High-Dimensional Graphical Model Selection Using $\ell_1$-Regularized Logistic Regression

We focus on the problem of estimating the graph structure associated with a discrete Markov random field. We describe a method based on `1regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an `1-constraint. Our framework applies to the high-dimensional setting, in which both the number of nodes p and maximum neighborhood sizes d are allowed to grow as a function of the number of observations n. Our main result is to establish sufficient conditions on the triple (n, p, d) for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. Under certain mutual incoherence conditions analogous to those imposed in previous work on linear regression, we prove that consistent neighborhood selection can be obtained as long as the number of observations n grows more quickly than 6d log d + 2d log p, thereby establishing that logarithmic growth in the number of samples n relative to graph size p is sufficient to achieve neighborhood consistency.