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

这个项目调查了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。众所周知，这些分布模拟了飓风、地震、流行病、野火、经济冲击等许多难以对冲的重大事件的影响。反过来，贝叶斯方法包含了解释如何将观察到的证据与主观信念相结合的统计理论。尽管前面提到的应用程序很重要，但贝叶斯推理的大多数计算方法通常都是为了有效地研究轻尾分布而设计的，这些分布对事件进行建模，在某种意义上更容易对冲。该项目的目标是研究存在于重尾目标分布的贝叶斯推断计算方法收敛速度核心的问题。本项目研究的方法将为设计更快、更有效的算法提供工具，以准确预测如上所述的高冲击事件。成功实现对重尾目标的有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牵头机构协议，这些将很难在单个项目中汇集。在这项提案中取得的成果将在课程中介绍，这些课程将加强扩大参与。PI将试图从代表不足的群体中招募人员。该项目的主要目标是研究表现出重尾特征的马尔可夫链平衡的收敛分析。虽然这一目标本质上是理论上的，但其动机来自于应用：现有的理论不适用于具有重尾目标的随机马尔可夫链蒙特卡罗(MCMC)算法，然而这种算法在实践中经常出现。尽管收敛对均衡分析具有基本的重要性，但还有一些重要的问题在文献中没有得到很好的研究。例如，谱间隙的存在被认为等同于马尔可夫链的几何收敛。然而，即使在几何收敛的情况下，遍历估计仍然可能表现出标准经验均值的重尾型的大偏差行为。在这个方向上的贡献将大大扩展Donsker-Varadhan的大偏差理论(这是概率论的基础)。相反，具有重尾平稳测度的马尔可夫链通常没有谱间隙，但仍可能表现出良好的收敛特性。设计快速收敛的马尔可夫链需要与MCMC中通常使用的标准朗之万扩散完全不同的动力学。PI将研究和建立在计算统计和机器学习(ML)的随机算法中出现的具有重尾平稳度量的马尔可夫链收敛到平衡的系统的理论处理。该项目将包括学生和博士后助理，他们将访问美国和英国的研究团队。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。科学成果将在计算统计学和ML的一些子领域产生重大影响，这些领域出现了这样的目标。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.