\section{Decoupled Setting}\label{sec:decoupled}
In the main paper, we assume both objective $\globalf$ and the constraints $\{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$ are revealed upon querying an input point. The setting is regarded as a coupling of the objective and constraints to differentiate from the decoupled setting, where the objective and constraints may be evaluated independently. In the decoupled setting, acquisition functions need to explicitly tradeoff the evaluation of the different aspects and, in addition to helping to pick the candidate $\instance_t \in \searchSpace$, suggest $g_t \in \{\globalf\}\cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$ for evaluation each time. This typically requires different acquisition from coupled setting \citep{gelbart2014bayesian}. However, we will show that our acquisition function and $\algname$ require minimum adaptation to the decoupled setting while bearing a similar performance guarantee. 
\subsection{Algorithm for Decoupled Setting}
When taking the $g_t \leftarrow \argmax_{g \in \gG} \alpha_{g, t}{(\instance_{g, t})}$ in \algoref{alg:main}, we explicitly choose the aspect that matters most at a certain iteration. Naturally, we could adapt \algname to the decoupled setting by querying $\instance_{g, t}$ on this unknown function $g_t \in \gG \subseteq \{\globalf\}\cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$ at iteration $t$. The modified algorithm is shown below.

\begin{algorithm*}%[H]
    \caption{\emph{Decoupled} \textbf{\underline{CO}}nstrained \textbf{\underline{B}}O through \textbf{\underline{A}}daptive \textbf{\underline{R}}egion of interest Acquisition (\algname-Decoupled)}
    \label{alg:decoupled}
    
        \begin{algorithmic}[1]
        \STATE {\bf Input}:Search space $\searchSpace$, initial observation $\Selected_0$, horizon $T$, confidence factor $\delta$, estimated $\epsilon_\cFunc$;
            \FOR{$t = 1\ to\ T$}
                \STATE Update the posteriors of $\GP_{\globalf,t}$ and $\GP_{\cFunc_\conIdx, t}$ according to \eqref{eq:posterior_mean} and \plaineqref{eq:posterior_covar}
                 
                \STATE Identify ROIs $\roi_t$, and undecided sets $U_{\cFunc_\conIdx, t}$ 
    
                \FOR{$\conIdx\in\conSpace$}
                    \IF{$U_{\cFunc_\conIdx, t} \neq \emptyset$}
                    \STATE Candidate for active Learning of each constraint: \\
                    $\instance_{\cFunc_\conIdx, t} \leftarrow \argmax_{\instance \in \discreteROI \cap U_{\cFunc_\conIdx, t}} \acqC{(\instance)}$ as in \eqref{eq:acqC}
                    \STATE  $\gG \leftarrow \gG \cup \cFunc_{k, t}$
                    \ENDIF
                \ENDFOR
                
                \STATE Candidate for optimizing the objective: \\
                $\instance_{\globalf, t} \leftarrow \argmax_{\instance \in \discreteROI} \acqF{(\instance)}$ as in \eqref{eq:acqF}
                \STATE $\gG \leftarrow \gG \cup \globalf$
    
                \STATE Maximize the acquisition values from different aspects: \\
                $g_t \leftarrow \argmax_{g \in \gG} \alpha_{g, t}{(\instance_{g_t, t})} $
    
                \STATE Pick the candidate to evaluate: $\instance_t \leftarrow \instance_{g_t, t}$  
    
                \STATE \emph{Update the observation set with the candidate and corresponding new observations on $g_t$}\\
                $ \Selected_t \leftarrow \Selected_{t-1} \cup \{(\instance_t, y_{g_t, t})\}$
                
            \ENDFOR
        \end{algorithmic}
\end{algorithm*}

\subsection{Theoretical guarantee and proof}
We first denote the maximum mutual information gain after $T$ rounds of evaluations as 
\begin{align}
    \maxInfoDe_T = \sum_{g \in \{\globalf\}\cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}}{\maxInfo_{g, T_g}}     
\end{align}

Where $T_g$ denotes the number of evaluations for $g \in \{\globalf\}\cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$ before $T$. Therefore we have $$T = \sum_{g \in \{\globalf\}\cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}}T_g$$
Then, we have the following guarantee for the performance of \algname-Decoupled.
{
\begin{theorem}\label{thm: decoupled-width}
    The width of the resulting confidence interval of the global optimum $f^*=f(\instance^*)$ has an upper bound. That is, under the same assumptions in \thmref{thm: width}, with $\beta=2\log(2(\conNum+1)\vert \discreteROI \vert \pi_t/ \delta)$ that is constant, and acquisition function in $\algoref{alg:decoupled}$, $\exists \epsilon_\globalf \leq \epsilon_\cFunc$, after at most $T \geq \frac{\beta \maxInfoDe_T C_1}{\epsilon_\globalf^2}$ iterations, we have $\Pr{\vert CI_{\globalf^*, T}\vert \leq \epsilon_\globalf, \globalf^* \in CI_{\globalf^*, T }} \geq 1 - \delta$
    Here $C_1=8/\log(1+\sigma^{-2})$. 
\end{theorem}
}
\begin{lem} \label{lem:acqBound-decoupled}
Under the conditions assumed in \thmref{thm: decoupled-width} except for \assref{apt: exist_star}, let $\alpha_t = \max_{g \in \gG } \alpha_{g,t}(\instance_{g, t})$ as in \algoref{alg:decoupled}, with $\beta=2\log(\frac{2(\conNum+1)\vert \discreteSet \vert T}{\delta})$ that is a constant, after at most $T \geq \frac{\beta {\maxInfoDe_T} C_1}{\epsilon_\globalf^2}$ iterations, $\alpha_T\leq \epsilon_\globalf$
Here $C_1=8/\log(1+\sigma^{-2})$.
\end{lem}

{Here is the critical difference to the proof of \thmref{thm: width}.}\\


\begin{proof}
{We first unify the notation in the acquisition functions.}\\
$\forall T\geq t\geq 1, \forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$, when $\discreteROI \cap U_{g, t}\neq \emptyset$,
\begin{align}\label{eq:nonEmpty_U_acqC_bound_decoupled}
    \max_{\instance \in \discreteROI \cap U_{g, t}} \UCBit_{g,t}(\instance) - \LCB_{g, t}(\instance)
    &\leq \alpha_t
\end{align}
$\forall T\geq t\geq 1, \forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$, when $\discreteROI \cap U_{\cFunc_\conIdx, t} = \emptyset$, let
\begin{align}\label{eq:empty_U_acqC_bound-decoupled}
    \max_{\instance \in \discreteROI \cap U_{g, t}} \UCBit_{g,t}(\instance) - \LCB_{g, t}(\instance)
    &= 0 \leq \alpha_t
\end{align}

$\forall T\geq t\geq 1, g=\globalf$, when $S_{\cFunc, t} = \emptyset$, we have
\begin{align}\label{eq:roi_acqf_bound_1-decoupled}
    \max_{\instance \in \discreteROI} \UCBit_{\globalf,t}(\instance) - \LCB_{\globalf, t}(\instance) \leq \alpha_t
\end{align}

$\forall T\geq t\geq 1, g=\globalf$, when $S_{\cFunc, t}\neq \emptyset$, we have
\begin{align}\label{eq:roi_acqf_bound_2-decoupled}
    \max_{\instance \in \discreteROI} \UCBit_{\globalf,t}(\instance) - \LCB_{\globalf, t. max} \leq \alpha_t
\end{align}


{By lemma 5.1, 5.2 and 5.4 of \citet{srinivas2009gaussian}, with $\beta=2\log(\frac{2(\conNum+1)\vert \discreteSet \vert T}{\delta})$, $\forall g \in \{\globalf\} \cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$ and $\forall x_t \in \discreteROI \subseteq \discreteSet$, we have $\sum_{t=1}^{T} (2\beta^{1/2}\sigma_{g, t-1} ,(\instance_t))^2\mathbf{1}(g_t = g) \leq C_1\beta\maxInfo_{g, T_g}$. By definition of $\alpha_t$ , we have the following}

\begin{align*}
    \sum_{t=1}^{T} \alpha_t^2 & = \sum_{t=1}^{T}\max_{g \in \{\globalf\}\cup\{\cFunc_\conIdx\}_{\conIdx\in\conSpace} } \alpha_{g_t,t}^2(\instance_{g_t, t})\\
    &\leq \sum_{t=1}^{T}\max_{g \in \{\globalf\}\cup\{\cFunc_\conIdx\}_{\conIdx\in\conSpace} } (2\beta^{1/2}\sigma_{g_t, t-1}(\instance_{g_t, t}))^2\\
    &\leq \sum_{g \in \{\globalf\}\cup\{\cFunc_\conIdx\}_{\conIdx\in\conSpace}}C_1\beta\maxInfo_{g, T_g}\\
    &= C_1\beta \maxInfoDe_T
\end{align*}

By Cauchy-Schwarz, we have 
$$
\frac{1}{T}(\sum_{t=1}^{T} \alpha_t)^2 \leq C_1\beta{\maxInfoDe_T}
$$

By the monotonocity assumed in $\assref{apt: mono_ci}$, $\forall g \in \{\cFunc_m\}_{\conIdx\in\conSpace}$, $\forall 1 \leq t_1 < t_2 \leq T$, $\forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$, we have $U_{g, t_2}\subseteq U_{g, t_1}$ and $\roi_{t_2} \subseteq \roi_{t_1}$, and most importantly, $\alpha_{t_2} \leq \alpha_{t_1}$. Therefore
$$
\alpha_T \leq \frac{1}{T}\sum_{t=1}^{T} \alpha_t \leq \sqrt{\frac{C_1\beta{\maxInfoDe_T}}{T}}
$$
As a result, after at most $T \geq \frac{\beta {\maxInfoDe_T} C_1}{\epsilon_\globalf^2}$ iterations, we have $\alpha_T\leq \epsilon_\globalf$.
\end{proof}

{The rest of the proof for \thmref{thm: decoupled-width} is essentially the same as proof for \thmref{thm: width} except for substituting \lemref{lem:acqBound} with \lemref{lem:acqBound-decoupled}.}
