  Logistic regression is one of the most popular methods in binary
classification, wherein estimation of model parameters is carried out by
solving the maximum likelihood (ML) optimization problem, and the ML estimator
is defined to be the optimal solution of this problem. It is well known that
the ML estimator exists when the data is non-separable, but fails to exist when
the data is separable. First-order methods are the algorithms of choice for
solving large-scale instances of the logistic regression problem. In this
paper, we introduce a pair of condition numbers that measure the degree of
non-separability or separability of a given dataset in the setting of binary
classification, and we study how these condition numbers relate to and inform
the properties and the convergence guarantees of first-order methods. When the
training data is non-separable, we show that the degree of non-separability
naturally enters the analysis and informs the properties and convergence
guarantees of two standard first-order methods: steepest descent (for any given
norm) and stochastic gradient descent. Expanding on the work of Bach, we also
show how the degree of non-separability enters into the analysis of linear
convergence of steepest descent (without needing strong convexity), as well as
the adaptive convergence of stochastic gradient descent. When the training data
is separable, first-order methods rather curiously have good empirical success,
which is not well understood in theory. In the case of separable data, we
demonstrate how the degree of separability enters into the analysis of $\ell_2$
steepest descent and stochastic gradient descent for delivering
approximate-maximum-margin solutions with associated computational guarantees
as well. This suggests that first-order methods can lead to statistically
meaningful solutions in the separable case, even though the ML solution does
not exist.
