Modern Statistical Optimization, Robustification and Inference with Applications to Big Data Analytics

大数据正在改变我们的世界，从金融工程到生物医学科学，它正在给各地的运营和分析带来革命性的变化。如何有效地从海量和噪声数据集中提取有用信息具有重要意义，这至少提出了三个重大挑战。实现这一目标的第一个挑战往往是计算方面的：数据的海量和复杂性要求在实践中使用高效的算法。第二个挑战来自于这样一个事实，即在现实世界中收集的数据，无论大小，往往被重尾误差或离群值所污染，使得传统的统计方法无法奏效。第三，大数据分析和有效的统计决策决策需要高效而准确的推理程序。我们试图在这项提案中解决这三个挑战。我们的第一个目标是探索非凸问题的统计优化的方向，并了解在广泛的连续和离散统计优化问题中的隐凸性。我们的目标是提供一个算法、理论和应用的统一工具箱，我们希望为大数据的优化提供新的基础理解和工具。它一旦完成，将在统计、机器学习、信号处理、图像恢复、词典学习和人工智能领域产生潜在和根本性的影响。我们的第二个目标是全面研究在低质量数据存在的情况下用于有意义数据分析的非渐近鲁棒性。关键的观察是偏差-稳健性权衡原则，我们认为这是许多应用中的普遍现象，如预测、分类、聚类和推理问题。这一目标在经过全面研究后，将对实践中的数据分析产生影响。我们将遵循现代软件原则，强调兼容性、可扩展性和可维护性，并为统计人员和从业人员开发开放源码包。最后，我们将前进到现代统计推断，在那里我们集中在可复制性问题上。我们致力于解决的科学问题是，来自大规模统计搜索的发现是否可以在未来的独立研究中复制。我们将通过将我们的方法应用于大规模成像遗传学数据集来验证我们的方法的有效性，例如通过荟萃分析增强神经成像遗传学以及基因组流行病学中心脏和衰老研究的队列。这一目标一旦完成，将影响各个科学学科的从业者，特别是神经科学和遗传学的从业者。

Big data is transforming our world, revolutionizing operations and analytics everywhere, from financial engineering to biomedical sciences. How to efficiently exact useful information from large and noisy datasets is of significant importance, and it has posed at least three grand challenges. The first challenge towards this goal is often computational: the massiveness and complexity of the data call for efficient algorithms to be used in practice. The second challenge arises from the fact that the data collected in real world, big or small, are often contaminated by heavy-tailed errors or outliers, making conventional statistical methods inadequate. Third, an efficient and accurate inference procedure is in demand for big data analytics and valid statistical decision makings. We attempt to addressing these three challenges in this proposal. Our first goal of this proposal is to explore the direction of statistical optimization for nonconvex problems and to understand the hidden convexity in a wide range of continuous and discrete statistical optimization problems. We aim to provide a unified toolbox of algorithms, theories and applications, and we expect to provide new fundamental understanding and tools for optimization in big data. It, upon completed, will have potential and fundamental impact in the area of statistics, machine learning, signal processing, imaging restoration, dictionary learning and artificial intelligence. Our second goal is to comprehensively study nonasymptotic robustification for meaningful data analytics in the presence of low-quality data. The key observation is the bias-robustness tradeoff principal, which we believe is a universal phenomenon in many applications, such as prediction, classification, clustering and inference problems. This goal, upon comprehensively studied, will impact data analytics in practice. We will follow modern software principals in the emphases of compatibility, extendability and maintainability, and develop open-source packages for statisticians and practitioners. Last, we will move forward to modern statistical inference, where we focus on the replicability issues. The scientific question we aim to address is whether the discoveries from a vast statistical search can be replicated in future and independent studies. We will verify the efficacy of our methods by applying them to large-scale imaging genetic datasets, such as the enhancing neuroimaging genetics through meta-analysis and the cohorts for heart and aging research in genomic epidemiology. This goal, upon completed, will impact practitioners in various scientific disciplines, especially in neuroscience and genetics.