\section{Simple Regret Lower Bound}\label{sec: lower bound for general graphs}
A closer inspection of \SRM\ given in Sec. \ref{sec: simple regret for general graphs} reveals that the algorithm only leverages causal side-information while pulling the observational arm. Hence, there remains a possibility that a better algorithm could be designed which uses the information shared between any two interventions. In this section, we show that this is not possible for a large and important class of causal graphs that we call tree-graphs and denote it as $\mathsf{T}$.   
Each graph in $\mathsf{T}$ is an $n$-ary tree, where each node can have $2$ to $n$ children. Additionally, all the leaves are connected to the outcome node $Y$. We also assume that all nodes of any graph in $\mathsf{T}$ are observable. 
Note that a causal bandit algorithm receives as input a causal graph $\mathcal{G}$ (corresponding to some CBN $\mathcal{C} = (\mathcal{G}, \mathbb{P})$) but the associated distribution $\mathbb{P}$ is unknown to the algorithm. Since there are multiple probability distributions that are compatible with a given $\mathcal{G}$ the algorithm is required to learn the unknown $\mathbb{P}$ through the arm pulls. We show in Thm. \ref{theorem: LB-Tree} that for any causal graph $\mathcal{G}$ in $\mathsf{T}$ and any positive integer $M\leq N$, there exists a distribution $\mathbb{P}$ such that $M = m(\mathcal{C})$, where $\mathcal{C}$ is CBN $(\mathcal{G}, \mathbb{P})$, and, any algorithm must explore at least $\Omega(M)$ arms to  minimize the worst-case expected simple regret.

\begin{theorem} \label{theorem: LB-Tree}
Corresponding to every causal graph $\mathcal{G} \in \mathsf{T}$, with $N$ intervenable nodes and any positive integer $M \leq N$, there exists a probability measure  $\mathbb{P}$ and CBN $\mathcal{C} = (\mathcal{G}, \mathbb{P})$ such that $m(\mathcal{C}) = M$ and the expected simple regret of any causal bandit algorithm \texttt{ALG} is $r_{\texttt{ALG}}(T) = \Omega\big(\sqrt{m(\mathcal{C})/T} \big)$.
\end{theorem}
The proof of Thm. \ref{theorem: LB-Tree} is in App. \ref{secappendix: proof of lower bound for tree}. 
Recall, from Sec. \ref{sec: simple regret for general graphs} that $m(\mathcal{C})$ is completely defined by $\mathbf{q}=(q_1,\ldots,q_N)$ and $\mathcal{G}$; in particular the definition of $m(\mathcal{C})$ does not depend on the entire probability distribution corresponding to CBN $\mathcal{C}$. We conclude this section by showing in Thm. \ref{theorem: LB-given-q} that the dependence of the regret on $\mathbf{q}$ in the definition of $m(\mathcal{C})$ is optimal for certain graphs. In Thm. \ref{theorem: LB-given-q}, a $\mathbf{q}$ is valid if there exists a probability measure $\mathbb{P}$ for the graph $\mathcal{G}$, which results in the given $\mathbf{q}$. The proof of Thm. \ref{theorem: LB-given-q} is in App. \ref{secappendix: lower bound for tree given q}.

\begin{theorem} \label{theorem: LB-given-q}
There exists a fully observable causal graph $\mathcal{G}$ with $N \geq 3$ nodes such that given any valid $\mathbf{q}$ corresponding to 
$\mathcal{G}$, there is a probability measure $\mathbb{P}$ conforming with $\mathbf{q}$ and CBN $\mathcal{C} = (\mathcal{G},\mathbb{P})$ for which  expected simple regret of any causal bandit algorithm is $\Omega\big(\sqrt{m(\mathcal{C})/T} \big)$.
\end{theorem}