DMS-EPSRC: Fast Martingales, Large Deviations, and Randomized Gradients for Heavy-tailed Distributions

本项目研究贝叶斯计算方法的理论基础，这些方法是研究重尾分布的关键。已知这些分布可以模拟可能难以对冲的高度后果性事件的影响，例如飓风，地震，流行病，野火，经济冲击等。 反过来，贝叶斯方法包含了统计理论的主体，解释了如何将联合收割机观察到的证据与主观信念相结合。尽管前面提到的应用程序的重要性，贝叶斯推理的大多数计算方法通常被设计为有效地研究轻尾分布，这在某种意义上更容易对冲的事件模型。该项目的目标是研究重尾目标分布贝叶斯推理计算方法收敛速度的核心问题。在这个项目中研究的方法将提供工具，设计更快，更有效的算法，以准确地预测高影响的事件，如上述。成功地对重尾目标进行高效和系统的贝叶斯推理需要广泛的专业知识和研究经验，如果没有DMS-EPSRC牵头机构的同意，这些知识和经验很难在单个项目中汇集起来。在这一建议中取得的成果将被介绍到有助于扩大参与的课程中。PI将尝试从代表性不足的群体中招募人员。该项目的主要目标是研究具有重尾特征的马尔可夫链的均衡收敛分析。虽然这一目标是理论上的性质，其动机来自应用：现有的理论不适用于随机马尔可夫链蒙特卡罗（MCMC）算法与重尾目标，但经常出现在实践中。尽管收敛到均衡分析的根本重要性，有一些重要的问题，还没有得到很好的研究文献。例如，已知谱隙的存在等价于马尔可夫链的几何收敛。然而，即使在几何收敛下，遍历估计仍然可能表现出标准经验均值的重尾型大偏差行为。在这个方向上的贡献将显着扩展大偏差理论（这是基本的概率）。相反，具有重尾平稳测度的马尔可夫链通常没有谱间隙，但可能表现出良好的收敛性。设计快速收敛的马尔可夫链需要完全不同于MCMC中通常使用的标准朗之万扩散的动力学。PI将研究并建立一个系统的理论处理马尔可夫链与计算统计和机器学习（ML）的随机算法中产生的重尾平稳措施的收敛到平衡。 该项目将涉及学生和博士后助理谁将访问研究团队都在美国在英国。这将进一步加强这些参与者的人力资源开发，因为他们将接触到广泛的合作者和想法网络。科学产出将在计算统计和ML的许多子领域产生超出应用概率的重大影响。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates the theoretical underpinnings of Bayesian computational methods that are key in studying heavy-tailed distributions. These distributions are known to model the impact of highly consequential events that may be difficult to hedge against, such as hurricanes, earthquakes, pandemics, wildfires, economic shocks, among many others. In turn, Bayesian methods encompass the body of statistical theory that explains how to combine observed evidence with subjective beliefs. Despite the importance of the applications mentioned earlier, most of the computational methods for Bayesian inference are typically designed to efficiently study light-tailed distributions, which model events that are in some sense easier to hedge against. The project's goal is to study questions that lie at the heart of the convergence speed of computational methods for Bayesian inference with heavy-tailed target distributions. The methods studied in this project will provide the tools to design faster and more efficient algorithms to accurately predict high impact events such as those described above. Successfully enabling efficient and systematic Bayesian inference for heavy-tailed targets requires a breadth of expertise and research experience which would be very difficult to assemble within a single project without the DMS-EPSRC Lead Agency agreement. The results obtained in this proposal will be introduced in courses that will enhance broadening participation. The PI will attempt to recruit personnel from under-represented groups.The main goal of the project is the study of the convergence analysis to equilibrium of Markov chains which exhibit heavy-tailed features. While this goal is theoretical in nature, its motivation comes from applications: the existing theory does not apply to randomized Markov chain Monte Carlo (MCMC) algorithms with heavy-tailed targets, which nevertheless arise frequently in practice. Despite the fundamental importance of convergence to equilibrium analysis, there are important questions that have not been well studied in the literature. For instance, the presence of a spectral gap is known to be equivalent to the geometric convergence of a Markov chain. However, even under geometric convergence, ergodic estimators may still exhibit large deviation behavior of the heavy-tailed type for standard empirical means. Contributions in this direction will significantly extend the Donsker-Varadhan theory of large deviations (which is fundamental in probability). Conversely, Markov chains with heavy-tailed stationary measures typically do not have a spectral gap but might nevertheless exhibit good convergence properties. Designing quickly convergence Markov chains requires dynamics that are completely different from the standard Langevin diffusion typically used in MCMC. The PI will investigate and build a systematic theoretical treatment of the convergence to equilibrium of Markov chains with heavy-tailed stationary measures arising in randomized algorithms of computational statistics and machine learning (ML). This project will involve students and a postdoctoral associates who will visit the research teams both in the US in the UK. This will further enhance the human resource development of these participants since they will be exposed to a broad network of collaborators and ideas. The scientific output will have a substantial impact beyond applied probability in a number of sub-areas of computational statistics and ML where such targets arise.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.