AF: Small: Rare Events - New Probabilistic and Algorithmic Techniques

AF：小：罕见事件 - 新的概率和算法技术

• 批准号: 1614023
• 负责人: Shachar Lovett
• 金额: $ 45万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1614023&HistoricalAwards=false
• 关键词: AF; Small; Rare; Events; New

Randomness is a powerful tool in mathematics and computer science. In mathematics, it underlies the "probabilistic method," a method to prove that certain mathematical objects with desired properties exist, by simply picking an object at random, and arguing that with a positive probability, the object has the required properties. In computer science, randomness is a powerful tool in algorithm design. Randomized algorithms are often simpler or perform better than their deterministic counterparts, and have found applications in many fields such as graph algorithms, linear programming, routing algorithms, coding theory, communication protocols, approximation algorithms, cryptography, and many more.One of the main reasons that randomness is ubiquitous in algorithm design, is that typically the probabilistic method shows that the desired events, which control the success of the algorithm, not only occur with a positive probability (which would be sufficient to prove existence), but that they in fact occur with overwhelming probability. As such, it immediately lends itself to the design of randomized algorithms.The focus of this project is on regimes where this is not the case. There are several probabilistic techniques which can prove the existence of "rare events." That is, they show that the desired events occur with a positive (yet tiny) probability. Although we do not have many such techniques, they have proven to be extremely valuable, with applications in many domains: combinatorial optimization, learning theory, approximation algorithms, distributed algorithms, computational geometry, numerical analysis, and more. The main reason is that such techniques, and more importantly, their algorithmic realizations, provide algorithm designers with new sets of tools that they can apply that go beyond "vanilla" probabilistic techniques.This project will focus both on the development of new mathematical and algorithmic tools and techniques, as well as on their assimilation by students and researchers. This involves mentoring students, creating and teaching relevant classes, writing expository surveys, and organizing workshops.On the technical side, the project focuses on two main domains. The first is discrepancy theory. Discrepancy theory studies irregularities within distributions, and has intimate relations with rounding techniques for integer programs, and more generally with combinatorial optimization. There are several important open problems in discrepancy theory (most notably the Komlos conjecture), which this project sets a concrete plan to resolve.The second domain is pseudo-randomness. Pseudo-randomness is the study of deterministic objects which attain certain properties satisfied by random objects. As such, this is a wide area of study, with many applications both in mathematics and computer science. In this project, we focus on pseudorandom objects which, for some bounded family of tests, behave exactly like random objects. While in some settings such objects are deeply understood and widely applied (for example, k-wise independence, which has applications in coding theory, data structures, de-randomization, and more), in many other settings they are much less understood. This project explores new approaches to better understand such objects and to develop new algorithmic applications of them.

随机性是数学和计算机科学中的强大工具。在数学中，它是“概率方法”的基础，该方法通过简单地随机挑选一个对象，并以正概率论证该对象具有所需属性，来证明某些具有所需属性的数学对象存在。在计算机科学中，随机性是算法设计中的强大工具。随机算法通常比确定性算法更简单或性能更好，并且在许多领域都有应用，例如图算法、线性规划、路由算法、编码理论、通信协议、近似算法、密码学等等。主要之一随机性在算法设计中普遍存在的原因是，通常概率方法表明，控制算法成功的期望事件不仅以正概率发生（这足以证明存在），而且它们以正概率发生。事实发生以压倒性的概率。因此，它立即适用于随机算法的设计。该项目的重点是非这种情况的制度。有几种概率技术可以证明“罕见事件”的存在。也就是说，它们表明期望的事件以正（但很小）的概率发生。尽管我们没有很多这样的技术，但它们已被证明是非常有价值的，在许多领域都有应用：组合优化、学习理论、逼近算法、分布式算法、计算几何、数值分析等等。主要原因是这些技术，更重要的是它们的算法实现，为算法设计者提供了一套新的工具，他们可以应用这些工具，超越“普通”概率技术。该项目将重点关注新数学和算法的开发工具和技术，以及学生和研究人员对它们的吸收。这包括指导学生、创建和教授相关课程、撰写说明性调查以及组织研讨会。在技术方面，该项目侧重于两个主要领域。第一个是差异理论。差异理论研究分布内的不规则性，并与整数程序的舍入技术密切相关，更普遍的是与组合优化。差异理论中有几个重要的开放问题（最著名的是科姆洛斯猜想），该项目制定了具体的计划来解决这些问题。第二个领域是伪随机性。伪随机性是对确定性对象的研究，这些确定性对象获得随机对象所满足的某些属性。因此，这是一个广泛的研究领域，在数学和计算机科学中都有许多应用。在这个项目中，我们关注伪随机对象，对于某些有界测试系列，其行为与随机对象完全相同。虽然在某些设置中，这些对象被深入理解并广泛应用（例如，k 向独立性，它在编码理论、数据结构、去随机化等领域有应用），但在许多其他设置中，它们的理解要少得多。该项目探索新方法来更好地理解此类对象并开发它们的新算法应用程序。