CAREER: Large Scale Stochastic Optimization and Statistics

随机优化提供了一个通用的框架来研究与预测相关的许多基本统计问题，如回归，分类和密度估计。此外，这是一个自然的框架，进口强大的算法从数值优化，特别是大规模的问题。这个项目的主要目标是了解统计和随机优化之间的基本相互作用。为了完成这一任务，调查员（a）从统计学中识别新的问题，特别是具有复杂结构的问题，这些问题可以被重新定义为随机优化问题;（B）开发新的算法，以最优和有效地解决大规模问题;（c）确定问题的基本特征，这些特征支配算法的性能及其基本限制;（d）探讨随机优化的外围问题，包括带随机约束的随机优化和带有限反馈的随机优化。在信息时代，数据的收集呈爆炸式增长，大规模数据集在生物学、网络、环境科学、社会学和市场营销等广泛的应用中无处不在。这导致迫切需要新的统计方法来分析这些规模空前的数据集。虽然数值优化技术可以用于多种场景，但它们的分析在很大程度上仍然与手头的统计任务无关。本研究的目的是提供一个统一的处理一些大规模的问题出现的统计学习和优化下的不确定性一般。因此，该项目不仅将产生新的和有效的算法，而且在一个新的理论框架，支持随机优化问题的分析，并使上述算法的进一步改进。

Stochastic optimization offers a general framework to study many fundamental statistical problems related to prediction such as regression, classification and density estimation. Furthermore, it is a natural framework to import powerful algorithms from numerical optimization, especially for large scale problems. The broad goal of this project is to understand the fundamental interactions between statistics and stochastic optimization. To accomplish this task the investigator (a) identifies new problems from statistics, especially with complex structure, that can be recast as stochastic optimization problems; (b) develops new algorithms that optimally and efficiently solve large scale problems; (c) determines essential characteristics of the problems that govern the performance of algorithms and their fundamental limitations; and (d) explores peripheral problems of stochastic optimization including stochastic optimization with stochastic constraints and stochastic optimization with limited feedback. The information era has witnessed an explosion in the collection of data and large scale data sets are ubiquitous in a wide range of applications including biology, networks, environmental science, sociology and marketing. This results in an acute need of new statistical methods to analyze these data sets of unprecedented size. While techniques from numerical optimization can be used in several scenarios, their analysis remains largely dissociated from that of the statistical task at hand. This research aims at providing a unified treatment of a number of large scale problems emerging from statistical learning and from optimization under uncertainty in general. Therefore, the project will not only result in new and effective algorithms, but also in a novel theoretical framework that supports the analysis of stochastic optimization problems and enables further improvements of said algorithms.