In statistics and machine learning, logistic regression is a widely-used
supervised learning technique primarily employed for binary classification
tasks. When the number of observations greatly exceeds the number of predictor
variables, we present a simple, randomized sampling-based algorithm for
logistic regression problem that guarantees high-quality approximations to both
the estimated probabilities and the overall discrepancy of the model. Our
analysis builds upon two simple structural conditions that boil down to
randomized matrix multiplication, a fundamental and well-understood primitive
of randomized numerical linear algebra. We analyze the properties of estimated
probabilities of logistic regression when leverage scores are used to sample
observations, and prove that accurate approximations can be achieved with a
sample whose size is much smaller than the total number of observations. To
further validate our theoretical findings, we conduct comprehensive empirical
evaluations. Overall, our work sheds light on the potential of using randomized
sampling approaches to efficiently approximate the estimated probabilities in
logistic regression, offering a practical and computationally efficient
solution for large-scale datasets.

在统计学和机器学习领域，逻辑回归是一种广泛应用于二分类任务的监督学习技术。本研究提出一种基于随机抽样的简单算法，针对逻辑回归问题，确保对模型的预测概率和整体差异都能得到高质量的近似。研究分析了当采用杠杆得分对观测数据进行抽样时，逻辑回归的预测概率的属性，并证明可以通过样本规模远小于总观测数据量来实现准确的近似。通过全面的实证评估验证了我们的理论发现，研究为在大规模数据集上高效近似逻辑回归的预测概率提供了实用和计算高效的解决方案。