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

DMS-EPSRC：重尾分布的快速鞅、大偏差和随机梯度

• 批准号: 2118199
• 负责人: Jose Blanchet
• 金额: $ 40万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-04-01 至 2025-03-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2118199&HistoricalAwards=false
• 关键词: DMS; EPSRC; Fast; Martingales; Large; DMS; EPSRC; Fast; Martingales; Large

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.

该项目研究了研究重尾分布的关键的贝叶斯计算方法的理论基础。已知这些分布可以模拟高度结果事件的影响，这些事件可能难以对冲，例如飓风，地震，大流行，野火，经济冲击等。 反过来，贝叶斯方法涵盖了统计理论的主体，该理论解释了如何将观察到的证据与主观信念相结合。尽管前面提到的应用非常重要，但贝叶斯推断的大多数计算方法通常旨在有效地研究轻尾分布，这些分布在某种意义上更容易对冲。该项目的目标是研究贝叶斯推断具有重尾目标分布的计算方法收敛速度的核心。该项目中研究的方法将提供工具来设计更快，更有效的算法，以准确预测上述高影响事件。成功地实现有效和系统的贝叶斯推论，对重尾目标进行了广泛的专业知识和研究经验，这将在一个项目中很难在一个项目中组装，而无需DMS-EPSRC领导者协议。在本提案中获得的结果将在课程中引入，以增强扩大参与。 PI将尝试从代表性不足的群体中招募人员。该项目的主要目的是研究融合分析与马尔可夫链平衡的研究，这些链条表现出重型特征。尽管该目标本质上是理论上的，但它的动机来自应用：现有理论不适用于具有重尾目标的随机Markov Chain Monte Carlo（MCMC）算法，但是实际上经常出现。尽管收敛到均衡分析的基本重要性，但文献中仍未对一些重要的问题进行了很好的研究。例如，已知光谱间隙的存在等效于马尔可夫链的几何收敛。然而，即使在几何融合下，对于标准的经验手段，厄贡估计量仍可能表现出重尾类型的较大偏差行为。在这个方向上的贡献将显着扩大大偏差的Donsker-Varadhan理论（这在概率上至关重要）。相反，具有重尾固定措施的马尔可夫连锁店通常没有光谱差距，但可能表现出良好的收敛性。快速设计Markov连锁链需要与MCMC中通常使用的标准Langevin扩散完全不同的动力学。 PI将通过在计算统计和机器学习（ML）随机算法（ML）中采取的重尾固定度量（ML），研究并建立对马尔可夫链均衡的融合的系统理论处理。 该项目将涉及学生和博士后员工，他们将访问英国美国的研究团队。这将进一步增强这些参与者的人力资源发展，因为他们将接触到广泛的合作者和思想网络。科学的产出将在许多计算统计和ML的次级分区中产生重大影响。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的审查标准通过评估来获得支持的。