CAREER: Robust and Efficient Algorithms for Statistical Estimation and Inference

统计和机器学习方法对于做出数据驱动的决策很有用。在存在由测量误差或数据收集过程本身固有的随机性引起的不确定性的情况下，这些方法特别有利。首席调查员(PI)将进行一项研究计划，以开发具有两个重要特征的统计和机器学习方法，即稳健性和效率。健壮算法的特点是，即使某些数据不是准确和干净的，而是“离群值”，例如完全不相关的或严重损坏的测量，它们也能够很好地执行。强大的技术可以帮助减少在人工监督的数据清理和预处理上花费的资源量。另一方面，有效的方法能够提取数据中包含的大多数有用信息，因此减少了数据指导决策中的不确定性。这项研究计划将与教育活动相结合，除其他外，将使本科生和研究生接触到统计学和机器学习方面的尖端方法，并让学生有机会在当地K-12学校担任个人导师和导师。研究计划的一部分致力于调查自我归一化和稳健统计技术之间的联系。特别是，PI将证明，与许多现有的方法不同，基于自归一化和的算法通常会产生有效的方法，例如在单变量和多变量均值估计的背景下。这类算法的分析与自归一化过程理论密切相关。研究的另一部分集中于增长阶U-统计量的渐近性质，以及这些性质对稳健和有效的经验风险最小化(ERM)的影响，ERM是现代数理统计和机器学习算法的关键原则之一。PI将引入一种新的稳健ERM方法，并将所产生算法的效率问题与U-统计理论中的纯数学问题联系起来。最后，这项研究将使用贝叶斯方法解决稳健统计中的不确定量化问题。具体地说，PI旨在开发基于增长阶U-统计量的标准后验分布的新的稳健模拟，并将调查这些稳健后验的渐近行为以及相应可信集合的渐近频率属性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Statistical and machine learning methods are useful for making data-driven decisions. These methods are particularly advantageous in the presence of uncertainty originating from measurement errors or randomness inherent to the data collection process itself. The principal investigator (PI) will pursue a research program to develop statistical and machine learning methods possessing two important characteristics, robustness and efficiency. Robust algorithms are characterized by their ability to perform well even when some of the data are not accurate and clean but instead are "outliers," such as completely irrelevant or grossly corrupted measurements. Robust techniques can help to reduce the amount of resources spent on human-supervised data cleaning and preprocessing. On the other hand, methods that are efficient are able to extract most of the useful information contained in the data, therefore reducing the amount of uncertainty in the data-guided decision. This research program will be integrated with educational activities that, among other things, will expose undergraduate and graduate students to cutting edge approaches in statistics and machine learning, and give students an opportunity to serve as individual tutors and mentors at local K-12 schools.One part of the research program is devoted to investigation of the connections between self-normalized sums and robust statistical techniques. In particular, the PI will demonstrate that, unlike many existing approaches, algorithms based on the self-normalized sums often give rise to efficient methods, for instance in the context of univariate and multivariate mean estimation. Analysis of such algorithms is closely related to the theory self-normalized processes. Another part of the research focuses on the asymptotic properties of U-statistics of growing order, and the implications of these properties for robust and efficient empirical risk minimization (ERM), one of the key principles underlying modern mathematical statistics and machine learning algorithms. The PI will introduce a new approach to robust ERM and will relate questions about efficiency of resulting algorithms to purely mathematical questions in the theory of U-statistics. Finally, the research will address uncertainly quantification in robust statistics using Bayesian methods. Specifically, the PI aims to develop new robust analogues of the standard posterior distribution based on U-statistics of growing order, and will investigate the asymptotic behavior of these robust posteriors as well as the asymptotic frequentist properties of the corresponding credible sets.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.