From e2e3bcb87f8090a1f1811af391f5edb5ec121df7 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Thu, 25 Oct 2012 15:16:10 -0700
Subject: Basic structure in paper.tex

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 paper.tex | 135 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 134 insertions(+), 1 deletion(-)

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 \section{Problem formulation}
 
+\subsection{Notations}
+
+\begin{itemize}
+    \item If $x\in\mathbf{R}^d$, $x^*$ will denote the transpose of vector $x$.
+        If $(x, y)\in\mathbf{R}^d\times\mathbf{R}^d$, $x^*y$ is the standard
+        inner product of $\mathbf{R}^d$.
+    \item We will often use the following order over symmetric matrices: if $A$
+        and $B$ are two $d\times d$ real symmetric matrices, we write that
+        $A\leq B$ iff:
+        \begin{displaymath}
+            \forall x\in\mathbf{R}^d,\quad x^*Ax \leq x^*Bx 
+        \end{displaymath}
+        That is, iff $B-A$ is symmetric semi-definite positive.
+    \item Let us recall this property of the ordered defined above: if $A$ and
+        $B$ are two symmetric definite positive matrices such that $A\leq B$,
+        then $A^{-1}\leq B^{-1}$.
+\end{itemize}
+
+\subsection{Data model}
+\begin{itemize}
+    \item set of $n$ users $\mathcal{N} = \{1,\ldots, n\}$
+    \item each user has a public vector of features $x_i\in\mathbf{R}^d$ and some
+        undisclosed variable $y_i\in\mathbf{R}$
+    \item we assume that the data has been normalized: $\forall
+        i\in\mathcal{N}$, $\norm{x_i}\leq 1$
+    \item \textbf{Ridge regression model:}
+        \begin{itemize}
+            \item $y_i = \beta^*x_i + \varepsilon_i$
+            \item $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$, 
+                $(\varepsilon_i)_{i\in \mathcal{N}}$ are mutually independent.
+            \item prior knowledge of $\beta$: $\beta\sim\mathcal{N}(0,\kappa I_d)$
+            \item $\mu = \frac{\kappa}{\sigma^2}$ is the regularization factor.
+            \item after the variables $y_i$ are disclosed, the experimenter
+                does \emph{maximum a posteriori estimation} which is equivalent
+                under Gaussian prior to solving the ridge regression
+                maximisation problem:
+                \begin{displaymath}
+                    \beta_{\text{max}} = \argmax \sum_i (y_i - \beta^*x_i)^2
+                    + \frac{1}{\mu}\sum_i \norm{\beta}_2^2
+                \end{displaymath}
+        \end{itemize}
+\end{itemize}
+
+\subsection{Economics}
+
+Value function: following the experiment design theory, the value of
+data is the decrease of uncertainty about the model. Common measure of
+uncertainty, the entropy. Thus, the value of a set $S$ of users is:
+\begin{displaymath}
+    V(S) = \mathbb{H}(\beta) - \mathbb{H}(\beta|Y_S)
+\end{displaymath}
+where $Y_S = \{y_i,\,i\in S\}$ is the set of observations.
+
+In our case (under Gaussian prior):
+\begin{align*}
+    \forall S\subset\mathcal{N},\; V(S)
+    & = \frac{1}{2}\log\det\left(I_d
+        + \mu\sum_{i\in S} x_ix_i^*\right)\\
+    & \defeq \frac{1}{2}\log\det A(S)
+\end{align*}
+
+\begin{lemma}[Marginal contribution]
+    \begin{displaymath}
+        \Delta_i V(S)\defeq V(S\cup\{i\}) - V(S) 
+        = \frac{1}{2}\log\left(1 + \mu x_i^*A(S)^{-1}x_i\right)
+    \end{displaymath}
+\end{lemma}
+
+\begin{proof}
+    We have:
+    \begin{align*}
+        V(S\cup\{i\}) & = \frac{1}{2}\log\det A(S\cup\{i\})\\
+        & = \frac{1}{2}\log\det\left(A(S) + \mu x_i x_i^*\right)\\
+        & = V(S) + \frac{1}{2}\log\det\left(I_d + \mu A(S)^{-1}x_i
+    x_i^*\right)\\
+        & = V(S) + \frac{1}{2}\log\left(1 + \mu x_i^* A(S)^{-1}x_i\right)
+    \end{align*}
+    where the last equality comes from Sylvester's determinant formula.
+\end{proof}
+\begin{itemize}
+    \item each user $i$ has a cost $c_i$
+    \item the auctioneer has a budget constraint $B$
+    \item optimisation problem:
+        \begin{displaymath}
+            OPT(V,\mathcal{N}, B) = \max_{S\subset\mathcal{N}} \left\{ V(S)\,|\,
+            \sum_{i\in S}c_i\leq B\right\}
+        \end{displaymath}
+\end{itemize}
+
+Problem, the costs are reported by users, they can lie. It is an auction
+problem. Goal design a mechanism (selection and payment scheme) which is:
+\begin{itemize}
+    \item truthful
+    \item individually rational
+    \item budget feasible
+    \item has a good approximation ratio
+\end{itemize}
+
 \section{Main result}
 
+\begin{algorithm}\label{mechanism}
+    \caption{Budget feasible mechanism for ridge regression}
+    \begin{algorithmic}[1]
+    \State $i^* \gets \argmax_{j\in\mathcal{N}}V(j)$
+    \State $x^* \gets \argmax_{x\in[0,1]^n} \{L_{\mathcal{N}\setminus\{i^*\}}(x)
+                                    \,|\, c(x)\leq B\}$
+        \Statex
+        \If{$L(x^*) < CV(i^*)$}
+            \State \textbf{return} $\{i^*\}$
+        \Else
+            \State $i \gets \argmax_{1\leq j\leq n}\frac{V(j)}{c_j}$
+            \State $S \gets \emptyset$
+            \While{$c_i\leq \frac{B}{2}\frac{V(S\cup\{i\})-V(S)}{V(S\cup\{i\})}$}
+                \State $S \gets S\cup\{i\}$
+                \State $i \gets \argmax_{j\in\mathcal{N}\setminus S}
+                \frac{V(S\cup\{j\})-V(S)}{c_j}$
+            \EndWhile
+            \State \textbf{return} $S$
+        \EndIf
+    \end{algorithmic}
+\end{algorithm}
+
+Choosing $C=...$ we have:
+
+\begin{theorem}
+    The mechanism is truthful, individually rational, budget feasible.
+    Furthermore, by choosing $C= ...$ we get an approximation ratio of:
+    \begin{multline*}
+        1 + C^* = \frac{5e-1 + C_\mu(4e-1)}{2C_\mu(e-1)}\\
+        + \frac{\sqrt{C_\mu^2(1+2e)^2
+        + 2C_\mu(14e^2+5e+1) + (1-5e)^2}}{2C_\mu(e-1)}
+    \end{multline*}
+\end{theorem}
+
 \section{General setup}
 
-\section{Conclusions}
+\section{Conclusion}
+
 \end{document}
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