A Non-Asymptotic Theory of Robustness

现代数据采集促进了具有复杂结构的大规模数据的收集，同时也给数据分析带来了一系列新的挑战，包括统计和计算。经验数据分布的重尾特征已在许多研究领域反复观察到，包括基因组学中的微阵列研究、医学中的神经影像学以及金融中的投资组合优化和风险管理。在功能性MRI研究中，参数统计方法往往不能产生有效的聚类推理，其主要原因是空间自相关函数不遵循假设的高斯形状；在金融领域，收益分配的幂律性质多年来已被证实为一种程式化的事实。最小二乘法虽然在实践中最常用，因为它作为一次性解决方案的简单性，但对样本分布的尾部很敏感，并且从非渐近的角度证明对于重尾数据是次优的。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。本研究的主要目标是了解鲁棒学习的有限样本特性，并开发计算效率高的程序，以推进鲁棒方法的实际应用。在这个项目中，PI将研究最小二乘方法的鲁棒替代方案，以解决两个基本问题：线性回归和协方差估计。为了实现对非对称和重尾数据的鲁棒性，主要思想是使用自适应Huber损失及其在矩阵空间上的扩展。从非渐近的角度来看，伴随的尺度参数应该适应样本量、维度和噪声水平，以便在稳定性增益和偏差代价之间进行最佳权衡。该项目的工作旨在(i)开发线性模型下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建一般和结构化大协方差矩阵估计器。该项目的独创性在于提供了关于鲁棒性的新视角，这不仅是对经典鲁棒统计的有益补充，而且对现代统计分析（包括高维估计和大规模推断）做出了重要贡献。该项目的理念与John Tukey的统计实践原则相呼应，强调了统计分析方法的重要性，这些方法对违反其使用的假设是稳健的，并允许数据影响分析方法的选择的可能性。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。

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.