Robust and Distributed Statistical Learning from Big Data

大数据在科学、工程、社会科学、人文等多个领域无处不在，对技术创新和经济发展具有重大影响。该项目旨在引入对大数据潜在质量差不敏感的稳健高维统计推断的有效方法，并开发大数据分析、计算和优化所需的分布式估计。这项研究将解决大数据在基因组学、遗传学、神经科学、机器学习、经济学和金融学中的几个健壮和分布式统计推断问题。该项目将促进我们对分子机制、生物过程、遗传关联、大脑功能以及经济和金融风险的理解。将通过本科生、研究生和博士后研究员的参与，以及开发公开可用的计算机代码，在坚实的理论支持下，实现大数据的稳健和分布式分析，从而实现研究和教育的融合。通过与产业界合作伙伴的密切合作，这项研究将加强学术界和产业界之间的合作。该项目将导致开发新的统计理论、方法和算法，以便从高维统计和大数据进行稳健的统计推断。第一个目标是通过适当收缩观测数据或损失函数，引入一种简单且广泛适用的稳健性推理原理。这减少了异常点和重尾分布的影响，并将矩条件从亚高斯分布减弱到回归的有界二阶矩或协方差估计的四阶矩。这项研究包括系统地发展高维均值稳健估计的理论和方法，以及在基因和转录本选择的大规模推理中控制错误发现率的方法的实施，协方差和精度矩阵的稳健正则化，以及它们在稳健主成分分析、因子分析和高维假设检验中的应用。此外，还将研究稳健稀疏回归、模型选择和低阶矩阵恢复。第二个目标是通过开发分布式估计和推理，使拟议的稳健程序适用于大数据环境。特别是，将使用分而治之的方法将计算分发到节点计算机，并解决隐私和数据所有权问题。通过海森矩阵的部分通信来减少由于基于似然模型的分布式计算而造成的信息损失的方法将被研究。两类重要的问题--踪迹回归和主成分分析--将被用来说明所提出的方法。

Big Data are ubiquitous in many areas of science, engineering, social sciences, and the humanities, and have significant impact in terms of technological innovation and economic development. This project seeks to introduce effective methods for robust high-dimensional statistical inference that are insensitive to the potential poor quality of big data, and to develop distributed estimation that is needed for Big Data analysis, computing, and optimization. The research will address several robust and distributed statistical inference problems for Big Data in genomics, genetics, neuroscience, machine learning, economics, and finance. The project will advance our understanding of molecular mechanisms, biological processes, genetic associations, brain functions, and economic and financial risk. Integration of research and education will be achieved through the involvement of undergraduate students, graduate students, and postdoctoral fellows, and the development of publicly available computer code for robust and distributed analysis of Big Data with sound theoretical support. Working closely with industrial partners, the research will lead to increased collaborations between academia and industry. The project will lead to the development of novel statistical theory, methods, and algorithms for robust statistical inference from high-dimensional statistics and Big Data. The first aim seeks to introduce a simple and widely applicable principle for robust inference via an appropriate shrinkage of observed data or loss functions. This reduces the influence of outliers and heavy-tailed distributions, and weakens the moment conditions from sub-Gaussian distributions to bounded second moments for regression or fourth moments for covariance estimation. The research includes plans to systematically develop the theory and methods for robust estimation of high-dimension means, and implementation of these methods to control false discovery rates in large scale inference for gene and transcripts selection, robust regularization of covariance and precision matrices, and their applications to robust principal component analysis, factor analysis and high-dimensional hypothesis testing. In addition, robust sparse regression, model selection, and low-rank matrix recovery will also be investigated. The second aim focuses on making the proposed robust procedures applicable to the Big Data environment via the development of distributed estimation and inference. In particular, divide-and-conquer methods will be used to distribute the computation to node machines and to solve privacy and data ownership issues. Approaches to reduce the information loss due to the distributed computation for likelihood based models via partial communication of the Hessian matrices will be investigated. Two important classes of problems, trace regression and principal component analysis, will be used to illustrate the proposed methods.