From d010cdae55a7e19d78a24e46c1db3058d69347fc Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Wed, 15 Aug 2012 18:27:30 -0700
Subject: Repository restructuration (has to be clean now that it is shared ;)

---
 slides.tex | 221 -------------------------------------------------------------
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-\documentclass{beamer}
-\usepackage[utf8x]{inputenc}
-\usepackage[greek,english]{babel}
-\usepackage[LGR,T1]{fontenc}
-\usepackage{lmodern}
-\usepackage{amsmath}
-\usepackage{graphicx}
-\usepackage{tikz}
-\newcommand{\tr}[1]{#1^*}
-\newcommand{\expt}[1]{\mathop{\mathbb{E}}\left[#1\right]}
-\newcommand{\mse}{\mathop{\mathrm{MSE}}}
-\usetheme{Boadilla}
-\usecolortheme{beaver}
-\title[Data monetization]{Data monetization: pricing user data\\with the Shapley Value}
-\author{Thibaut \textsc{Horel}}
-\institute[Technicolor]{Work with {\greektext Στρατής \textsc{Ιωαννίδης}}}
-\setbeamercovered{transparent}
-
-\AtBeginSection[]
-{
-\begin{frame}<beamer>
-\frametitle{Outline}
-\tableofcontents[currentsection]
-\end{frame}
-}
-
-\begin{document}
-
-\begin{frame}
-\maketitle
-\end{frame}
-
-\section{Problem}
-\begin{frame}{Problem Overview}
-\begin{center}
-\includegraphics[width=0.8\textwidth]{schema.pdf}
-\end{center}
-\end{frame}
-
-\begin{frame}{Motivation}
-
-Why is it useful?
-
-\begin{itemize}
-\item sell/buy user data
-\item compensate users in your database
-\end{itemize}
-
-\end{frame}
-
-\begin{frame}{Challenges}
-
-We need to understand:
-\begin{itemize}
-\item the recommender system
-\item the revenue structure
-\item the relation between the revenue and the value of the database
-\item how to compensate individual users
-\end{itemize}
-\end{frame}
-
-\section{Shapley value}
-
-\begin{frame}{Individual value}
-  \begin{itemize}
-  \item $S$: set of players
-  \item $S'\subset S$: coalition
-  \item $V: \mathcal{P}(S)\to \mathbf{R}$: \alert{value} function:
-    \begin{itemize}
-    \item $V(\emptyset) = 0$
-    \item $V(S')$: value of the coalition $S'$
-    \end{itemize}
-  \end{itemize}
-
-  \alert{Question:} How to split the value of a subset across all its constituents?
-
-  \alert{Goal:} $\phi_i(S',V)$: value of player $i$ in $S'$
-\end{frame}
-
-\begin{frame}{Solution attempts}
-  \begin{itemize}
-    \visible<1->{\item $\phi_i(S',V) = V(\{i\})$?} \visible<2->{``The whole is greater than the sum of its parts''
-      $$V(S')\neq \sum_{i\in S'} V(\{i\})$$}
-    \visible<3->{\item $\phi_i(S',V) = V(S') - V(S'\setminus\{i\})$: the \alert{marginal contribution}?} \visible<4->{depends on the time the player joins the coalition}
-  \end{itemize}
-\end{frame}
-
-\begin{frame}{Axioms}
-  \begin{description}
-    \item[Efficiency:] No money lost in the splitting process
-      $$V(S) = \sum_{i\in S} \phi_i(S,V)$$
-\pause
-    \item[Symmetry:] Equal pay for equal contribution
-
-      \alert{if} $\forall S'\subset S\setminus\{i,j\}, V(S'\cup\{i\}) = V(S'\cup\{j\})$
-
-      \alert{then} $\phi_i(S,V) = \phi_j(S,V)$
-\pause
-    \item[Fairness:] $j$'s contribution to $i$ equals $i$'s contribution to $j$
-      $$\phi_i(S,V) - \phi_i(S\setminus\{j\}) = \phi_j(S,V)- \phi_j(S\setminus\{i\})$$
-  \end{description}
-\end{frame}
-
-\begin{frame}{Solution: the Shapley Value}
-
-  \begin{theorem}[Shapley, 1953]
-    There is only one function satisfying Efficiency, Symmetry and
-    Fairness:
-    $$\phi_i(S,V) = \frac{1}{\vert S\vert!}\sum_{\pi\in\mathfrak{S}(S)}\Delta_i\big(V,S(i,\pi)\big)$$
-  \end{theorem}
-  \begin{itemize}
-  \item $\mathfrak{S}(S)$: set of permutations of $S$
-  \item $S(i,\pi)$: set of players before $i$ in $\pi$
-  \item $\Delta_i\big(V,T\big) =
-    V\big(T\big)-V\big(T\setminus\{i\})$: marginal contribution of $i$
-    to $T$
-  \end{itemize}
-\end{frame}
-
-\begin{frame}{Properties}
-  \begin{itemize}
-  \item
-    \alert{if} $V$ is superadditive: $V(S\cup T) \geq V(S) + V(T)$
-\pause
-
-    \alert{then} the Shapley Value is individually rational:
-    $\phi_i(S,V) > V(\{i\})$ 
-\pause
-  \item 
-    \alert{if} $V$ is supermodular: $\Delta_i(V,S)\leq\Delta_i(V,T),
-    \forall S\subset T$
-\pause
-
-    \alert{then} the Shapley Value lies in the core of
-    the game:
-    $$\sum_{i\in S'}\phi_i(S,V) \geq V(S'), \forall S'\subset S$$
-\pause
-  \item Hervé Moulin \& Scott Shenker (1999): Group-strategyproof auction
-  \end{itemize}
-\end{frame}
-
-\section{Example scenario}
-
-\begin{frame}{Setup}
-  \begin{itemize}
-  \item  database $X$: $n\times p$ matrix
- 
-  \item $x_i\in \mathbf{R}^p$ $i$-th row of $X$: data of user $i$
-
-  \item for each user: $y_i$ variable of interest, could be:
-    \begin{itemize}
-    \item already observed
-    \item about to be observed 
-    \end{itemize}
-  \end{itemize}
-\end{frame}
-
-\begin{frame}{Recommender system}
-\alert{Goal:} predict the variable of interest $y$ of a new user $x$
-
-\alert{Linear model:} relation between data and variable of interest:
-$$y = \langle\beta,x\rangle + \varepsilon, \quad \varepsilon\simeq\mathcal{N}(0,\sigma^2)$$
-
-\alert{Linear regression:} estimator $\tilde{\beta} =
-(\tr{X}X)^{-1}\tr{X}Y$
-
-\alert{Prediction:} $\tilde{y} = \langle\tilde{\beta},x\rangle$
-\end{frame}
-
-\begin{frame}{Revenue structure}
-\alert{Prediction error:} $\mse(X,x) = \expt{(\tilde{y} - y)^2} =
-\sigma^2\tr{x}(\tr{X}X)^{-1}x + \sigma^2$  
-\vspace{1cm}
-\begin{block}{Property}
-  \emph{The prediction error decreases as the size of the database increases}
-\end{block}
-\vspace{1cm}
-\alert{$\Rightarrow$} choose a decreasing function of the prediction
-error
-\pause
-
-\alert{Problem:} $\mse$ not submodular
-\end{frame}
-
-\begin{frame}{Revenue structure (2)}
-Exponentially decreasing revenue:
-$$V(X,x) = \exp\big(-\mse(X,x)\big)$$
-
-Value of database $X$:
-$$V(X) = \int_{x\in\mathbb{R}^d} V(X,x) dx$$
-\end{frame}
-
-\begin{frame}{Properties}
-\begin{block}{Closed-form expression}
-$$V(X) = C\sqrt{\det(\tr{X}X)}$$ 
-\end{block}
-
-\vspace{1cm}
-\pause
-\begin{block}{Supermodularity}
-$$X\mapsto\log\big(V(X)\big)$$ is supermodular
-\end{block}
-\pause
-\alert{$\Rightarrow$} we can apply the Shapley value theory!
-\end{frame}
-
-\begin{frame}{Conclusion}
-
-\begin{itemize}
-\item general framework to study the pricing problem
-\item the case of the linear regression
-\end{itemize}
-\vspace{1cm}
-\pause
-Future directions
-\begin{itemize}
-\item different revenue models
-\item fluid Shapley Value (Aumann-Shapley) for a continuum of players
-\end{itemize}
-\end{frame}
-\end{document}
\ No newline at end of file
-- 
cgit v1.2.3-70-g09d2

