\begin{proposition} \label{prop: Glasso_optimal_rate}
	Let $\bS_{p,f,n}$ be a $d'\times d'$ (where $d'=p$ or $d'=f$) random matrix such that with probability $1-\frac{2}{n^2}$, $|\bS_{p,f,n}-\bSigma_*|_\infty \leq r_{p,f,n}$. Assume $\bSigma_*\in S_{++}^{d'}$ has uniformly bounded spectrum as $p,f\to\infty$ (analog to Assumption 1). Choose $\lambda_{p,f,n} = c \cdot r_{p,f,n}$ for some absolute constant $c>0$. Consider the Glasso operator $\bG(\cdot,\cdot)$ defined in (\ref{G_operator}). Let $s=s_{\bTheta_*}$ be the sparsity parameter associated with $\bTheta_*:=\bSigma_*^{-1}$. Assume $\sqrt{d'+s} \cdot r_{p,f,n}=o(1)$. Then, with probability $1-\frac{2}{n^2}$,
	\begin{equation*}
		\nn \bG(\bS_{p,f,n},\lambda_{p,f,n})-\bTheta_* \nn_F \leq \frac{2\sqrt{2} (1+c)}{\lambda_{min}(\bSigma_*)^2} \sqrt{d'+s} \cdot r_{p,f,n}
	\end{equation*}
	as $p,f,n\to\infty$.
\end{proposition}
\begin{IEEEproof}
	The proof follows from a slight modification of Thm. 1 in \cite{Rothman}, or Thm. 3 in \cite{TimeVaryingGraphs}. This modification is due to the different $r_{p,f,n}$.
\end{IEEEproof}
