\section{Preliminary Lemmas}
We state some standard concentration bounds that are used in our proofs. Their proofs can be found in the citations provided.
\begin{lemma}[Chernoff Bounds, Section 4.2 in \cite{Mitzenmacher2005}]{\label{lemma: Chernoff bound}}
Let $Z$ be any random variable. Then for any $t > 0$,
\begin{enumerate}
    \item $\mathbb{P}(Z \geq \mathbb{E}[Z]+t) \leq \min_{\lambda > 0} \mathbb{E}[e^{\lambda(Z - \mathbb{E}[Z])}]e^{-\lambda t}$
    \item $\mathbb{P}(Z \leq \mathbb{E}[Z]-t) \leq \min_{\lambda > 0} \mathbb{E}[e^{\lambda(\mathbb{E}[Z] - Z)}]e^{-\lambda t}$
\end{enumerate}
\end{lemma}
\vspace{0.1cm}
\begin{lemma}[Hoeffding's Lemma, Lemma 2.6 in \cite{Massart2007}]{\label{lemma: Hoeffding's lemma}}
Let $Z$ be a bounded random variable with $Z \in [a, b]$. Then,
\[
\mathbb{E}[\exp(\lambda(Z-\mathbb{E}[Z])] \leq \exp\Big(\frac{\lambda^2(b-a)^2}{8}\Big)
\]
for all $\lambda \in \mathbb{R}$.
\end{lemma}
\vspace{0.1cm}
\begin{lemma}[Chernoff-Hoeffeding inequality, \cite{Chernoff1952, Hoeffding1963}]\label{lemma: chernoff-hoeffding inequality}
Suppose $X_1, \ldots, X_T$ are independent random variables taking values in the interval $[0,1]$, and let $X = \sum_{t\in [T]} X_t$ and $\overline{X} = \frac{1}{T}(\sum_{t\in [T]} X_t)$. Then for any $\varepsilon \geq 0$ the following holds:
\begin{enumerate}
    \item $\mathbb{P}(\overline{X} - \mathbb{E}[\overline{X}] \geq \varepsilon ) \leq
    e^{-2\varepsilon^2T}$
    \item $\mathbb{P}(\overline{X} - \mathbb{E}[\overline{X}] \leq -\varepsilon ) \leq
    e^{-2\varepsilon^2T}$
\end{enumerate}
\end{lemma}

\section{Example of CBN with $m(\mathcal{C}) \ll N$}
\label{appendix:example}
Consider a CBN $\mathcal{C} = (\mathcal{G},\mathbb{P})$ with $N$ intervenable nodes and in-degree at most $k-1$, and let $k$ be such that $2^k \ll N$. Further, let $\mathbb{P}$ be such that for at most $2^k$ nodes, chosen in the reverse topological order, the conditional probability of a node being 1 given its parents is Bernoulli with parameter $1/2^{k+1}$, and for the remaining nodes the conditional probability of a node being 1 given its parents is Bernoulli with parameter $1/2$. Now, using the definition of $m(\mathcal{C})$ provided in Section \ref{sec: simple regret for general graphs}, it is easy to see that $m(\mathcal{C}) \leq 2^k \ll N$.