论文标题
关于多级逻辑回归的梯度下降下的最大似然估计和收敛率
On the existence of the maximum likelihood estimate and convergence rate under gradient descent for multi-class logistic regression
论文作者
论文摘要
我们重新审视了多级逻辑回归的最大似然估计的问题。我们表明，确保其存在的一种方法是通过为示例数据集中的每个类分配积极概率。不需要数据可分离性的概念，这与每个数据样本属于一个类的多类逻辑回归的经典设置相反。当将梯度下降用作优化器时，我们还将收敛速率的一般且建设性的估计值提供给最大似然估计。我们的估计涉及最大似然函数的Hessian的条件数。本文中使用的方法取决于简单的操作者理论框架。
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.
