Bayesian Optimization of CVaR with Unknown Hyperparameters

We study Bayesian optimization of the Conditional Value-at-Risk (CVaR) of an expensive black-box function that depends on a decision and an environmental variable. Existing no-regret algorithms assume that the correct Gaussian process (GP) kernel and its hyperparameters are known. In practice, however, they must be estimated from limited data, often yielding a misspecified surrogate. This is especially damaging for CVaR because misspecification can distort the estimated tail-risk and thus lead to catastrophic decisions. We propose HE-CV-UCB, the first no-regret algorithm for CVaR Bayesian optimization with unknown GP hyperparameters. Given a finite candidate set containing a well-specified model, HE-CV-UCB adaptively eliminates those inconsistent with the observed data, actively queries where the survivors disagree, and returns a design that is robust across the surviving candidates. We prove that it attains cumulative regret matching the oracle rate of CV-UCB up to a factor depending only on the candidate set's size. Across synthetic benchmarks and three real-world applications, HE-CV-UCB achieves competitive performance and frequently outperforms existing CVaR Bayesian optimization methods.