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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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