Add second version of the notes

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+\documentclass{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath,amsthm,amsfonts}
+\usepackage{comment}
+\newtheorem{lemma}{Lemma}
+\newtheorem{fact}{Fact}
+\newtheorem{example}{Example}
+\newcommand{\var}{\mathop{\mathrm{Var}}}
+\newcommand{\condexp}[2]{\mathop{\mathbb{E}}\left[#1|#2\right]}
+\newcommand{\expt}[1]{\mathop{\mathbb{E}}\left[#1\right]}
+\newcommand{\norm}[1]{\lVert#1\rVert}
+\newcommand{\tr}[1]{#1^*}
+\newcommand{\ip}[2]{\langle #1, #2 \rangle}
+\newcommand{\mse}{\mathop{\mathrm{MSE}}}
+\newcommand{\trace}{\mathop{\mathrm{tr}}}
+\begin{document}
+
+\section{Problem}
+
+\begin{itemize}
+\item  $D = (x_i)_{1\leq i\leq n}$
+\item $(x_i)_{1\leq i\leq n}$ sampled in an i.i.d fashion from $\mu$
+\end{itemize}
+
+There is a function $F$ and you are interested in estimating the value
+$F(\mu)$. We assume that you have an estimation scheme $\tilde{F}$,
+which given a set of data points $S$ returns an estimation
+$\tilde{F}(S)$ which is optimal in some sense. Your also given a
+revenue function $R$ which is a decreasing function of the estimation
+error. Then the value $V$ of the databse is defined by:
+\begin{displaymath}
+  V(D) = \max_{S\subseteq D} R\left(| F(\mu) - \tilde{F} |\right)
+\end{displaymath}
+
+\begin{example}
+  
+\end{example}
+
+\begin{fact}
+
+  \begin{itemize}
+  \item If $R$ is decreasing then $V$ is increasing in the size of $D$.
+  \item If $R$ is concave then $V$ is supermodular.
+  \end{itemize}
+\end{fact}
+
+\begin{proof}
+  
+\end{proof}
+
+\end{document}
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