  Logistic regression is commonly used for modeling dichotomous outcomes. In
the classical setting, where the number of observations is much larger than the
number of parameters, properties of the maximum likelihood estimator in
logistic regression are well understood. Recently, Sur and Candes have studied
logistic regression in the high-dimensional regime, where the number of
observations and parameters are comparable, and show, among other things, that
the maximum likelihood estimator is biased. In the high-dimensional regime the
underlying parameter vector is often structured (sparse, block-sparse,
finite-alphabet, etc.) and so in this paper we study regularized logistic
regression (RLR), where a convex regularizer that encourages the desired
structure is added to the negative of the log-likelihood function. An advantage
of RLR is that it allows parameter recovery even for instances where the
(unconstrained) maximum likelihood estimate does not exist. We provide a
precise analysis of the performance of RLR via the solution of a system of six
nonlinear equations, through which any performance metric of interest (mean,
mean-squared error, probability of support recovery, etc.) can be explicitly
computed. Our results generalize those of Sur and Candes and we provide a
detailed study for the cases of $\ell_2^2$-RLR and sparse
($\ell_1$-regularized) logistic regression. In both cases, we obtain explicit
expressions for various performance metrics and can find the values of the
regularizer parameter that optimizes the desired performance. The theory is
validated by extensive numerical simulations across a range of parameter values
and problem instances.
