  We consider the problem of estimating the graph structure associated with a
discrete Markov random field. We describe a method based on
$\ell_1$-regularized logistic regression, in which the neighborhood of any
given node is estimated by performing logistic regression subject to an
$\ell_1$-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
results provide 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 assumptions on the population Fisher
information matrix, we prove that consistent neighborhood selection can be
obtained for sample sizes $n = \Omega(d^3 \log p)$, with the error decaying as
$\order(\exp(-C n/d^3))$ for some constant $C$. If these same assumptions are
imposed directly on the sample matrices, we show that $n = \Omega(d^2 \log p)$
samples are sufficient.
