From cd7c3fe857f1ec8972959be2a5a9b73c6f4167ef Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 11 Feb 2013 01:06:19 -0800
Subject: Small fixes to section 3 and 4

---
 problem.tex | 24 ++++++++++++++++--------
 1 file changed, 16 insertions(+), 8 deletions(-)

(limited to 'problem.tex')

diff --git a/problem.tex b/problem.tex
index cec7b47..668cd57 100644
--- a/problem.tex
+++ b/problem.tex
@@ -2,7 +2,8 @@
 \subsection{Linear Regression and Experimental Design}
 
 The theory of experimental design \cite{pukelsheim2006optimal,atkinson2007optimum,chaloner1995bayesian} considers the following formal setting. %  studies how an experimenter \E\  should select the parameters of a set of experiments she is about to conduct. In general, the optimality of a particular design depends on the purpose of the experiment, \emph{i.e.}, the quantity \E\  is trying to learn or the hypothesis she is trying to validate. Due to their ubiquity in statistical analysis, a large literature on the subject focuses on learning \emph{linear models}, where \E\ wishes to  fit  a linear function to the data she has collected.
-Putting cost considerations aside, suppose that an experimenter \E\ wishes to conduct $k$ among $n$ possible experiments. Each experiment $i\in\mathcal{N}\defeq \{1,\ldots,n\}$ is associated with a set of parameters (or features) $x_i\in \reals^d$, normalized so that $\|x_i\|_2\leq 1$. Denote by $S\subseteq \mathcal{N}$, where $|S|=k$, the set of experiments selected; upon its execution, experiment $i\in S$ reveals an output variable (the ``measurement'') $y_i$,  related to the experiment features $x_i$ through a linear function, \emph{i.e.},
+%Putting cost considerations aside
+Suppose that an experimenter \E\ wishes to conduct $k$ among $n$ possible experiments. Each experiment $i\in\mathcal{N}\defeq \{1,\ldots,n\}$ is associated with a set of parameters (or features) $x_i\in \reals^d$, normalized so that $\|x_i\|_2\leq 1$. Denote by $S\subseteq \mathcal{N}$, where $|S|=k$, the set of experiments selected; upon its execution, experiment $i\in S$ reveals an output variable (the ``measurement'') $y_i$,  related to the experiment features $x_i$ through a linear function, \emph{i.e.},
 \begin{align}\label{model}
      \forall i\in\mathcal{N},\quad y_i = \T{\beta} x_i + \varepsilon_i
 \end{align}
@@ -71,6 +72,7 @@ Our analysis will focus on the case of a \emph{homotropic} prior, in which the p
 %$V(S)+V(T)\geq V(S\cup T)+V(S\cap T)$ for all $S,T\subseteq \mathcal{N}$; it is also monotone, \emph{i.e.}, $V(S)\leq V(T)$ for all $S\subset T$.
 
 \subsection{Budgeted-Feasible Experimental Design: Full Information Case}
+
 We depart from the above classic experimental design setting by assuming that each experiment is associated with a cost $c_i\in\reals_+$. Moreover, the experimenter $\E$ is limited by a budget $B\in \reals_+$. 
 The cost $c_i$ can capture, \emph{e.g.}, the amount the subject $i$ deems sufficient to
 incentivize her participation in the experiment. 
@@ -84,20 +86,26 @@ In the full-information case, the experiment costs are common knowledge; as such
 \end{align}\label{edp}
 \end{subequations}
 \end{center}
-\EDP{}, as defined above, is  NP-hard; to see this, note that \textsc{Knapsack} reduces to EDP with dimension $d=1$ by mapping the weight of each item, say, $w_i$, to an experiment with $x_i=w_i$. Moreover, \eqref{modified} is submodular, monotone and satisfies $V(\emptyset) = 0$.
+\EDP{}, as defined above, is  NP-hard; to see this, note that \textsc{Knapsack}
+reduces to EDP with dimension $d=1$ by mapping the weight of each item, say,
+$w_i$, to an experiment with $x_i^2=w_i$.% Moreover, \eqref{modified} is submodular, monotone and satisfies $V(\emptyset) = 0$.
 
  Note that \eqref{modified} is a submodular set function, \emph{i.e.}, 
-$V(S)+V(T)\geq V(S\cup T)+V(S\cap T)$ for all $S,T\subseteq \mathcal{N}$. It is also monotone, \emph{i.e.}, $V(S)\leq V(T)$ for all $S\subset T$, with $V(\emptyset)=0$. Finally, it is also non-negative, \emph{i.e.}, $V(S)\geq 0$ for all $S\subset\mathcal{N}$, as the matrix $\T{X_S}X_S$ is positive semi-definite for all $S\subseteq \mathcal{N}$.
+$V(S)+V(T)\geq V(S\cup T)+V(S\cap T)$ for all $S,T\subseteq \mathcal{N}$. It is
+also monotone, \emph{i.e.}, $V(S)\leq V(T)$ for all $S\subset T$, with
+$V(\emptyset)=0$. Finally, it is also non-negative, \emph{i.e.}, $V(S)\geq 0$
+for all $S\subseteq\mathcal{N}$, since the matrix $\T{X_S}X_S$ is positive semi-definite for all $S\subseteq \mathcal{N}$.
 %Finally, we define the strategic version  of \EDP{} (denoted by \SEDP) by adding the additional constraints of truthfulness, individual rationality, and normal payments, as described in the previous section.
  We denote by
 \begin{equation}\label{eq:non-strategic}
     OPT = \max_{S\subseteq\mathcal{N}} \Big\{V(S) \;\Big| \;
     \sum_{i\in S}c_i\leq B\Big\}
 \end{equation}
-for the optimal value achievable in the full-information case. 
+the optimal value achievable in the full-information case. 
+
 \subsection{Experimental Design Problem: Strategic Case}
-When the experiment subjects are strategic, the experimental design problem becomes a special case of  a \emph{budget feasible reverse auction}
-, as introduced by \citeN{singer-mechanisms}.  %comprises a set of items $\mathcal{N}=\{1,\ldots,n\}$ as well as a single buyer. Each item has a cost $c_i\in\reals_+$. Moreover, the buyer has a positive value function  $V:2^{\mathcal{N}}\to \reals_+$, as well as a budget $B\in \reals_+$. In the \emph{full information case}, the costs $c_i$ are common
+
+When the experiment subjects are strategic, the experimental design problem becomes a special case of  a \emph{budget feasible reverse auction}, as introduced by \citeN{singer-mechanisms}.  %comprises a set of items $\mathcal{N}=\{1,\ldots,n\}$ as well as a single buyer. Each item has a cost $c_i\in\reals_+$. Moreover, the buyer has a positive value function  $V:2^{\mathcal{N}}\to \reals_+$, as well as a budget $B\in \reals_+$. In the \emph{full information case}, the costs $c_i$ are common
 %knowledge; the objective of the buyer in this context is to select a set $S$
 %maximizing the value $V(S)$ subject to the constraint $\sum_{i\in S} c_i\leq
 %B$. We write:
@@ -117,11 +125,11 @@ $S:\reals_+^n \to 2^\mathcal{N}$ and (b) a \emph{payment function}
 $p:\reals_+^n\to \reals_+^n$. Given the
 vector or costs $c=[c_i]_{i\in\mathcal{N}}$, the allocation function $f$ determines the set in
 $ \mathcal{N}$ of items to be purchased, while the payment function
-returns a vector of payments $[p_i]_{i\in \mathcal{N}}$.
+returns a vector of payments $[p_i(c)]_{i\in \mathcal{N}}$.
  Let $s_i(c) = \id_{i\in S(c)}$ be the binary indicator of  $i\in f(c)$. Mechanism design in budget feasible reverse auctions seeks mechanisms that have the following properties \cite{singer-mechanisms,chen}:
 \begin{itemize}
 \item \emph{Normalization.} Unallocated experiments receive zero payments, \emph{i.e.},
- \begin{align}\notin S(c)\text{ implies }p_i(c)=0.\label{normal}\end{align}
+ \begin{align}s_i(c)=0\text{ implies }p_i(c)=0.\label{normal}\end{align}
 \item\emph{Individual Rationality.} Payments for allocated experiments exceed costs, \emph{i.e.},
  \begin{align} p_i(c)\geq c_i\cdot s_i(c).\label{ir}\end{align}
 \item \emph{No Positive Transfers.} Payments are non-negative,\emph{i.e.},
-- 
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