\section{Preliminaries}
\label{prelim}

\label{prelim:subgauss}
For $\bx \in \mathbb{R}^d$, let $\| \bx\|_2$ denote the Euclidean norm. For $\mb{A} \in \mathbb{R}^{d \times d}$, let the operator norm of $\mb{A}$ be denoted by $\| \mb{A}\| = \mathsf{max}_{\| \bx\|_2 = 1} \| \mb{A}\bx\|_2$.
We present the following theorem from Chapter 6 of \cite{Wainwright-HDP}:
\begin{theorem}
\label{thm:prelim-random}
    Consider $\bX_1, \bX_2, \dots, \bX_m$ in $\mathbb{R}^d$ iid from $N(\bm{\mu},\bm{\Sigma})$. Then, for any $\zeta > 0$, we have with probability $1-2 e^{-m\zeta^2/2}$,
    \begin{multline}
    \left\vert\left\vert \frac{1}{m} \sum_{i=1}^{m} \left(\bX_i \bX_i^T - \mathbb{E}[\bX_i \bX_i^T]\right) \right\vert\right\vert 
     \\ \leq \left(2\sqrt{\frac{d}{m}} + 2\zeta + \left(\sqrt{\frac{d}{m}} + \zeta\right)^2\right) \\
     \quad \quad \cdot (\| \bm{\mu}\|_2^2 + \lambda_{\tn{max}}(\bm{\Sigma}))
        \label{eqn:prelim-random}
    \end{multline}
\end{theorem}

\subsection{Real functions from a class}\label{sec:functions}
For a class $\mc{F}$ of real-valued functions (regressors) over $\mbc{X}$ with values in $[0, 1]$, and any $\mbc{X}' \subseteq \mbc{X}$ s.t. $|\mbc{X}'| = N$, let $\mc{C}_p(\xi, \mc{F}, \mbc{X}')$ denote a minimum cardinality subset (cover) of $\mc{F}$ such that for each $f^* \in \mc{F}$, there exists $f \in \mc{C}_p(\xi, \mc{F}, \mbc{X}')$ s.t. $\left(\E_{\bx \in \mbc{X}'}\left[\left|f^*(\bx) - f(\bx)\right|^p\right]\right)^{1/p} \leq \xi$ for $p \in [1,\infty)$, and $\max_{\bx\in \mbc{X}'}\left|f^*(\bx) - f(\bx)\right| \leq \xi$ for $p =\infty$.

The maximum size of $\mc{C}_p(\xi, \mc{F}, \mbc{X}')$ over all choices of  $|\mbc{X}'| = N$ is defined to be  $N_p(\xi, \mc{F}, N)$.
We refer the reader to Sections 10.2-10.4 of \cite{Anthony-Bartlett} for more details.

The \textit{pseudo-dimension} of $\mathcal{F}$, $\mathsf{Pdim}(\mathcal{F})$ is a measure of the complexity of $\mathcal{F}$. As described in Sec. 10.4 and 12.3 of \cite{Anthony-Bartlett} the pseudo dimension can be used to bound the size of covers of $\mathcal{F}$ as follows:
\begin{equation}
N_1(\xi,\mathcal{F}, N) \leq N_{\infty} (\xi, \mathcal{F}, N) \leq (eN/\xi p)^p \label{eqn:coversize}
\end{equation}
where $p = \mathsf{Pdim}(\mathcal{F})$ and $N \geq p$.
