Logistic regression with peergroup effects via inference in higherorder Ising models
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Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:36533663, 2020.
Abstract
Spin glass models, such as the SherringtonKirkpatrick, Hopfield and Ising models, are all wellstudied members of the exponential family of discrete distributions, and have been influential in a number of application domains where they are used to model correlation phenomena on networks. Conventionally these models have quadratic sufficient statistics and consequently capture correlations arising from pairwise interactions. In this work we study extensions of these models to models with higherorder sufficient statistics, modeling behavior on a social network with peergroup effects. In particular, we model binary outcomes on a network as a higherorder spin glass, where the behavior of an individual depends on a linear function of their own vector of covariates and some polynomial function of the behavior of others, capturing peergroup effects. Using a {\em single}, highdimensional sample from such model our goal is to recover the coefficients of the linear function as well as the strength of the peergroup effects. The heart of our result is a novel approach for showing strong concavity of the log pseudolikelihood of the model, implying statistical error rate of $\sqrt{d/n}$ for the Maximum PseudoLikelihood Estimator (MPLE), where $d$ is the dimensionality of the covariate vectors and $n$ is the size of the network (number of nodes). Our model generalizes vanilla logistic regression as well as the models studied in recent works of \cite{chatterjee2007estimation,ghosal2018joint,DDP19}, and our results extend these results to accommodate higherorder interactions.
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