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.

翻译：本文重新探讨了多类逻辑回归中最大似然估计的存在性问题。我们证明，确保其存在性的一种方法是为样本数据集中的每个类别赋予正概率。这与经典多类逻辑回归设置（其中每个数据样本仅属于一个类别）不同，无需引入数据可分性概念。此外，我们提供了当使用梯度下降作为优化器时，最大似然估计收敛速率的通用且具有构造性的估计。该估计涉及对最大似然函数海森矩阵条件数进行界定。本文所采用的方法基于一个简单的算子理论框架。