Optimal Poisson Subsampling for Softmax Regression

Softmax regression, which is also called multinomial logistic regression, is widely used in various fields for modeling the relationship between covariates and categorical responses with multiple levels. The increasing volumes of data bring new challenges for parameter estimation in softmax regression, and the optimal subsampling method is an effective way to solve them. However, optimal subsampling with replacement requires to access all the sampling probabilities simultaneously to draw a subsample, and the resultant subsample could contain duplicate observations. In this paper, the authors consider Poisson subsampling for its higher estimation accuracy and applicability in the scenario that the data exceed the memory limit. The authors derive the asymptotic properties of the general Poisson subsampling estimator and obtain optimal subsampling probabilities by minimizing the asymptotic variance-covariance matrix under both A- and L- optimality criteria. The optimal subsampling probabilities contain unknown quantities from the full dataset, so the authors suggest an approximately optimal Poisson subsampling algorithm which contains two sampling steps, with the first step as a pilot phase. The authors demonstrate the performance of our optimal Poisson subsampling algorithm through numerical simulations and real data examples.