The logistic regression model is one of the most popular data generation model in noisy binary classification problems. In this work, we study the sample complexity of estimating the parameters of the logistic regression model up to a given $\ell_2$ error, in terms of the dimension and the inverse temperature, with standard normal covariates. The inverse temperature controls the signal-to-noise ratio of the data generation process. While both generalization bounds and asymptotic performance of the maximum-likelihood estimator for logistic regression are well-studied, the non-asymptotic sample complexity that shows the dependence on error and the inverse temperature for parameter estimation is absent from previous analyses. We show that the sample complexity curve has two change-points in terms of the inverse temperature, clearly separating the low, moderate, and high temperature regimes.

翻译：Logistic回归模型是含噪二分类问题中最流行的数据生成模型之一。本文研究在标准正态协变量条件下，Logistic回归模型参数在给定$\ell_2$误差范围内的样本复杂度问题，该复杂度与维度和逆温度相关。逆温度控制数据生成过程的信噪比。尽管Logistic回归最大似然估计的泛化界与渐近性能已有深入研究，但现有分析缺乏关于参数估计误差与逆温度依赖关系的非渐近样本复杂度结果。我们证明样本复杂度曲线关于逆温度存在两个变点，清晰划分出低温、中温与高温三个区域。