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：逻辑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，基于维度和逆温度参数，将逻辑回归模型参数估计至给定$\ell_2$误差所需的样本复杂度。其中逆温度控制数据生成过程的信噪比。尽管逻辑回归最大似然估计的泛化界与渐近性能已有充分研究，但关于参数估计随误差和逆温度变化的非渐近样本复杂度分析仍属空白。我们证明样本复杂度曲线根据逆温度存在两个变点，清晰划分出低温、中温和高温三个区域。