U-statistic Reduction with Accurate Risk Control

u型统计是在许多现代统计学习工具中发挥核心作用的一类统计。然而，沉重的计算成本困扰着它们的实际应用。尽管自20世纪70年代以来，对u统计量的计算约简进行了大量的研究，但对约简u统计量的统计推断中的准确风险控制这一重要问题的研究很少。此外，计算速度如何与风险控制准确性相权衡仍未确定。该项目将弥补这一重大差距，提供急需的基础设施技术，使统计人员能够安全地扩展其基于u统计的学习工具。该项目的结果将为广泛的研究领域和应用提供深远的好处，包括非参数统计、机器学习、社会学、计算机视觉和生物医学科学。该项目还将为研究生提供研究培训。本项目将研究经典的“无噪声”u统计和网络u统计，这是有噪声u统计的一个重要子集。该研究将引入创新的理论分析技术，导致计算统计权衡的尖锐特征，并制定基于重采样和次采样的新方法，优于现有方法。该项目旨在建立一个总体框架，对不同流行的u统计量缩减方案进行原则性分析。研究结果将总结为易于实现和调优的实用、分步指南。PI的团队还将为公众用户开发和传播用户友好的软件。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

U-statistics are a class of statistics that play central roles in many modern statistical learning tools. However, the heavy computational cost haunts their practical application. Despite considerable research efforts since the 1970s towards computational reduction for U-statistics, there exists little study on the important problem of accurate risk control in statistical inference for reduced U-statistics. Also, how computational speed trades off with risk control accuracy remains uncharacterized. This project will bridge this significant gap, providing the urgently needed infrastructual techniques that enable statisticians to securely scale up their U-statistic-based learning tools. The results of this project will provide profound benefits to a wide spectrum of research areas and applications, including nonparametric statistics, machine learning, sociology, computer vision and biomedical sciences. The project will also provide research training for graduate students.This project will study both classical "noiseless" U-statistics and network U-statistics, an important subset of noisy U-statistics. The research will introduce innovative theoretical analysis techniques that lead to a sharp characterization of computational-statistical trade-offs and formulate new methods outperforming existing ones based on resampling and subsampling. The project aims to establish a general framework for principled analysis of different popular U-statistic reduction schemes. The research findings will be summarized into practical, step-by-step guides for easy implementation and tuning. The PI's team will also develop and disseminate user-friendly software for public users.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.