\section{Proofs}\label{sec:proof}
\subsection{Proof of \lemref{lem: roi}}
\newtheorem{lem0}{Lemma}
\begin{lem0}
    Under the assumptions above, the regions of interest $\roi_{t}$, as defined in \eqref{eq:roi}, contain the global optimum with high probability. Formally, for all $\delta \in (0,1)$, $T\geq t\geq 1$, and any finite discretization $\discreteSet$ of $\searchSpace$ that contains the optimum $\instance^* = \argmax_{\instance\in \searchSpace}f(\instance)$ where $\cFunc_\conIdx(\instance^*) > \epsilon_{\cFunc}$ for all $\conIdx\in\conSpace$ and $\beta_t=2\log(2(\conNum+1)\vert \discreteSet \vert \pi_t/ \delta)$ with $\sum_{t\geq 1}^{T}\pi_t^{-1} = 1$,  we have $\Pr{\instance^* \in \discreteROI} \geq 1-\delta$.
\end{lem0}

\begin{proof}
     With probability at least $1-1/2\delta$, $\forall \instance\in \discreteSet, \forall T \geq t\geq 1, \forall g \in \{\globalf\} \cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$,
    $$\vert g(\instance) - \mu_{g, t-1}(\instance)\vert \leq \beta_{t}^{1/2}\sigma_{g, t-1}(\instance)$$
    
    Note that we also take the union bound on $ g \in \{\globalf\} \cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$. 
    
    This is similarly derived as lemma 5.1 of \citet{srinivas2009gaussian} or lemma 1 of \citet{zhang2023learning}. {\emph{Different from previous proofs}, we do not require the lemma to hold for $\forall t \geq 1$. Instead, we require it to hold for $\forall T \geq t \geq 1$. This alleviates the need of the convergence of the series $\sum_{t\geq 1}\pi_t^{-1} = 1$ to $\sum_{t\geq 1}^{T}\pi_t^{-1} = 1$ when taking the union bound. Specifically, we could set $\pi_t = {T}$, which essentially makes $\beta_t=2\log(\frac{2(\conNum+1)\vert \discreteSet \vert T}{\delta})$ a constant. Hence, we use the $\beta$ in the following instead of $\beta_t$ as traditionally used to highlight this difference.} 

    {First, by definition $S_{\cFunc, t} \defeq \bigcap_m^{\conSpace} S_{\cFunc_\conIdx, t}$, we have $\forall t \leq T, \instance \in \discreteSet \cap S_{\cFunc, t}, \forall \conIdx\in\conSpace$}
    $$
    \Pr{\cFunc_\conIdx(\instance) \geq \LCB_{\cFunc_\conIdx, t}(\instance) > 0} \geq 1-1/2\delta
    $$
    meaning with probability at $1-\delta$, $\instance$ lies in the feasible region. At the same time, we have, $\forall t \leq T$, $\forall \conIdx\in\conSpace$, given $\cFunc_\conIdx(\instance) > 0$
    $$
    \Pr{\UCB_{\globalf, t}(\instance^*) \geq \globalf(\instance^*) \geq \globalf(\instance) \geq \LCB_{\globalf, t}(\instance) \text{ }} \geq 1-1/2\delta
    $$
    Given the mutual independency between the objective $\globalf$ and the constraints $\cFunc_\conIdx$, and by the definition of the threshold $\LCB_{\globalf,t, \max}$, we have
    {$\forall t \leq T$, when $\exists \instance \in \discreteSet \cap S_{\cFunc, t}$,}
    $$
    \Pr{\UCB_{\globalf, t}(\instance^*) > \LCB_{\globalf, t, \max} } \geq 1-\delta
    $$

    Note when $\discreteSet \cap S_{\cFunc, t} = \emptyset$, $\LCB_{\globalf,t, \max}=-\infty$, we have $\Pr{\UCB_{\globalf, t}(\instance^*) > \LCB_{\globalf, t, \max} } = 1$.

    {In summary, we've shown that with probability at least $1-\delta$, $\instance^* \in \discreteSet \cap \roi_{\globalf, t}$.}

    Next, by the definition of $\instance^* = \argmax_{\instance\in \searchSpace}f(\instance)$ $s.t.$ $\cFunc_\conIdx(\instance^*) > \epsilon_{\cFunc}$ we have $\forall t \leq T, \forall \conIdx\in\conSpace$
    % $$
    % \Pr{\UCBit_{\cFunc_\conIdx, t}(\instance^*) = \mu_{\cFunc_\conIdx, t-1}(\instance^*) + \beta_{t}^{1/2}\sigma_{\cFunc_\conIdx, t-1}(\instance^*) \geq \cFunc_\conIdx(\instance^*) > 0} \geq 1-1/2\delta
    % $$
    $$
    \Pr{\UCBit_{\cFunc_\conIdx, t}(\instance^*) \geq \cFunc_\conIdx(\instance^*) > 0} \geq 1-1/2\delta
    $$
    {meaning with probability at least $1-1/2\delta$, $\instance^* \in \discreteSet \cap \roi_{\cFunc_\conIdx, t}$. And in general, we have $\forall t \leq T, \forall \conIdx\in\conSpace$
    $$
    \Pr{\instance^* \in \discreteSet \cap \roi_{t}} \geq 1-\delta
    $$}
\end{proof}
    
\subsection{Proof of \thmref{thm: width}}
The following lemmas show that the maximum of the acquisition functions \eqref{eq:acqF} and \plaineqref{eq:acqC} are both bounded after sufficient evaluations.

\begin{lem} \label{lem:acqBound}
Under the conditions assumed in \thmref{thm: width} except for \assref{apt: exist_star}, let $\alpha_t = \max_{g \in \gG } \alpha_{g,t}(\instance_{g, t})$ as in \algoref{alg:main}, with $\beta=2\log(\frac{2(\conNum+1)\vert \discreteSet \vert T}{\delta})$ that is a constant, after at most $T \geq \frac{\beta \widehat{\maxInfo_T} C_1}{\epsilon_\globalf^2}$ iterations, $\alpha_T\leq \epsilon_\globalf$
Here $C_1=8/\log(1+\sigma^{-2})$.
\end{lem}

{The inequation $T \geq \frac{\beta \widehat{\maxInfo_T} C_1}{\epsilon_\globalf^2}$ has $T$ on both side, which follows the convention in \cite{gotovos2013active}.}

\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}
    \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}
    \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}
    \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}
    \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 $\sum_{t=1}^{T} (2\beta^{1/2}\sigma_{g, t-1},(\instance_t))^2 \leq C_1\beta\maxInfo_{g, T}$. By definition of $\alpha_t$ , we have the following}
{
\begin{align*}
    \sum_{t=1}^{T} \alpha_t^2
    &\leq \sum_{t=1}^{T} \max_{g \in \{\globalf\}\cup\{\cFunc_\conIdx\}_{\conIdx\in\conSpace} } (2\beta^{1/2}\sigma_{g, t-1}(\instance_{g, t}))^2\\
    &\leq \sum_{t=1}^{T} \sum_{g \in \{\globalf\}\cup\{\cFunc_\conIdx\}_{\conIdx\in\conSpace} } (2\beta^{1/2}\sigma_{g, t-1}(\instance_{t}))^2\\
    &\leq \sum_{g \in \{\globalf\}\cup \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}}C_1\beta\maxInfo_{g, T}\\
    &= C_1\beta\widehat{\maxInfo_T}
\end{align*}}

The last line holds due to the definition in \eqref{eq:gammaT}. By Cauchy-Schwarz, we have 
$$
\frac{1}{T}(\sum_{t=1}^{T} \alpha_t)^2 \leq C_1\beta\widehat{\maxInfo_T}
$$


{With $\assref{apt: mono_ci}$, $\forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$, $\forall 1 \leq t_1 < t_2 \leq T$, $\forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$, 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
$$
\alpha_T \leq \frac{1}{T}\sum_{t=1}^{T} \alpha_t \leq \sqrt{\frac{C_1\beta\widehat{\maxInfo_T}}{T}}
$$
As a result, after at most $T \geq \frac{\beta \widehat{\maxInfo_T} C_1}{\epsilon_\globalf^2}$ iterations, we have $\alpha_T\leq \epsilon_\globalf$.}
\end{proof}

With \lemref{lem:acqBound}, we could first prove that after adequately $T$ rounds of evaluations such that $\epsilon_\globalf \leq \min_{\conIdx\in\conSpace}\epsilon_\conIdx$ is sufficiently small, with certain probability, $\instance^* \in S_{\cFunc, T}$. Then $\LCB_{\globalf,t, \max}\ne -\infty$, and therefore the width of $[\max_{\instance \in \discreteROI}\LCB_{\globalf, T}(\instance),
\max_{\instance \in \discreteROI}\UCBit_{\globalf,T}(\instance)]$, which is a the high confidence interval of $f^*$, is bounded by $\epsilon_\globalf$.

\begin{proof}
    We first prove that after at most $T \geq \frac{\beta \widehat{\maxInfo_T} C_1}{\epsilon_\globalf^2}$ iterations, $\Pr{\instance^* \in \discreteROI \cap S_{\cFunc, T}} \geq 1-1/2\delta$.
    Given \eqref{eq:nonEmpty_U_acqC_bound} and \plaineqref{eq:empty_U_acqC_bound} and \lemref{lem:acqBound}, we have $\forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$, 
    $$
    \max_{\instance \in \discreteSet_{\hat{X}_T} \cap U_{g, T}} \UCBit_{g,T}(\instance) - \LCB_{g, T}(\instance) \leq \epsilon_\globalf \leq \min_{\conIdx\in\conSpace}\epsilon_\conIdx
    $$ 
    According to the definition of $U_{g, T}$, $\forall \instance \in\discreteSet_{\hat{X}_T} \cap U_{g, T}, \forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$
    $$\UCBit_{g,T}(\instance) \leq \epsilon_\globalf + \LCB_{g, T}(\instance) \leq \epsilon_\globalf \leq \min_{\conIdx\in\conSpace}\epsilon_\conIdx $$

    According to \assref{apt: exist_star}, and \lemref{lem: roi}, $\forall \conIdx\in\conSpace$, we have 

    \begin{align*}
        \Pr{\UCBit_{\cFunc_\conIdx, T}(\instance^*) > \max_{\instance\in \discreteROI \cap U_{\cFunc_\conIdx, t}} \UCBit_{\cFunc_\conIdx, T}(\instance)} \geq 1-1/2\delta
    \end{align*}
    Given =  $\discreteSet_{\hat{X}_T}\cap S_{\cFunc, T} = \discreteROI \cap \roi_{\cFunc,T} \backslash
    \cup_{\conIdx\in\conSpace}U_{\cFunc_\conIdx, T}$, when $t = T$, we have
    \begin{align}\label{eq: feasibility}
        \Pr{\instance^* \in \discreteSet_{\hat{X}_T} \cap S_{\cFunc, T}} \geq 1-1/2\delta        
    \end{align}


    As a result
    $$
    \Pr{\LCB_{\globalf,T, \max} \ne -\infty}  \geq 1-1/2\delta
    $$
    Next, we prove the upper bound for the width of the high-confidence interval of $f^*$. Given that $\LCB_{\globalf,T, \max} \ne -\infty$, we have
    \begin{align*}
        \textit{\quad}\max_{\instance \in \discreteSet_{\hat{X}_T}}\UCBit_{\globalf, T}(\instance) - \max_{\instance \in \discreteSet_{\hat{X}_T}}\LCB_{\globalf, T}(\instance)
        \leq \max_{\instance \in \discreteSet_{\hat{X}_T}}\UCBit_{\globalf, T}(\instance) - \LCB_{\globalf, T, \max}
        \leq \alpha_T
        \leq \epsilon_\globalf
    \end{align*}
    Combining it with the observation that with probability $1-1/2\delta$,

    $$
    \max_{\instance \in \discreteSet_{\hat{X}_T}}\LCB_{\globalf, T}(\instance) < \globalf(\instance^*)\leq \max_{\instance \in \discreteSet_{\hat{X}_T}}\UCBit_{\globalf, T}(\instance)
    $$
    we attain the final result that after $T \geq \frac{\beta \widehat{\maxInfo_T} C_1}{\epsilon^2}$ iterations,
    $$
    \Pr{\vert CI_{\globalf^*, T}\vert \leq \epsilon, \globalf^* \in CI_{\globalf^*, T }} \geq 1 - \delta
    $$

\end{proof}

\subsection{Proof of \corref{cor: zero-regret}}
\begin{proof}
    We simply need to show that after $t \geq \frac{\beta \widehat{\maxInfo_t} C_1}{{\epsilon}_{\globalf}^2}$ iterations, with probability at least $1-\delta$, $\instance^*$ is the only member in $\discreteROI$.

    Similar to \thmref{thm: width}, we have $\Pr{\vert CI_{\globalf^*, t}\vert \leq \epsilon_{\globalf}, \globalf^* \in CI_{\globalf^*, t}} \geq 1-\delta$. At the same time, given the proof of \lemref{lem:acqBound}, we have $\forall \instance \in \discreteROI$,  $2\beta^{1/2}\sigma_{f, t-1}(\instance) \leq \epsilon_{\globalf} $.

    Then if $\exists \instance \neq \instance^*$ and $\instance \in \discreteROI$, we have $\globalf^* - \globalf(\instance) > 2\epsilon_{\globalf}$, while $$\Pr{\globalf^* - \globalf(\instance) \leq \vert CI_{\globalf^*, t}\vert + \UCB_{\globalf, t}(\instance) - \LCB_{\globalf, t}(\instance) \leq 2\beta^{1/2}\sigma_{f, t-1}(\instance) + \epsilon_{\globalf} \leq 2\epsilon_{\globalf}} \geq 1-\delta$$ 
    This contradiction means with probability at least $1-\delta$, $\instance^*$ is the only member in $\discreteROI$, and $\instance_t = \instance^*$.
    As a result, $\Pr{\regret_{t} = 0} \geq 1-\delta$, when $T\geq t \geq \frac{\beta \widehat{\maxInfo_t} C_1}{{\epsilon}_{\globalf}^2}$.
\end{proof}
\subsection{Proof of \corref{cor: cum-regret}}

\begin{proof}
    We follow the same path as the proof of \corref{cor: zero-regret}.

    Similar to \thmref{thm: width}, we have $\Pr{\vert CI_{\globalf^*, t}\vert \leq \alpha_t \leq \epsilon_{\globalf}, \globalf^* \in CI_{\globalf^*, t}} \geq 1-\delta$. At the same time, given the proof of \lemref{lem:acqBound}, we have $\forall \instance \in \discreteROI$,  $2\beta^{1/2}\sigma_{f, t-1}(\instance) \leq \alpha_t \leq \epsilon_{\globalf} $.

    Then $\forall \instance \neq \instance^*$ and $\instance \in \discreteROI$, we have $$\Pr{\globalf^* - \globalf(\instance) \leq \vert CI_{\globalf^*, t}\vert + \UCB_{\globalf, t}(\instance) - \LCB_{\globalf, t}(\instance) \leq 2\alpha_t \leq 2\epsilon_{\globalf}} \geq 1-\delta$$ 
    Then by assumption, $\forall \instance \in \discreteROI$, $\forall \conIdx\in\conSpace$, we have probability at least $1-\delta$, $\cFunc_\conIdx(\instance) \geq \epsilon_{\cFunc}$, and hence $\instance \notin U_{\cFunc_\conIdx, t}$. According to the algorithm, it regresses to GP-UCB by \cite{srinivas2009gaussian} between $t'$ and $T$.
    \begin{align*}
        \sum_{t=t'}^{T}(\reward(\instance^*) - \reward(\instance_t))^2 &\leq \beta{C_1}(\maxInfo_{T} - \maxInfo_{t'})\\
        &\leq \beta{C_1}\maxInfo_{T}(1-t'/T)
    \end{align*}
     By Cauchy-Schwarz, we have
     \begin{align*}
        \sum_{t=t'}^{T}(\reward(\instance^*) - \reward(\instance_t)) &\leq \sqrt{(T-t') \sum_{t=t'}^{T}(\reward(\instance^*) - \reward(\instance_t))^2}\\
        &\leq \sqrt{\frac{(T-t')^2}{T}\beta{C_1}\maxInfo_{T}}\\
        &\leq \sqrt{(T-t')\beta{C_1}\maxInfo_{T}}
    \end{align*}
    
\end{proof}



\subsection{Proof of \corref{cor: feasibility}}


\begin{proof}
    We assume $\discreteSet_{\hat{X}_T} \neq \emptyset$ and prove by contradiction.
    Given \eqref{eq:nonEmpty_U_acqC_bound} and \plaineqref{eq:empty_U_acqC_bound} and \lemref{lem:acqBound}, we have $\forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$,
    $$
    \max_{\instance \in \discreteSet_{\hat{X}_T} \cap U_{g, T}} \UCBit_{g,T}(\instance) - \LCB_{g, T}(\instance) \leq \epsilon'_{\cFunc}
    $$ 
    According to the definition of $U_{g, T}$, $\forall \instance \in \discreteSet_{\hat{X}_T} \cap U_{g, T}, \forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$, with probability at least $1-1/2\delta$, we have
    $$\cFunc_\conIdx(\instance) \leq \UCBit_{\cFunc_\conIdx, T}(\instance) \leq {\epsilon'}_{\cFunc} + \LCB_{g, T}(\instance) \leq {\epsilon'}_{\cFunc} +\cFunc_\conIdx(\instance) $$

    Then we have $\forall \instance \in \discreteSet_{\hat{X}_T} \cap U_{g, T}$, $\exists \conIdx\in\conSpace$
    \begin{align*}
        \Pr{\cFunc_\conIdx(\instance) \leq {\epsilon'}_{\cFunc} +\cFunc_\conIdx(\instance)  < 0} \geq 1-1/2\delta
    \end{align*}
    This contradiction means $\forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$,  $\discreteSet_{\hat{X}_T} \cap U_{g, T} = \emptyset$ with probability as least $1-1/2\delta$.

    According to the definition of $S_{g, T}$, $\forall \instance \in \discreteSet_{\hat{X}_T} \cap S_{g, T}, \forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$
    $$
    \LCB_{g, T}(\instance) \geq \epsilon'_{\cFunc}
    $$
    Then we have $\forall \instance \in \discreteSet_{\hat{X}_T} \cap S_{g, T}$, $\exists g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$
    \begin{align*}
        \Pr{-\epsilon'_{\cFunc} \geq \cFunc_\conIdx(\instance) \geq \LCB_{g, T}(\instance) \geq \epsilon'_{\cFunc}} \geq 1-1/2\delta
    \end{align*}
    This contradiction means $\forall g \in \{\cFunc_\conIdx\}_{\conIdx\in\conSpace}$,  $\discreteROI \cap S_{g, T} = \emptyset$ with probability as least $1-1/2\delta$.

    Combining the above contradictions, we have at least when $t=T$,
    \begin{align*}
        \Pr{\discreteSet_{\roi_T} = \emptyset} \geq 1-\delta        
    \end{align*}
\end{proof}
