {2010},
COMMENTED_isbn = {9781605589077},
COMMENTED_publisher = {Omnipress},
COMMENTED_address = {Madison, WI, USA},
abstract = {Many applications require optimizing an unknown, noisy function that is expensive to evaluate. We formalize this task as a multi-armed bandit problem, where the payoff function is either sampled from a Gaussian process (GP) or has low RKHS norm. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze GP-UCB, an intuitive upper-confidence based algorithm, and bound its cumulative regret in terms of maximal information gain, establishing a novel connection between GP optimization and experimental design. Moreover, by bounding the latter in terms of operator spectra, we obtain explicit sublinear regret bounds for many commonly used covariance functions. In some important cases, our bounds have surprisingly weak dependence on the dimensionality. In our experiments on real sensor data, GP-UCB compares favorably with other heuristical GP optimization approaches.},
booktitle = {Proceedings of the 27th International Conference on Machine Learning},
pages = {1015–1022},
numpages = {8},
COMMENTED_location = {Haifa, Israel},
COMMENTED_series = {ICML'10}
}

@misc{optimization_functions,
    url = {https://www.sfu.ca/~ssurjano/optimization.html},
    author = {Surjanovic, Sonja and Bingham, Derek},
    title = {Virtual Library of Simulation Experiments: Test Functions and Datasets. Optimization Test Problems.},
    year = {2013},
    note = {Online; accessed 02-January-2023}
}

@InProceedings{Tay2022,
  title = 	 {Efficient Distributionally Robust {B}ayesian Optimization with Worst-case Sensitivity},
  author =       {Tay, Sebastian Shenghong and Foo, Chuan Sheng and Daisuke, Urano and Leong, Richalynn and Low, Bryan Kian Hsiang},
  booktitle = 	 {Proceedings of the 39th International Conference on Machine Learning},
  pages = 	 {21180--21204},
  year = 	 {2022},
  COMMENTED_editor = 	 {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
  volume = 	 {162},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {17--23 Jul},
  COMMENTED_publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v162/tay22a/tay22a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v162/tay22a.html},
  abstract = 	 {In distributionally robust Bayesian optimization (DRBO), an exact computation of the worst-case expected value requires solving an expensive convex optimization problem. We develop a fast approximation of the worst-case expected value based on the notion of worst-case sensitivity that caters to arbitrary convex distribution distances. We provide a regret bound for our novel DRBO algorithm with the fast approximation, and empirically show it is competitive with that using the exact worst-case expected value while incurring significantly less computation time. In order to guide the choice of distribution distance to be used with DRBO, we show that our approximation implicitly optimizes an objective close to an interpretable risk-sensitive value.}
}

@InProceedings{Tay2023,
  title = 	 {No-regret Sample-efficient Bayesian Optimization for Finding Nash Equilibria with Unknown Utilities},
  author =       {Tay, Sebastian Shenghong and Nguyen, Quoc Phong and Foo, Chuan Sheng and Low, Bryan Kian Hsiang},
  booktitle = 	 {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
  pages = 	 {3591--3619},
  year = 	 {2023},
  COMMENTED_editor = 	 {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
  volume = 	 {206},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {25--27 Apr},
  COMMENTED_publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v206/tay23a/tay23a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v206/tay23a.html},
  abstract = 	 {The Nash equilibrium (NE) is a classic solution concept for normal-form games that is stable under potential unilateral deviations by self-interested agents. Bayesian optimization (BO) has been used to find NE in continuous general-sum games with unknown costly-to-sample utility functions in a sample-efficient manner. This paper presents the first no-regret BO algorithm that is sample-efficient in finding pure NE by leveraging theory on high probability confidence bounds with Gaussian processes and the maximum information gain of kernel functions. Unlike previous works, our algorithm is theoretically guaranteed to converge to the optimal solution (i.e., NE). We also introduce the novel setting of applying BO to finding mixed NE in unknown discrete general-sum games and show that our theoretical framework is general enough to be extended naturally to this setting by developing a no-regret BO algorithm that is sample-efficient in finding mixed NE. We empirically show that our algorithms are competitive w.r.t. suitable baselines in finding NE.}
}

@InProceedings{Toscano-Palmerin2016,
author="Toscano-Palmerin, Saul
and Frazier, Peter I.",
COMMENTED_editor="Owen, Art B.
and Glynn, Peter W.",
title="Stratified Bayesian Optimization",
booktitle="Monte Carlo and Quasi-Monte Carlo Methods",
year="2018",
publisher="Springer International Publishing",
COMMMENTED_address="Cham",
pages="145--166",
abstract="We consider derivative-free black-box global optimization of expensive noisy functions, when most of the randomness in the objective is produced by a few influential scalar random inputs. We present a new Bayesian global optimization algorithm, called Stratified Bayesian Optimization (SBO), which uses this strong dependence to improve performance. Our algorithm is similar in spirit to stratification, a technique from simulation, which uses strong dependence on a categorical representation of the random input to reduce variance. We demonstrate in numerical experiments that SBO outperforms state-of-the-art Bayesian optimization benchmarks that do not leverage this dependence.",
COMMENTED_isbn="978-3-319-91436-7"
}

@article{Toscano-Palmerin2022,
abstract = {Nonconvex derivative-free time-consuming objectives are often optimized using ``black-box"" optimization. These approaches assume very little about the objective. While broadly applicable, they typically require more evaluations than methods exploiting more problem structure. Often, such time-consuming objectives are actually the sum or integral of a larger number of functions, each of which consumes significant time when evaluated individually. This arises in designing aircraft, choosing parameters in ride-sharing dispatch systems, and tuning hyperparameters in deep neural networks. We develop a novel Bayesian optimization algorithm that leverages this structure to improve performance. Our algorithm is average-case optimal by construction when a single evaluation of the integrand remains within our evaluation budget. Achieving this one-step optimality requires solving a challenging value of information optimization problem, for which we provide a novel efficient discretization-free computational method. We also prove consistency for our method in both continuum and discrete finite domains for objective functions that are sums. In numerical experiments comparing against previous state-of-the-art methods, including those that also leverage sum or integral structure, our method performs as well or better across a wide range of problems and offers significant improvements when evaluations are noisy or the integrand varies smoothly in the integrated variables.},
archivePrefix = {arXiv},
arxivId = {1803.08661},
author = {Toscano-Palmerin, Saul and Frazier, Peter I.},
COMMENTED_doi = {10.1137/19M1303125},
eprint = {1803.08661},
file = {:C\:/Users/dweichert/Downloads/Literatur_BO/toscano_palmerin.pdf:pdf},
COMMENTED_issn = {10526234},
journal = {SIAM Journal on Optimization},
keywords = {Bayesian optimization,Gaussian process,black-box optimization},
mendeley-groups = {RobustBO_UAI},
number = {2},
pages = {417--444},
title = {{Bayesian Optimization With Expensive Integrands}},
volume = {32},
year = {2022}
}

@inproceedings{Wang17,
  author    = {Zi Wang and
               Stefanie Jegelka},
  COMMENTED_editor    = {Doina Precup and
               Yee Whye Teh},
  title     = {Max-value Entropy Search for Efficient Bayesian Optimization},
  booktitle = {Proceedings of the 34th International Conference on Machine Learning},
  series    = {Proceedings of Machine Learning Research},
  volume    = {70},
  pages     = {3627--3635},
  COMMENTED_publisher = {{PMLR}},
  year      = {2017},
  COMMENTED_url       = {http://proceedings.mlr.press/v70/wang17e.html},
  timestamp = {Wed, 29 May 2019 08:41:45 +0200},
  biburl    = {https://dblp.org/rec/conf/icml/WangJ17.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{Weichert2021,
  COMMENTED_doi = {10.48550/ARXIV.2107.13772},
  COMMENTED_url = {https://arxiv.org/abs/2107.13772},
  author = {Weichert, Dorina and Kister, Alexander},
  COMMENTED_keywords = {Machine Learning (cs.LG), Optimization and Control (math.OC), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Mathematics, FOS: Mathematics},
  title = {Bayesian Optimization for Min Max Optimization},
  COMMENTED_publisher = {arXiv},
  booktitle={Workshop on Real World Experiment Design and Active Learning at ICML 2020},
  year = {2020},
  COMMENTED_copyright = {arXiv.org perpetual, non-exclusive license}
}

@inproceedings{Yang2023,
abstract = {Bayesian Optimization (BO) is a sample-efficient optimization algorithm widely employed across various applications. In some challenging BO tasks, input uncertainty arises due to the inevitable randomness in the optimization process, such as machining errors, execution noise, or contextual variability. This uncertainty deviates the input from the intended value before evaluation, resulting in significant performance fluctuations in the final result. In this paper, we introduce a novel robust Bayesian Optimization algorithm, AIRBO, which can effectively identify a robust optimum that performs consistently well under arbitrary input uncertainty. Our method directly models the uncertain inputs of arbitrary distributions by empowering the Gaussian Process with the Maximum Mean Discrepancy (MMD) and further accelerates the posterior inference via Nystrom approximation. Rigorous theoretical regret bound is established under MMD estimation error and extensive experiments on synthetic functions and real problems demonstrate that our approach can handle various input uncertainties and achieve state-of-the-art performance.},
archivePrefix = {arXiv},
arxivId = {2310.20145},
author = {Yang, Lin and Lyu, Junlong and Lyu, Wenlong and Chen, Zhitang},
eprint = {2310.20145},
file = {:C\:/Users/dweichert/Downloads/Literatur_BO/Yang_Neurips23.pdf:pdf},
mendeley-groups = {RobustBO_UAI},
title = {{Efficient Robust Bayesian Optimization for Arbitrary Uncertain inputs}},
booktitle = {Advances in Neural Information Processing Systems},
COMMENTED_url = {http://arxiv.org/abs/2310.20145},
year = {2023}
}

@article{Zhu1997,
author = {Zhu, Ciyou and Byrd, Richard H. and Lu, Peihuang and Nocedal, Jorge},
title = {Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization},
year = {1997},
issue_date = {Dec. 1997},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {23},
number = {4},
COMMENTED_issn = {0098-3500},
COMMENTED_url = {https://doi.org/10.1145/279232.279236},
COMMENTED_doi = {10.1145/279232.279236},
abstract = {L-BFGS-B is a limited-memory algorithm for solving large nonlinear optimization problems subject to simple bounds on the variables. It is intended for problems in which information on the Hessian matrix is difficult to obtain, or for large dense problems. L-BFGS-B can also be used for unconstrained problems and in this case performs similarly to its predessor, algorithm L-BFGS (Harwell routine VA15). The algorithm is implemented in Fortran 77.},
journal = {ACM Transactions on Mathematical Software},
COMMENTED_month = {dec},
pages = {550–560},
numpages = {11},
keywords = {variable metric method, nonlinear optimization, limited-memory method, large-scale optimization}
}

@ARTICLE{complexity_nelder_mead,
  author={Garmanjani, Roholla and Vicente, Luís Nunes},
  journal={IMA Journal of Numerical Analysis}, 
  title={Smoothing and worst-case complexity for direct-search methods in nonsmooth optimization}, 
  year={2013},
  volume={33},
  number={3},
  pages={1008-1028},
  COMMENTED_keywords={derivative-free optimization;direct search;smoothing function;worst-case complexity;nonsmooth;nonconvex},
  COMMENTED_doi={10.1093/imanum/drs027}
}

@inproceedings{Rahimi2007,
 author = {Rahimi, Ali and Recht, Benjamin},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {J. Platt and D. Koller and Y. Singer and S. Roweis},
 pages = {},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Random Features for Large-Scale Kernel Machines},
 COMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2007/file/013a006f03dbc5392effeb8f18fda755-Paper.pdf},
 volume = {20},
 year = {2007}
}

@inproceedings{Malkomes2018,
 author = {Malkomes, Gustavo and Garnett, Roman},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {S. Bengio and H. Wallach and H. Larochelle and K. Grauman and N. Cesa-Bianchi and R. Garnett},
 pages = {},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Automating Bayesian optimization with Bayesian optimization},
 COMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2018/file/2b64c2f19d868305aa8bbc2d72902cc5-Paper.pdf},
 volume = {31},
 year = {2018}
}


@InProceedings{Gardner2019,
  title = 	 {{Discovering and Exploiting Additive Structure for Bayesian Optimization}},
  author = 	 {Gardner, Jacob and Guo, Chuan and Weinberger, Kilian and Garnett, Roman and Grosse, Roger},
  booktitle = 	 {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics},
  pages = 	 {1311--1319},
  year = 	 {2017},
  COMMENTED_editor = 	 {Singh, Aarti and Zhu, Jerry},
  volume = 	 {54},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {20--22 Apr},
  COMMENTED_publisher =    {PMLR},
  COMMENTED_pdf = 	 {http://proceedings.mlr.press/v54/gardner17a/gardner17a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v54/gardner17a.html},
  abstract = 	 {Bayesian optimization has proven invaluable for black-box optimization of expensive functions. Its main limitation is its exponential complexity with respect to the dimensionality of the search space using typical kernels. Luckily, many objective functions can be decomposed into additive subproblems, which can be optimized independently. We investigate how to automatically discover such (typically unknown) additive structure while simultaneously exploiting it through Bayesian optimization. We propose an efficient algorithm based on Metropolis-Hastings sampling and demonstrate its efficacy empirically on synthetic and real-world data sets. Throughout all our experiments we reliably discover hidden additive structure whenever it exists and exploit it to yield significantly faster convergence.}
}

@inproceedings{Belakaria2019,
 author = {Belakaria, Syrine and Deshwal, Aryan and Doppa, Janardhan Rao},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
 pages = {},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Max-value Entropy Search for Multi-Objective Bayesian Optimization},
 COMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/82edc5c9e21035674d481640448049f3-Paper.pdf},
 volume = {32},
 year = {2019}
}


@InProceedings{Takeno2022,
  title = 	 {Sequential and Parallel Constrained Max-value Entropy Search via Information Lower Bound},
  author =       {Takeno, Shion and Tamura, Tomoyuki and Shitara, Kazuki and Karasuyama, Masayuki},
  booktitle = 	 {Proceedings of the 39th International Conference on Machine Learning},
  pages = 	 {20960--20986},
  year = 	 {2022},
  COMMENTED_editor = 	 {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
  volume = 	 {162},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_month = 	 {17--23 Jul},
  publisher =    {PMLR},
  COMMENTED_pdf = 	 {https://proceedings.mlr.press/v162/takeno22a/takeno22a.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v162/takeno22a.html},
  abstract = 	 {Max-value entropy search (MES) is one of the state-of-the-art approaches in Bayesian optimization (BO). In this paper, we propose a novel variant of MES for constrained problems, called Constrained MES via Information lower BOund (CMES-IBO), that is based on a Monte Carlo (MC) estimator of a lower bound of a mutual information (MI). Unlike existing studies, our MI is defined so that uncertainty with respect to feasibility can be incorporated. We derive a lower bound of the MI that guarantees non-negativity, while a constrained counterpart of conventional MES can be negative. We further provide theoretical analysis that assures the low-variability of our estimator which has never been investigated for any existing information-theoretic BO. Moreover, using the conditional MI, we extend CMES-IBO to the parallel setting while maintaining the desirable properties. We demonstrate the effectiveness of CMES-IBO by several benchmark functions and real-world problems.}
}


@InProceedings{Gardner2014,
  title = 	 {Bayesian Optimization with Inequality Constraints},
  author = 	 {Gardner, Jacob and Kusner, Matt and Xu, (Eddie) Zhixiang and Weinberger, Kilian and Cunningham, John},
  booktitle = 	 {Proceedings of the 31st International Conference on Machine Learning},
  pages = 	 {937--945},
  year = 	 {2014},
  COMMENTED_editor = 	 {Xing, Eric P. and Jebara, Tony},
  volume = 	 {32},
  COMMENTED_number =       {2},
  series = 	 {Proceedings of Machine Learning Research},
  COMMENTED_address = 	 {Bejing, China},
  COMMENTED_month = 	 {22--24 Jun},
  publisher =    {PMLR},
  COMMENTED_pdf = 	 {http://proceedings.mlr.press/v32/gardner14.pdf},
  COMMENTED_url = 	 {https://proceedings.mlr.press/v32/gardner14.html},
  abstract = 	 {Bayesian optimization is a powerful framework for minimizing expensive objective functions while using very few function evaluations.  It has been successfully applied to a variety of problems, including hyperparameter tuning and experimental design.  However, this framework has not been extended to the inequality-constrained optimization setting, particularly the setting in which evaluating feasibility is just as expensive as evaluating the objective.  Here we present constrained Bayesian optimization, which places a prior distribution on both the objective and the constraint functions.  We evaluate our method on simulated and real data, demonstrating that constrained Bayesian optimization can quickly find optimal and feasible points, even when small feasible regions cause standard methods to fail.},
}

@conference{gelbart2014,
  year = {2014},
  author = {Gelbart, Michael A. and Snoek, Jasper and Adams, Ryan P.},
  title = {Bayesian Optimization with Unknown Constraints},
  booktitle = {Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence (UAI)},
  COMMENTED_keywords = {Bayesian optimization, UAI},
  COMMENTED_note = {arXiv:1403.5607 [stat.ML]}
}  

@inproceedings{Swersky2013,
 author = {Swersky, Kevin and Snoek, Jasper and Adams, Ryan P},
 booktitle = {Advances in Neural Information Processing Systems},
 COMMENTED_editor = {C.J. Burges and L. Bottou and M. Welling and Z. Ghahramani and K.Q. Weinberger},
 pages = {},
 COMMENTED_publisher = {Curran Associates, Inc.},
 title = {Multi-Task Bayesian Optimization},
 COMMENTED_url = {https://proceedings.neurips.cc/paper_files/paper/2013/file/f33ba15effa5c10e873bf3842afb46a6-Paper.pdf},
 volume = {26},
 year = {2013}
}

@article{Forrester2007MultifidelityOV,
  title={Multi-fidelity optimization via surrogate modelling},
  author={Alexander I. J. Forrester and Andr{\'a}s S{\'o}bester and Andy J. Keane},
  journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  year={2007},
  volume={463},
  pages={3251 - 3269},
  COMMENTED_url={https://api.semanticscholar.org/CorpusID:12107319}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% not used

@article{Thompson1933,
author = {Thompson, William R.},
journal = {Biometrika},
number = {3},
pages = {285--294},
publisher = {[Oxford University Press, Biometrika Trust]},
title = {{On the Likelihood that One Unknown Probability Exceeds Another in View of the Evidence of Two Samples}},
volume = {25},
year = {1933}
}

@article{Frazier2018,
    title={A Tutorial on Bayesian Optimization},
    author={Peter I. Frazier},
    year={2018},
    journal = {arXiv preprint arXiv:1807.02811},
    eprint={1807.02811},
    archivePrefix={arXiv},
    primaryClass={stat.ML}
}

@article{Cunningham2011,
    title={Gaussian Probabilities and Expectation Propagation},
    author={John P. Cunningham and Philipp Hennig and Simon Lacoste-Julien},
    year={2011},
    journal = {arXiv preprint arXiv:1111.6832},
    eprint={1111.6832},
    archivePrefix={arXiv},
    primaryClass={stat.ML}
}

% related work section
@article{UrRehman2015,
author = {ur Rehman, Samee and Langelaar, Matthijs},
journal = {Structural and Multidisciplinary Optimization},
number = {2},
pages = {319--336},
title = {{Efficient global robust optimization of unconstrained problems affected by parametric uncertainties}},
volume = {52},
year = {2015}
}

@inproceedings{Wabersich2015,
  author    = {Kim P. Wabersich and
               Marc Toussaint},
  title     = {Automatic testing and minimax optimization of system parameters for
               best worst-case performance},
  booktitle = {2015 {IEEE/RSJ} International Conference on Intelligent Robots and
               Systems},
  pages     = {5533--5539},
  year      = {2015},
  timestamp = {Wed, 16 Oct 2019 14:14:51 +0200},
  biburl    = {https://dblp.org/rec/conf/iros/WabersichT15.bib},
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  author    = {Shion Takeno and
               Hitoshi Fukuoka and
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               Toshiyuki Koyama and
               Motoki Shiga and
               Ichiro Takeuchi and
               Masayuki Karasuyama},
  title     = {Multi-fidelity Bayesian Optimization with Max-value Entropy Search
               and its Parallelization},
  booktitle = {Proceedings of the 37th International Conference on Machine Learning,
               {ICML} 2020, 13-18 July 2020, Virtual Event},
  series    = {Proceedings of Machine Learning Research},
  volume    = {119},
  pages     = {9334--9345},
  publisher = {{PMLR}},
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@inproceedings{Liu2020,
  author    = {Sijia Liu and
               Songtao Lu and
               Xiangyi Chen and
               Yao Feng and
               Kaidi Xu and
               Abdullah Al{-}Dujaili and
               Mingyi Hong and
               Una{-}May O'Reilly},
  title     = {Min-Max Optimization without Gradients: Convergence and Applications
               to Black-Box Evasion and Poisoning Attacks},
  booktitle = {Proceedings of the 37th International Conference on Machine Learning,
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  title     = {Practical Bayesian optimization in the presence of outliers},
  booktitle = {International Conference on Artificial Intelligence and Statistics,
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               Spain},
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@inproceedings{Bogunovic2020,
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               {AISTATS} 2020, 26-28 August 2020, Online [Palermo, Sicily, Italy]},
  series    = {Proceedings of Machine Learning Research},
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               Jae{-}Gil Lee and
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  title     = {Stable Bayesian Optimization},
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@inproceedings{Curi2020,
  author    = {Sebastian Curi and
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  editor    = {Hugo Larochelle and
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               Raia Hadsell and
               Maria{-}Florina Balcan and
               Hsuan{-}Tien Lin},
  title     = {Adaptive Sampling for Stochastic Risk-Averse Learning},
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               on Neural Information Processing Systems 2020, NeurIPS 2020, December
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  author    = {Matthew Staib and
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  editor    = {Kamalika Chaudhuri and
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  title     = {Distributionally Robust Submodular Maximization},
  booktitle = {The 22nd International Conference on Artificial Intelligence and Statistics,
               {AISTATS} 2019, 16-18 April 2019, Naha, Okinawa, Japan},
  series    = {Proceedings of Machine Learning Research},
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@inproceedings{Nguyen2020,
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               Sunil Gupta and
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               Santu Rana and
               Svetha Venkatesh},
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               Roberto Calandra},
  title     = {Distributionally Robust Bayesian Quadrature Optimization},
  booktitle = {The 23rd International Conference on Artificial Intelligence and Statistics,
               {AISTATS} 2020, 26-28 August 2020, Online [Palermo, Sicily, Italy]},
  series    = {Proceedings of Machine Learning Research},
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  title     = {Robust Optimization for Non-Convex Objectives},
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@inproceedings{Bogunovic2017,
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  series    = {Proceedings of Machine Learning Research},
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@inproceedings{Wilder2018,
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@InProceedings{Han2022,
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  @misc{Weichert2022,
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@misc{Wang2021,
  doi = {10.48550/ARXIV.2111.08862},
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  publisher = {arXiv},
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@misc{Yang2021,
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