\section{Introduction}
Bayesian optimization (BO) has been extensively studied over the past few decades as a powerful framework for addressing expensive black-box optimization tasks in machine learning, engineering, and science. In many real-world applications, these optimization tasks often involve black-box constraints that are costly to evaluate \citep{digabel2015taxonomyconstraintssimulationbasedoptimization}. Examples include choosing from a plethora of untested medical therapies under safety constraints \citep{sui2015safe}; determining optimal pumping rates in hydrology to minimize operational costs under constraints on plume boundaries  
\citep{gramacy2016modeling}; or tuning hyperparameters of a neural network under memory constraints \citep{gelbart2014bayesian}.
It is common to model constraints analogously to the objectives via Gaussian processes (GP) and then utilize an acquisition function to trade off the learning and optimization to decide subsequent query points.


%\begin{adjustwidth}{0cm}{0cm}
\begin{table*}[!h]
  \centering
  \resizebox{0.8\textwidth}{!}{%
    \begin{tabularx}{\textwidth}{@{}X X X X@{}}
      \toprule
      \makecell{\textbf{Approach}} & \makecell{\textbf{Feasible Region} \\ \textbf{Identification}} & 
      \makecell{\textbf{Regret Guarantees}} & \makecell{\textbf{Constraint Learning}\\ \textbf{and Trade-off}} \\
      \midrule
      \makecell{Extension of\\ Unconstrained Methods} & 
        Not explicitly guaranteed & 
        No established convergence rate with respect to constraints & 
        No adaptive trade-off between objective and constraints \\
      \midrule
      \makecell{Violation-Tolerant\\ Objectives} & 
        May select infeasible points despite decreasing violations & 
        Provides upper bounds on regret, but only when infeasible points yield meaningful rewards & 
        Penalizes constraint violations separately from the objective \\
      \midrule
      \makecell{Active Learning\\ of Constraints} & 
        Typically limited to one constraint at a time & 
        Guarantees available only for level-set estimation & 
        Splits constraint learning and objective optimization into separate phases \\
      \midrule
      \makecell{\textbf{\algname}\\(Proposed)} & 
        \textbf{Guaranteed identification of the feasible region} & 
        \textbf{Non-asymptotic regret convergence rate} & 
        \textbf{Integrated, adaptive trade-off between objective optimization and constraint learning} \\
      \bottomrule
    \end{tabularx}
  }
  \caption{Comparison of existing CBO approaches and \algname (Proposed). The first three rows serve as a high-level summary of the respective approaches discussed in the preceding paragraphs of Section~\ref{sec:related_work}, where detailed discussion and citations are provided.}
  \label{tab:comparison}
\end{table*}
% \end{adjustwidth}%

Recently, significant advancements have been made in several directions to address constrained BO (CBO). 
For instance, extended Expected Improvement approaches \citep{bernardo2011optimization, gelbart2014bayesian, gardner2014bayesian, zhang2021constrained, bachoc2020gaussian} learn the constraints passively and calibrate the acquisition with feasibility. The augmented lagrangian (AL) methods \citep{gramacy2016modeling, picheny2016bayesian, ariafar2019admmbo} convert constrained optimization into unconstrained optimization with additional hyperparameters. The entropy-based methods \citep{takeno2022sequential} optimize the lower bound of the mutual information concerning the underlying optimum within the feasible region. 

In general, the existing methods extend the unconstrained BO methods to learn the constraints and optimize the objective simultaneously with heuristics, approximation, or relaxation of the objective. Nevertheless, these treatments often hinder the rigorous theoretical analysis, particularly regarding regret, which is well-established for unconstrained BO tasks \cite{srinivas2009gaussian, chowdhury2017kernelized}. 
The key challenge lies in the fact that an optimum outside the feasible region or insufficient exploration within the feasible region can lead to unbounded regret. 
{Therefore, it is crucial for CBO algorithms to rigorously guarantee the \emph{accurate and efficient} identification of the entire feasible region.  This ensures that the algorithm remains within the feasible region and selects near-optimal candidates with high probability after sufficient iterations.} Unfortunately, this guarantee is absent in existing methods, which motivates our approach to incorporate active learning of unknown constraints in designing a CBO algorithm with a regret guarantee.

In this paper, we propose a novel framework that integrates active learning for level-set estimation (AL-LSE) \citep{gotovos2013active, nguyen2021information} with Bayesian optimization for constrained problems. Our approach leverages the theoretical strengths of both paradigms, enabling a rigorous performance analysis of the proposed CBO method in finding the interior optimum. A brief illustration of the framework design is shown in \figref{fig:pipeline}. The remainder of this paper is organized as follows. We first provide a detailed overview of recent advancements in various facets of CBO. Next, we formally state the CBO problem and introduce the definition of probabilistic regret as a performance metric that facilitates rigorous analysis. Following the problem statement, we present our novel CBO framework, offer the corresponding performance analysis, and provide empirical evidence for its efficacy. Finally, we summarize the key contributions of our work and discuss potential directions for future research.
