Optimizing Conditional Value-At-Risk of Black-Box Functions

Quoc Phong Nguyen, Zhongxiang Dai, Bryan Kian Hsiang Low, Patrick Jaillet

Keywords: Bayesian optimization, conditional value-at-risk, CVaR, UCB, Thompson sampling

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.

Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.

TL;DR: To optimize the conditional value-at-risk of a black-box function, we develop two Bayesian optimization algorithms with performance guarantees, one of which can handle batch queries.

Code: https://github.com/qphong/BayesOpt-LV