A Non-Asymptotic Theory of Robustness

现代数据采集为收集具有复杂结构的大规模数据提供了便利，同时也给统计和计算方面的数据分析带来了一系列新的挑战。经验数据分布的重尾特征在许多研究领域都得到了反复观察，包括基因组学中的微阵列研究，医学中的神经成像，以及金融中的投资组合优化和风险管理。在功能磁共振研究中，参数统计方法往往不能产生有效的聚类式推断，其主要原因是空间自相关函数不符合假设的高斯形状；在金融学中，收益分布的幂律性质多年来已被验证为风格化的事实。最小二乘法虽然由于其一次性解的简单性而在实践中最常用，但它对样本分布的尾部很敏感，从非渐近的观点来看，对于重尾数据被证明是次优的。在这个项目中，PI将为各种问题开发稳健的统计程序，从均值估计、线性回归、高维稀疏回归到大协方差矩阵估计。这项研究的主要目标是了解稳健学习的有限样本特性，并开发计算高效的程序来促进稳健方法的实际应用。在这个项目中，PI将研究用于两个基本问题的稳健的替代方法：线性回归和协方差估计。为了实现对非对称和重尾数据的稳健性，主要思想是利用自适应Huber损失及其在矩阵空间上的扩展。从非渐近的观点来看，伴随的尺度参数应该与样本大小、维度和噪声水平相适应，以便在稳定性收益和偏差成本之间进行最佳折衷。该项目的工作旨在(I)发展线性模型下的稳健估计和推理的新方法，并使用概率中的集中不等式、统计学中的M估计的有限样本理论和最优化中的凸分析等技术来研究它们的数学基础，以及(Ii)在对数据的最小假设下构造一般的和结构化的大协方差矩阵估计。该项目的独创性在于提供了关于稳健性的新观点，这不仅是对经典稳健统计的有益补充，而且对包括高维估计和大规模推断在内的现代统计分析也作出了重要贡献。该项目的理念呼应了John Tukey的统计实践原则，强调了具有统计分析方法的重要性，这些方法对违反其使用的假设具有健壮性，并允许数据影响分析方法选择的可能性。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Modern data acquisitions have facilitated the collection of large-scale data with complex structures, and meanwhile, have introduced a series of new challenges to data analysis both statistically and computationally. The heavy-tailed character of the distribution of empirical data has been repeatedly observed in many fields of research, including microarray studies in genomics, neuroimaging in medicine, and portfolio optimization and risk management in finance. In functional MRI studies, the parametric statistical methods often fail to produce valid cluster-wise inference, where the principal cause is that the spatial autocorrelation functions do not follow the assumed Gaussian shape; in finance, the power-law nature of the distribution of returns has been validated as a stylized fact over the years. The least squares method, albeit being most commonly used in practice due to its simplicity as a once-for-all solution, is sensitive to the tails of sample distributions and is proven to be suboptimal for heavy-tailed data from a non-asymptotic viewpoint. In this project, the PI will develop robust statistical procedures for various problems, ranging from mean estimation, linear regression, high-dimensional sparse regression to large covariance matrix estimation. The main goals of this research are to understand the finite-sample properties of robust learning, and to develop computationally efficient procedures that advance the practical use of robust methods.In this project, the PI will study robust alternatives to the method of least squares for two fundamental problems: linear regression and covariance estimation. To achieve robustness against asymmetric and heavy-tailed data, the main idea is to use the adaptive Huber loss and its extension on the matrix space. From a non-asymptotic viewpoint, the accompanying scale parameter should adapt to the sample size, dimension and noise level for optimal tradeoff between the gain in stability and cost in bias. The work on the project aims to (i) develop new methods for robust estimation and inference under linear models, and investigate their mathematical underpinnings using techniques from concentration inequality in probability, finite-sample theory for M-estimation in statistics and convex analysis in optimization, and (ii) construct both general and structured large covariance matrix estimators under minimal assumptions on the data. The originality of the project resides in providing new perspectives on robustness, which not only represent useful complements to classical robust statistics but also make important contributions to modern statistical analysis, including high dimensional estimation and large-scale inference. The philosophy of the project echos John Tukey's principles for statistical practice by highlighting the importance of having methods of statistical analysis that are robust to violations of the assumptions underlying their use and allowing the possibility of data's influencing the choice of method by which they are analyzed.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.