A Non-Asymptotic Theory of Robustness

现代数据采集促进了结构复杂的大规模数据的收集，同时也给数据分析带来了一系列统计和计算上的新挑战。经验数据分布的重尾特征在许多研究领域都被反复观察到，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。在功能磁共振成像研究中，参数统计方法往往无法产生有效的聚类推理，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，收益率分布的幂律性质多年来一直被验证为程式化的事实。最小二乘法，虽然是最常用的在实践中，由于其简单的一次性解决方案，是敏感的样本分布的尾部，并被证明是次优的重尾数据从非渐近的观点。在这个项目中，PI将为各种问题开发强大的统计程序，从均值估计，线性回归，高维稀疏回归到大型协方差矩阵估计。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使用。在本项目中，PI将研究两个基本问题的最小二乘法的鲁棒替代方案：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，伴随的尺度参数应适应样本大小，尺寸和噪声水平之间的最佳权衡的增益在稳定性和成本的偏差。该项目的工作旨在（i）开发线性模型下稳健估计和推断的新方法，并使用概率中的浓度不等式、统计学中M估计的有限样本理论和优化中的凸分析技术研究其数学基础，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。该项目的独创性在于提供了鲁棒性的新视角，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。该项目的哲学呼应约翰Tukey的统计实践的原则，通过强调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鲁棒性，并允许数据的可能性影响其分析方法的选择。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识产权进行评估来支持。优点和更广泛的影响审查标准。

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.