AF: Medium: Collaborative Research: Information Compression in Algorithm Design and Statistical Physics

AF：媒介：协作研究：算法设计和统计物理中的信息压缩

• 批准号: 1514434
• 负责人: Alistair Sinclair
• 金额: $ 28.63万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-15 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1514434&HistoricalAwards=false
• 关键词: AF; Medium; Collaborative; Research; Information

The existence of connections between probabilistic algorithms, statistical physics and information theory has been known for decades and has yielded a number of unexpected breakthroughs. Recent discoveries of the PIs and other researchers give clear indications that these connections go much deeper than previously thought. A key new idea is the realization that stochastic local search algorithms can be judged by their capacity to compress the randomness they consume, with convergence following as a consequence of compressibility. Further exploration of this idea is expected to have significant impact, both conceptual and technical, in multiple scientific fields. This includes algorithm design by information theoretic methods, the study of phase transitions in statistical mechanical systems based on information bottleneck arguments, and non-constructive proofs of existence of combinatorial objects. The project will offer a wide range of research opportunities at various levels of sophistication for graduate and undergraduate students in three state universities.Information compression arguments have recently found striking applications in computer science and combinatorics. A glowing example is Moser's proof of the algorithmic Lovasz Local Lemma, which suggested an entirely new way of reasoning about randomized algorithms. Inspired by the work of Moser, one of the PIs with a collaborator has very recently created a general framework for analyzing stochastic local search algorithms using information compression. The framework is purely algorithmic, completely bypassing the Probabilistic Method. Besides helping to analyze the running times of existing algorithms it can also be used as a powerful new tool for designing novel, non-obvious randomized algorithms. The proposed research further develops this framework with the aim of unearthing completely new applications in computer science and combinatorics, while establishing mathematically rigorous connections to statistical physics. Concrete examples of such applications to be investigated include new tools for bounding the mixing time of Markov chains and algebraic connections between randomized algorithms and the classical theory of phase transitions in statistical physics.

概率算法、统计物理学和信息论之间的联系几十年来一直为人所知，并产生了许多意想不到的突破。 PI 和其他研究人员最近的发现清楚地表明，这些联系比之前想象的要深入得多。一个关键的新想法是认识到随机局部搜索算法可以通过其压缩所消耗的随机性的能力来判断，并且收敛性是可压缩性的结果。对这一想法的进一步探索预计将在多个科学领域产生概念和技术方面的重大影响。这包括信息论方法的算法设计、基于信息瓶颈论证的统计机械系统中的相变研究以及组合对象存在的非构造性证明。该项目将为三所州立大学的研究生和本科生提供各种复杂程度的广泛研究机会。信息压缩论证最近在计算机科学和组合数学中得到了惊人的应用。一个出色的例子是 Moser 对算法 Lovasz 局部引理的证明，它提出了一种关于随机算法的全新推理方式。受到 Moser 工作的启发，其中一位 PI 与合作者最近创建了一个通用框架，用于使用信息压缩来分析随机本地搜索算法。该框架是纯粹的算法，完全绕过概率方法。除了帮助分析现有算法的运行时间之外，它还可以用作设计新颖的、非显而易见的随机算法的强大新工具。拟议的研究进一步开发了该框架，旨在挖掘计算机科学和组合学中的全新应用，同时与统计物理学建立数学上严格的联系。要研究的此类应用的具体示例包括用于限制马尔可夫链混合时间的新工具以及随机算法与统计物理学中相变经典理论之间的代数联系。