  We revisit the problem of the existence of the maximum likelihood estimate
for multi-class logistic regression. We show that one method of ensuring its
existence is by assigning positive probability to every class in the sample
dataset. The notion of data separability is not needed, which is in contrast to
the classical set up of multi-class logistic regression in which each data
sample belongs to one class. We also provide a general and constructive
estimate of the convergence rate to the maximum likelihood estimate when
gradient descent is used as the optimizer. Our estimate involves bounding the
condition number of the Hessian of the maximum likelihood function. The
approaches used in this article rely on a simple operator-theoretic framework.
