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.

从分布中绘制样品是贝叶斯统计，机器学习，统计物理学以及许多涉及随机模型的领域等领域的核心计算挑战。在所有方法中，马尔可夫链蒙特卡洛（MCMC）算法是最广泛使用的采样算法，具有广泛的应用，尤其是在高维贝叶斯推断中。自从Metropolis等人的基础工作以来，已经提出，研究和实施MCMC算法。在1953年，在实践中使用的算法的许多收敛属性尚不清楚。 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从本科生到博士学生，为下一代统计学家和数据科学家提供指导机会。本项目将解决以高维推断中MCMC算法为中心的三个特定研究问题。首先，该项目打算严格对MCMC算法的效率进行采样对数 - 循环分布的效率，并提供对混合时间的简洁非反应小型max分析。采样中的对数符号分布与凸功能在优化中一样重要，并且在不确定对数字循环分布上采样算法的基本限制的情况下，人们无法期望建立基础理论基础。将严格研究广泛使用的算法，例如汉密尔顿蒙特卡洛，吉布斯采样和撞击。其次，由于集中不平等是了解MCMC采样算法效率的重要组成部分，因此该项目将通过新的技术工具（例如随机定位）对高维对数concave分布的浓度产生良好的了解。最后，该项目将统一通过本地化方案研究离散状态和连续采样算法的现有理论工具。拟议的研究旨在通过对MCMC采样算法及其在离散和连续情况下的最佳设置进行全面了解，以推进该领域。该项目将提供广泛的跨学科举措，以增强统计科学的本科生和研究生的专业发展。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子的智力优点和更广泛影响的审查标准来评估值得通过评估来支持的。