\section{Related Work}
\label{sec:related_work}

\paragraph{Extension of unconstrained methods}
While the majority of BO research focuses on unconstrained problems \citep{bernardo2011optimization, frazier2018tutorial, gramacy2020surrogates, binois2022survey, garnett_bayesoptbook_2023}, several works also address black-box constraints. The pioneering work by \citet{schonlau1998global} first extended Expected Improvement (EI) to the constrained setting, and subsequent developments \citep{gelbart2014bayesian, gardner2014bayesian, feliot2017bayesian, letham2019constrained, wang2024constrained} refined this approach by defining the acquisition function at a given point as the product of the expected improvement and the probability that the point is feasible. In addition, the posterior sampling method (Thompson sampling) was extended to scalable CBO (SCBO) by \citet{eriksson2021scalable}, generalizing the unconstrained TuRBO approach \citep{eriksson2019scalable} by incorporating additional samples from the constraint posteriors to weight the objective samples. Methods based on information criteria \citep{hernandez2014predictive, wang2017max} have also been extended to the constrained setting \citep{hernandez2015predictive, perrone2019constrained, takeno2022sequential}, although these approaches rely heavily on sampling-based approximations. Another line of work transforms the CBO task into an unconstrained problem via the augmented Lagrangian framework \citep{gramacy2016modeling, picheny2016bayesian, ariafar2019admmbo}, allowing vanilla BO to be applied as a subroutine, particularly in decoupled settings. In general, these methods do not guarantee the identification of the feasible region during optimization, and consequently, they lack a convergence rate for regret that accounts for feasibility.

\paragraph{Violation-tolerant objectives}
In addition to the aforementioned empirical approaches, recent works \citep{zhou2022kernelized, lu2022no, pmlr-v211-guo23a, xu2023constrained} have considered a relaxed CBO objective to facilitate theoretical analysis of convergence rates. These works assume that queries outside the feasible region still incur a reward and incorporate constraint violations either as a weighted penalty within the regret or analyze them separately from the objective’s regret. Although they provide upper bounds on both the constraint violations and the regret, these methods do not adequately address the issue of potentially infinite regret arising from evaluations outside the feasible region. Since diminishing constraint violations do not guarantee the eventual selection of a feasible point, the analysis in these works diverges from our objective without nontrivial modifications.

\paragraph{Active learning of constraints}
The concept of data selection in active learning dates back to MacKay’s work \citep{mackay1992information}, and stepwise uncertainty reduction (SUR) has been used to estimate failure probabilities in industrial settings \citep{bect2012sequential}. The principled approach of active learning for level-set estimation (AL-LSE) was introduced by \citet{gotovos2013active} to perform classification over the sample space, offering theoretical guarantees on convergence rates. Since both AL-LSE and BO employ Gaussian processes to model underlying functions, \citet{bogunovic2016truncated} unified these problems using truncated variance reduction and by selecting a kernel that ensures submodularity of the variance reduction. However, a direct application of level-set estimation methods is limited by their focus on a single unknown function and the lack of a straightforward extension to balance the learning of multiple unknown functions. While \citet{malkomes2021beyond, komiyama2022bridging} proposed novel acquisition functions that prioritize diversity in the active search, these approaches do not provide a mechanism for adaptively trading off between constraint learning and objective optimization. A similar challenge is encountered in \citet{antonio2021sequential}, where the algorithm decouples the learning of the feasible region from the optimization of the objective by addressing them in two separate phases. These decoupled, two-phase approaches are fundamentally different from our goal of an integrated algorithm that adaptively balances learning and optimization in every step.
