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diff --git a/presentation/WWW15/beamer_3.tex b/presentation/WWW15/beamer_3.tex new file mode 100644 index 0000000..0243b88 --- /dev/null +++ b/presentation/WWW15/beamer_3.tex @@ -0,0 +1,603 @@ +\documentclass[10pt]{beamer} + +\usepackage{amssymb, amsmath, graphicx, amsfonts, color, amsthm, wasysym, framed} + +\newtheorem{proposition}{Proposition} + +\title{Learning from Diffusion processes} +\subtitle{What cascades really teach us about networks} +\author{Jean Pouget-Abadie, Thibaut Horel} +\institute[]{\includegraphics[scale=.35]{figures/SEASLogo_RGB.png}} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Introduction} + +\begin{itemize} +\item If we observe who catches the flu and when over several years, can we guess {\bf who is friends with whom?} +\item If we observe behaviors spreading on Facebook (\emph{Ice bucket challenge, Je suis Charlie}), can we know {\bf who is most likely to influence you?} +\item If we observe memes/ideas on internet, can we tell {\bf which blogs/commmunities take their information from which source?} +\end{itemize} + +\end{frame} + +%% Intuition says yes --> this problem can be formalized and is known as the + +%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Introduction} + +\begin{definition}{{\bf Network Inference Problem} {\small \cite{GomezRodriguez:2010}} } + + +If \emph{only} the diffusion process is observed, but the network is \emph{unknown}: +\begin{itemize} +\item Can we learn the network? +\item For which types of diffusion process? +\item After how many observations? +\end{itemize} +\end{definition} + +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Prior work} +\begin{itemize} +\item {\small \cite{GomezRodriguez:2010, gomezbalduzzi:2011, Abrahao:13, Daneshmand:2014}} : Continuous-time Independent Cascade-like diffusion model +\item {\small \cite{Netrapalli:2012}}: Discrete-time Independent Cascade Model (formalized in {\small \cite{Kempe:03}}) with correlation decay +\item {\small \cite{?}} +\end{itemize} +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Discrete-time Independent Cascade Model} + +\begin{figure} +\includegraphics[scale=.3]{figures/ic_illustration.pdf} +\caption{Weighted, directed graph} +\end{figure} + +\begin{itemize} + +\item At $t=0$, each node is {\color{blue} infected} with probability $p_{\text{init}}$ and {\bf susceptible} with probability $1-p_{\text{init}}$ + +\item Each {\color{blue} infected} node $i$ has a ``one-shot'' probability $p_{i,j}$ of infecting each of its susceptible neighbors $j$ at $t+1$. + +\item A node stays {\color{blue} infected} for one round, then it {\color{red} dies} + +\item Process continues until random time $T$ when no more {\bf susceptible} nodes are left +\end{itemize} + +%Notes: Revisit the celebrated independent cascade model -> Influence maximisation is tractable, requires knowledge of weights +\end{frame} + +% %%%%%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Discrete-time Independent Cascade Model} + +% \begin{figure} +% \includegraphics[scale=.5]{figures/weighted_graph.png} +% \caption{Weighted, directed graph} +% \end{figure} + +% \begin{block}{Example} +% \begin{itemize} +% \item At $t=0$, the {\color{orange} orange} node is infected, and the two other nodes are susceptible. $X_0 = $\{{\color{orange} orange}\} +% \item At $t=1$, the {\color{orange}} node infects the {\color{blue} blue} node and fails to infect the {\color{green} green} node. The {\color{orange} orange} node dies. $X_1 = $\{{\color{blue} blue}\} +% \item At $t=2$, {\color{blue} blue} dies. $X_2 = \emptyset$ +% \end{itemize} +% \end{block} + +% \end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Discrete-time Independent Cascade Model} + +\begin{figure} +\includegraphics[scale=.5]{figures/weighted_graph.png} +\caption{Weighted, directed graph} +\end{figure} + +\begin{itemize} +\item If the {\color{orange} orange} node and the {\color{green} green} node are infected at $t=0$, what is the probability that the {\color{blue} blue} node is infected at $t=1$? +$$1 - \mathbb{P}(\text{not infected}) = 1 - (1 - .72)(1-.04)$$ +\end{itemize} + + +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Independent Cascade Model} +% \begin{figure} +% \includegraphics[scale=.5]{figures/weighted_graph.png} +% \caption{Weighted, directed graph} +% \end{figure} + +% \begin{itemize} +% \item In general, for each susceptible node $j$: +% $$\mathbb{P}(j \text{ becomes infected at t+1}|X_{t}) = 1 - \prod_{i \in {\cal N}(j) \cap X_{t}} (1 - p_{i,j})$$ +% \end{itemize} + +% \end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Independent Cascade Model} + +\begin{itemize} +\item $X^t$: set of {\color{red} infected} nodes + +\item Probability that node $j$ gets infected at $t+1$: +\begin{framed} + \begin{align*} + \tag{IC} + \mathbb{P}\big[X^{t+1}_j = 1\,|\, X^{t}\big] + & = 1 - \prod_{i = 1}^m {(1 - p_{i,j})}^{X^t_i} \\ + & = 1 - \prod_{i = 1}^m e^{\Theta_{i,j}X^t_i} \\ + & = 1 - e^{\Theta_j \cdot X^t} + \end{align*} + \end{framed} +% \item $f: z \mapsto 1 - e^z$ +% \item $(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$ +\item $\Theta_{i,j} \equiv \log(1-p_{i,j})$ +\end{itemize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Voter Model} +\begin{figure} +\includegraphics[scale=.3]{figures/vt_illustration.pdf} +\end{figure} +\begin{itemize} +\item Probability that node $j$ is infected at $t+1$: +\begin{equation*} + \tag{VT} + \mathbb{P}\big[X^{t+1}_j = 1\,|\, X^{t}\big] + = \sum_{i \in X^t} p_{i,j} = p_j \cdot X^t +\end{equation*} +\end{itemize} + +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Generalized Linear Cascades} + +{\bf Generalized Linear Cascade Model} +\begin{itemize} + \item $f: \mathbb{R} \rightarrow [0,1]$: inverse link function + \item Probability depends on $f$-transform of {\bf scalar product}: + \begin{framed} + $$\mathbb{P}(X^{t+1}_j = 1 | X^t) = f(\Theta_j \cdot X^t)$$ + \end{framed} + \item {\bf Decomposable} node by node conditionally on infected nodes + \item Examples: + \begin{itemize} + \item IC model : $f: z \mapsto 1- e^z$ + \item VT model : $f: z \mapsto z$ + \item Discretized version of continuous diffusion model $f: z \mapsto 1-e^{-z}$ + \end{itemize} +\end{itemize} + +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Setup} +\begin{figure} +\centering +\includegraphics[scale=.6]{../images/drawing.pdf} +\caption{$\mathbb{P}(j \in X_{t+1}| X_t) = f(X_t\cdot \theta)$} +\end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} +\frametitle{Sparse Recovery} +\begin{itemize} +\item Solving for $Ax = b$ when $A$ is non-degenerate is possible if: +\begin{itemize} +\item A is {\bf almost invertible} +\item x is {\bf sparse} +\end{itemize} +\item If $x$ is solution to $\min L(x)$ where $L$ is convex, then {\small \cite{Negahban:2009}} solving for: +\begin{framed} +\begin{equation*} +\min_x L(x) + \lambda \| x \| +\end{equation*} +\end{framed} +gives finite-sample guarantees under certain assumptions. +\end{itemize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Result} +{\bf Assumptions} +\begin{itemize} +\item $f$ and $1-f$ are $(1)$ log-concave and have $(2)$ log-gradient bounded by $\frac{1}{\alpha}$ +\item $\nabla^2{\cal L}$ verifies the $(S, \gamma)-({\bf RE})$ condition +\end{itemize} +\vspace{1cm} +{\bf Algorithm} +\begin{itemize} +\item Solve $MLE$ program with $\lambda = 2 \sqrt{\frac{\log m}{\alpha n}}$: +\begin{framed} +\begin{equation*} + \hat \theta_i \in \arg \max_{\theta} {\cal L}_i(\theta_i | x^1, + \dots x^n) - \lambda \|\hat \theta_i\|_1 +\end{equation*} +\end{framed} +\end{itemize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Theorem} +If previous assumptions are met, with high probability, + \begin{framed} + \begin{equation*} + \|\hat \theta - \theta^*\|_2 \leq \frac{6}{\gamma} \sqrt{\frac{s \log + m}{\alpha n}} + \end{equation*} + \end{framed} + where + \begin{itemize} + \item $s$ is degree of node, + \item $m$ is number of nodes, + \item $n$ is the number of observations, + \item $\alpha$ is the gradient bound, + \item $\gamma$ is the $({\bf RE})$-constant + \end{itemize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Restricted Eigenvalue Condition} +\begin{block}{Almost sparse vectors} +\begin{itemize} +\item ${\cal C} := \{ X : \|X\|_2 = 1, \|X_{\bar S}\|_1 \leq 3 \| X_S\|_1 \}$ +\end{itemize} +\end{block} +\begin{definition} +$A$ verifies the $(S, \gamma)$-({\bf RE}) condition \cite{bickel2009simultaneous} if: +$$\forall X \in {\cal C}, X^T A X \geq \gamma$$ +\end{definition} +\begin{block}{Properties} +\begin{itemize} +\item If $\mathbb{E}[A]$ verifies the $(S, \gamma)$-{(\bf RE)} condition, + then $A$ verifies the $(S, \gamma/2)$-{(\bf RE)} + condition~{\small \cite{vandegeer:2009}} +\end{itemize} +\end{block} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Restricted Eigenvalue Condition} +\begin{itemize} +\item If $f$ and $1-f$ are $c$-strictly log-concave, then if the {\bf Gram matrix} verifies the $\gamma$-({\bf RE})-condition, then the Hessian verifies the $c\gamma$-({\bf RE})-condition. +\end{itemize} + +\begin{align*} +\mathbb{E}[X^T X] = \left( \begin{array}{ccc} a_1 & b_{1, 2} & b_{1, m} \\ +\dots & \dots & \dots \\ +b_{1, m} & \dots & a_m \end{array}\right) +\end{align*} +where +\begin{itemize} +\item $a_i \equiv$ ``average time node $i$ is infected'' +\item $b_{i,j} \equiv$ ``average time node $i$ and node $j$ are infected \emph{together}'' +\end{itemize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} +\frametitle{Conclusion} +\begin{itemize} +\item Introduced Generalized Linear Cascades +\item Better finite sample guarantees +\item Interpretable conditions +\item Lower bound+approx. sparse case developped in full paper~\cite{Pouget:2015} +\end{itemize} +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Independent Cascade Model} +% For each susceptible node $j$, the event that it becomes {\color{blue} infected} conditioned on previous time step is a Bernoulli variable: +% $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$ +% \begin{itemize} +% \item $\theta_{i,j} := \log(1 - p_{i,j})$ +% \item $\theta_j := (0, 0, 0, \theta_{4,j}, 0 \dots, \theta_{k,j}, \dots)$ +% \item $f : x \mapsto 1 - e^x$ +% \begin{align*} +% \mathbb{P}(j\in X_{t+1}|X_{t}) & = 1 - \prod_{i \in {\cal N}(j) \cap X_{t}} (1 - p_{i,j}) \\ +% & = 1 - \exp \left[ \sum_{i \in {\cal N}(j) \cap X_{t}} \log(1 - p_{i,j}) \right] \\ +% & = 1 - \exp \left[ X_{t} \cdot \theta_{j}\right] +% \end{align*} +% \end{itemize} +% \end{frame} + + + +%%%%%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Independent Cascade Model} +% For each susceptible node $j$, the event that it becomes {\color{blue} infected} conditioned on previous time step is a Bernoulli variable: +% $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$ + +% \begin{block}{Decomposability} +% \begin{itemize} +% \item Conditioned on $X_t$, the state of the nodes at the next time step are mutually independent +% \item We can learn the parents of each node independently +% \end{itemize} +% \end{block} + +% \begin{block}{Sparsity} +% \begin{itemize} +% \item $\theta_{i,j} = 0 \Leftrightarrow \log(1 - p_{i,j}) = 0 \Leftrightarrow p_{i,j} = 0$ +% \item If graph is ``sparse'', then $p_{j}$ is sparse, then $\theta_j$ is sparse. +% \end{itemize} +% \end{block} +% \end{frame} + + + +%%%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Learning from Diffusion Processes} +% \begin{block}{Problem Statement} +% \begin{itemize} +% \item We are given a graph ${\cal G}$, and a diffusion process $f$ parameterized by $\left((\theta_j)_j, p_{\text{init}}\right)$. +% \item Suppose we {\bf only} observe $(X_t)$ from the diffusion process. +% \item Under what conditions can we learn $\theta_{i,j}$ for all $(i,j)$? +% \end{itemize} +% \end{block} +% \end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Sparse Recovery} +% \begin{figure} +% \centering +% \includegraphics[scale=.6]{../images/drawing.pdf} +% \caption{$\mathbb{P}(j \in X_{t+1}| X_t) = f(X_t\cdot \theta)$} +% \end{figure} +% \end{frame} + + + +%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Learning from Diffusion Processes} + +% % \begin{figure} +% % \includegraphics[scale=.4]{../images/sparse_recovery_illustration.pdf} +% % \caption{Generalized Cascade Model for node $i$} +% % \end{figure} + +% \begin{block}{Likelihood Function} +% \begin{align*} +% {\cal L}(\theta_1, \dots, \theta_m| X_1, \dots X_n) = \sum_{i=1}^m \sum_{t} & X_{t+1}^i \log f(\theta_i \cdot X_t) + \\ +% & (1 - X_{t+1}^i) \log(1 - f(\theta_i \cdot X_t)) +% \end{align*} +% \end{block} + +% \begin{block}{Penalized log-likelihood} +% For each node $i$, $$\theta_i \in \arg \max \frac{1}{n_i}{\cal {L}}_i(\theta_i | X_1, X_2, \dots, X_{n_i}) - \lambda \|\theta_i\|_1$$ +% \end{block} + +% \end{frame} + +%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Conditions} +% \begin{block}{On $f$} +% \begin{itemize} +% \item $\log f$ and $\log (1-f)$ have to be concave +% \item $\log f$ and $\log (1-f)$ have to have bounded gradient +% \end{itemize} +% \end{block} + +% \begin{block}{On $(X_t)$} +% \begin{itemize} +% \item Want ${\cal H}$, the hessian of ${\cal L}$ with respect to $\theta$, to be well-conditioned. +% \item $ n < dim(\theta) \implies {\cal H}$ is degenerate. +% \item {\bf Restricted Eigenvalue condition} = invertible on ``almost sparse'' vectors. +% \end{itemize} +% \end{block} + +% \end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%% + + +% %%%%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Main Result} +% Adapting a result from \cite{Negahban:2009}, we have the following theorem: + +% \begin{theorem} +% For node $i$, assume +% \begin{itemize} +% \item the Hessian verifies the $(S,\gamma)$-RE condition where $S$ is the set of parents of node $i$ (support of $\theta_i$) +% \item $f$ and $1-f$ are log-concave +% \item $|(\log f)'| < \frac{1}{\alpha}$ and $|(\log 1- f)'| < \frac{1}{\alpha}$ +% \end{itemize} then with high probability: +% $$\| \theta^*_i - \hat \theta_i \|_2 \leq \frac{6}{\gamma}\sqrt{\frac{s\log m}{\alpha n}}$$ +% \end{theorem} + +% \begin{corollary} +% By thresholding $\hat \theta_i$, if $n > C' s \log m$, we recover the support of $\theta^*$ and therefore the edges of ${\cal G}$ +% \end{corollary} + +% \end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%% + + + +%%%%%%%%%%%%%%%%%%%%%%% + + + + + +%%%%%%%%%%%%%%%%%%%%%% + +% \begin{frame} +% \frametitle{Voter Model} +% \begin{itemize} + +% \item {\color{red} Red} and {\color{blue} Blue} nodes. At every step, each node $i$ chooses one of its neighbors $j$ with probability $p_{j,i}$ and adopts that color at $t+1$ + +% \item If {\color{blue} Blue} is `contagious' state: + +% \begin{equation} +% \nonumber +% \mathbb{P}(i \in X^{t+1}|X^t) = \sum_{j \in {\cal N}(i)\cap X^t} p_{ji} = X^t \cdot \theta_i +% \end{equation} +% \end{itemize} +% \end{frame} + +%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Future Work} +\begin{itemize} +\item Lower bound restricted eigenvalues of expected gram matrix +\item Confidence Intervals +\item Show that $n > C' s \log m$ measurements are necessary w.r.t. expected hessian. +\item Linear Threshold model $\rightarrow$ 1-bit compressed sensing formulation +\item Better lower bounds +\item Active Learning +\end{itemize} +\end{frame} + + +%%%%%%%%%%%%%%%%% +{\scriptsize +\bibliography{../../paper/sparse} +\bibliographystyle{apalike} +} +%%%%%%%%%%%%%%%%%%% + + +\begin{frame} +\frametitle{Analysis} + +\begin{block}{Guarantees} +\begin{itemize} +\item Positive result despite correlated measurements \smiley +\item Several measurements per cascade +\item Good finite-sample guarantee +\end{itemize} +\end{block} + +\begin{block}{Assumptions} +\begin{itemize} +\item The Hessian must verify the $(S,\gamma)$-RE condition \frownie +\item Can we make a conditional statement on $\Theta$ and not $X_t$? +\end{itemize} +\end{block} + +\end{frame} + + + +%%%%%%%%%%%%%%%%%%%% + + + + + +\begin{frame} +\frametitle{Restricted Eigenvalue Condition} + +\begin{block}{From Hessian to Gram Matrix} +\begin{itemize} +\item If $f$ and $1 -f$ are strictly log-concave with constant $c$, then if $(S, \gamma)$-RE holds for the gram matrix $\frac{1}{n}X X^T$ , then $(S, c \gamma)$-RE holds for ${\cal H}$ +\end{itemize} +\end{block} + +\begin{block}{From Gram Matrix to Expected Gram Matrix} +\begin{itemize} +\item If $n > C' s^2 \log m$ and $(S, \gamma)$-RE holds for $\mathbb{E}(\frac{1}{n}XX^T)$, then $(S, \gamma/2)$-RE holds for $\frac{1}{n}XX^T$ w.h.p +\item $\mathbb{E}(\frac{1}{n}XX^T)$ only depends on $p_{\text{init}}$ and $(\theta_j)_j$ +\end{itemize} +\end{block} + +\begin{block}{Expected Gram Matrix} +\begin{itemize} +\item Diagonal : average number of times node is infected +\item Outer-diagonal : average number of times pair of nodes is infected {\emph together} +\end{itemize} +\end{block} + +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Approximate Sparsity} +\begin{itemize} +\item $\theta^*_{\lceil s \rceil}$ best s-sparse approximation to $\theta^*$ +\item $\|\theta^* - \theta^*_{\lceil s \rceil} \|_1$: `tail' of $\theta^*$ +\end{itemize} +\begin{theorem} +Under similar conditions on $f$ and a relaxed RE condition, there $\exists C_1, C_2>0$ such that with high probability: +\begin{equation} +\|\hat \theta_i - \theta^*_i\|_2 \leq C_1 \sqrt{\frac{s\log m}{n}} + C_2 \sqrt[4]{\frac{s\log m}{n}}\|\theta^* - \theta^*_{\lceil s \rceil} \|_1 +\end{equation} +\end{theorem} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{Lower Bound} +\begin{itemize} +\item Under correlation decay assumption for the IC model, ${\Omega}(s \log N/s)$ cascades necessary for graph reconstruction (Netrapalli et Sanghavi SIGMETRICS'12) +\item Adapting (Price \& Woodruff STOC'12), in the approximately sparse case, any algorithm for any generalized linear cascade model such that: +$$\|\hat \theta - \theta^*\|_2 \leq C \|\theta^* - \theta^*_{\lfloor s \rfloor}\|_2$$ +requires ${\cal O}(s \log (n/s)/\log C)$ measurement. +\end{itemize} +\end{frame} + +\end{document}
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