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diff --git a/finale/mid_report.tex b/finale/mid_report.tex index 9873abd..d22a557 100644 --- a/finale/mid_report.tex +++ b/finale/mid_report.tex @@ -46,30 +46,72 @@ The Network Inference Problem (NIP) is the machine learning challenge of recovering the edges and edge weights of an unknown weighted graph from the observations of a random contagion process propagating over this graph. - While estimators with provable convergence rate guarantees have been - obtained under various formulations of the NIP, a rigorous statistical - treatment of the problem is still lacking. In this work, we build upon the - unified NIP formulation of [] to explore the connections between the - topological properties of the graph to be learnt and the resulting quality - of the estimators. Specifically, we analyze which properties of the graph - render NIP unfeasible or hard, and which properties can be exploited to - improve the quality of the estimators. + While estimators for the edge weights with provable convergence guarantees + have been obtained under various formulations of the NIP, a rigorous + statistical treatment of the problem is still lacking. In this work, we + build upon the unified NIP formulation of \cite{pouget} to explore the + connections between the topological properties of the graph to be learnt + and the resulting quality of the estimators. Specifically, we analyze which + properties of the graph render NIP unfeasible or hard, and which properties + can be exploited to improve the quality of the estimators. \end{abstract} \section{Introduction} -Central question: \emph{relating properties of the graph to the difficulty of -doing graph inference from cascades.} +A random contagion process propagating over a graph provides valuable +information about the presence and weights of its edges. This fact is exploited +in many real-life applications, including learning how people influence each +other by observing how they communicate or share information in social +networks, learning protein activation graphs in biology or learning correlations +between financial assets. -Two subquestions: +Formally, this problem can be formulated as a machine learning task where one +specifies a probabilistic contagion model whose parameters are the edge weights +of the unknown graph. Then, given observations drawn from this random process, +one seeks to construct an estimator of the edge weights. This problem is known +in the literature as the Network Inference Problem + +A growing body of works has focused on obtaining convergence guarantees for +estimators of the edge weights under various formulations of the Network +Inference Problem. In \cite{pouget}, we noted that many of these formulations +could be interpreted as specific instances of a common \emph{Generalized Linear +Cascade Model} (GLMC) and showed how to apply results from the sparse recovery +literature to obtain close-to-optimal convergence guarantees. + +However, in \cite{pouget} as in previous works, the convergence guarantees rely +on many assumptions about the non-degeneracy of the probabilistic model or the +observations. These assumptions are hard to interpret and weaken possible +applications of the obtained results to real-life applications. Furthermore, +previous works focused on obtaining generic convergence guarantees and it is +not obvious that the developed frameworks can be adapted in presence of domain +specific knowledge about the graph to be learned. The goal of the current work +is to address these shortcomings by asking the following question: \emph{how do + properties of the graph to be learned impact the task of solving the + Network Inference Problem for this graph?}. Specifically, we wish to + understand: \begin{itemize} - \item which properties of the graph make it impossible or hard to learn - (identifiability, convergence rate)? - \item which properties of the graph can be exploited to learn better or - faster (use a Bayesian prior)? + \item which properties of the graph render the Network Inference Problem + unfeasible or hard to solve. This relates to the question of + identifiability of the contagion model, and understanding how the + constants which appear on convergence rate bounds for edge weights + estimators relate to topological properties of the graph. + \item how domain specific knowledge about the graph to be learned can be + exploited to make the Network Inference Problem easier to solve. + A natural framework for this is a Bayesian one, where the domain + specific knowledge is encoded as a Bayesian prior. To the best of our + knowledge, this approach has not been explored beyond MAP estimation + under a sparse (Laplace) prior in the context of discrete-time models. \end{itemize} -\subsection*{Related work} +The organization of the current report is as follows: we conclude the +introduction with a review of related works. The Generalized Linear Cascade +(GLC) model from \cite{pouget} is presented in Section 2. Section 3. explores +the question of identifiablity of the model. Section 4. presents both the +frequentist and Bayesian approach for inference of the GLC model as well as +a novel active learning formulation of the problem. The code used in our +experiments can be found in the Appendix. + +\paragraph{Related works.} The study of edge prediction in graphs has been an active field of research for over a decade~\cite{Nowell08, Leskovec07, AdarA05}.~\cite{GomezRodriguez:2010} introduced the {\scshape Netinf} algorithm, which approximates the likelihood of @@ -77,21 +119,27 @@ cascades represented as a continuous process. The algorithm, relying on Maximum-Likelihood and convex relaxations was improved in later work~\cite{gomezbalduzzi:2011}, but is not known to have any theoretical guarantees beside empirical validation on synthetic networks. -\cite{Netrapalli:2012} studied the discrete-time version of the independent -cascade model and obtained the first ${\cal O}(s^2 \log m)$ recovery guarantee -on general networks, by using unregularized MLE and making a {\it correlation -decay\/} assumption, which limits the number of new infections at every step. -They also suggest a {\scshape Greedy} algorithm, with provably good performance -in tree graphs, which maintains a counter of each node's antecedents, i.e. the -set of nodes infected at a prior time step to its infection time. The work -of~\cite{Abrahao:13} studies the same continuous-model framework as -\cite{GomezRodriguez:2010} and obtains an ${\cal O}(s^9 \log^2 s \log m)$ -support recovery algorithm, without the \emph{correlation decay} assumption. -\cite{du2013uncover,Daneshmand:2014} propose Lasso regularization of MLE for -recovering the weights of the graph under a continuous-time independent cascade -model. The work of~\cite{du2014influence} is slightly orthogonal to ours since -they suggest learning the \emph{influence} function, rather than the parameters -of the network directly. + +Let us denote by $m$ the number of nodes in the graph and for a given node, we +denote by $s$ its in-degree. In what follows, we call a \emph{recovery +guarantee} an upper bound on the number of observations required (expressed as +a function of $s$ and $m$) to learn the incoming edges of a node to a fixed +arbitrary accuracy level. \cite{Netrapalli:2012} studied the discrete-time +version of the independent cascade model and obtained the first ${\cal O}(s^2 +\log m)$ recovery guarantee on general networks, by using unregularized MLE and +making a {\it correlation decay\/} assumption, which limits the number of new +infections at every step. They also suggest a {\scshape Greedy} algorithm, +with provably good performance in tree graphs, which maintains a counter of +each node's antecedents, i.e. the set of nodes infected at a prior time step to +its infection time. The work of~\cite{Abrahao:13} studies the same +continuous-model framework as \cite{GomezRodriguez:2010} and obtains an ${\cal +O}(s^9 \log^2 s \log m)$ support recovery algorithm, without the +\emph{correlation decay} assumption. \cite{du2013uncover,Daneshmand:2014} +propose Lasso regularization of MLE for recovering the weights of the graph +under a continuous-time independent cascade model. The work +of~\cite{du2014influence} is slightly orthogonal to ours since they suggest +learning the \emph{influence} function, rather than the parameters of the +network directly. More recently, the continuous process studied in previous work has been reformulated as a Hawkes process, with recent papers @@ -385,6 +433,6 @@ given $(x_t)_{t\geq1}$}$ \Comment{Update posterior on $\theta$} \end{algorithm} \bibliography{sparse}{} -\bibliographystyle{alpha} +\bibliographystyle{abbrv} \end{document} |