High Dimensional Logistic Regression Under Network Dependence
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Logistic regression is one of the most fundamental methods for modeling the probability of a binary outcome based on a collection of covariates. However, the classical formulation of logistic regression relies on the independent sampling assumption, which is often violated when the outcomes interact through an underlying network structure. This necessitates the development of models that can simultaneously handle both the network peer-effect (arising from neighborhood interactions) and the effect of high-dimensional covariates. In this paper, we develop a framework for incorporating such dependencies in a high-dimensional logistic regression model by introducing a quadratic interaction term, as in the Ising model, designed to capture pairwise interactions from the underlying network. The resulting model can also be viewed as an Ising model, where the node-dependent external fields linearly encode the high-dimensional covariates. We propose a penalized maximum pseudo-likelihood method for estimating the network peer-effect and the effect of the covariates, which, in addition to handling the high-dimensionality of the parameters, conveniently avoids the computational intractability of the maximum likelihood approach. Consequently, our method is computationally efficient and, under various standard regularity conditions, our estimate attains the classical high-dimensional rate of consistency. In particular, our results imply that even under network dependence it is possible to consistently estimate the model parameters at the same rate as in classical logistic regression, when the true parameter is sparse and the underlying network is not too dense. As a consequence of the general results, we derive the rates of consistency of our estimator for various natural graph ensembles, such as bounded degree graphs, sparse Erdős-Rényi random graphs, and stochastic block models.
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