CAREER: Towards Tight Guarantees of Markov Chain Sampling Algorithms in High Dimensional Statistical Inference

职业：高维统计推断中马尔可夫链采样算法的严格保证

• 批准号: 2237322
• 负责人: Yuansi Chen
• 金额: $ 45万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2237322&HistoricalAwards=false
• 关键词: CAREER; Towards; Tight; Guarantees; Markov

Drawing samples from a distribution is a core computational challenge in fields such as Bayesian statistics, machine learning, statistical physics, and many other areas involving stochastic models. Among all methods, Markov Chain Monte Carlo (MCMC) algorithms stand out as the most widely used class of sampling algorithms with a broad range of applications, notably in high dimensional Bayesian inference. While MCMC algorithms have been proposed, studied, and implemented since the foundational work of Metropolis et al. in 1953, many convergence properties of algorithms used in practice are not well understood. Practitioners in Bayesian statistics are often faced with a series of key challenges to be addressed rigorously: the choice of algorithm hyper-parameters, the estimated computational cost and the choice of the best algorithm, etc. This project focuses on developing theoretical guarantees of MCMC sampling algorithms that arise in large-scale Bayesian statistical inference problems.The project will also offer numerous interdisciplinary research training, outreach and mentoring opportunities for the next generation of statisticians and data scientists at all levels, from undergraduate to doctoral students.This project will address three specific research problems centered around MCMC algorithms in high dimensional inference. First, the project intends to rigorously rank the efficiency of MCMC algorithms for sampling log-concave distributions and to provide succinct non-asymptotic mini-max analysis of mixing time. Log-concave distributions in sampling are as important as convex functions in optimization, and one cannot expect to build a foundational theory basis without determining the fundamental limits of sampling algorithms on log-concave distributions. Widely-used algorithms such as Hamiltonian Monte Carlo, Gibbs sampling and hit-and-run will be studied rigorously. Second, as concentration inequalities constitute an essential component in understanding the efficiency of MCMC sampling algorithms, the project will develop a fine-grained understanding of concentration of high dimensional log-concave distributions via new technical tools such as stochastic localization. Finally, the project will unify the existing theoretical tools for studying discrete-state and continuous-state sampling algorithms through localization schemes. The proposed research aims to advance the field with a comprehensive understanding of MCMC sampling algorithms and their optimal settings in both discrete and continuous cases. The project will provide a wide range of interdisciplinary initiatives to enhance professional development of undergraduate and graduate students in statistical sciences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

从分布中抽取样本是贝叶斯统计、机器学习、统计物理学以及涉及随机模型的许多其他领域等领域的核心计算挑战。在所有方法中，马尔可夫链蒙特卡罗 (MCMC) 算法脱颖而出，成为使用最广泛的一类采样算法，具有广泛的应用，特别是在高维贝叶斯推理中。自 Metropolis 等人的基础工作以来，MCMC 算法已经被提出、研究和实现。 1953 年，实践中使用的算法的许多收敛特性还没有得到很好的理解。贝叶斯统计的从业者经常面临一系列需要严格解决的关键挑战：算法超参数的选择、估计计算成本和最佳算法的选择等。本项目重点开发MCMC采样的理论保证该项目还将为从本科生到博士生的各个级别的下一代统计学家和数据科学家提供大量跨学科研究培训、推广和指导机会该项目将解决以高维推理中的 MCMC 算法为中心的三个具体研究问题。首先，该项目打算对采样对数凹分布的 MCMC 算法的效率进行严格排名，并提供混合时间的简洁非渐近最小最大分析。采样中的对数凹分布与优化中的凸函数一样重要，如果不确定采样算法对对数凹分布的基本限制，就不能指望建立基础理论基础。哈密​​顿蒙特卡罗、吉布斯采样和肇事逃逸等广泛使用的算法将得到严格研究。其次，由于浓度不等式是理解 MCMC 采样算法效率的重要组成部分，因此该项目将通过随机定位等新技术工具，对高维对数凹分布的浓度进行细粒度的理解。最后，该项目将统一现有的理论工具，通过定位方案研究离散状态和连续状态采样算法。拟议的研究旨在通过全面了解 MCMC 采样算法及其在离散和连续情况下的最佳设置来推进该领域的发展。该项目将提供广泛的跨学科举措，以促进统计科学领域本科生和研究生的专业发展。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。