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

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，这些分布可以模拟可能难以对冲的高度后果事件的影响，例如飓风、地震、流行病、野火、经济冲击等。 反过来，贝叶斯方法涵盖了统计理论的主体，解释了如何将观察到的证据与主观信念结合起来。尽管前面提到的应用很重要，但大多数贝叶斯推理的计算方法通常都是为了有效地研究轻尾分布而设计的，这种分布对在某种意义上更容易对冲的事件进行建模。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。该项目研究的方法将提供设计更快、更有效的算法的工具，以准确预测如上所述的高影响事件。成功地对重尾目标实现高效、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，这些专业知识和研究经验很难整合到单个项目中。该提案中获得的结果将被引入课程中，以扩大参与范围。 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。虽然这个目标本质上是理论性的，但其动机来自应用：现有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，但这种算法在实践中经常出现。尽管收敛对于均衡分析具有根本重要性，但仍有一些重要问题尚未在文献中得到充分研究。例如，已知谱间隙的存在等同于马尔可夫链的几何收敛。然而，即使在几何收敛下，遍历估计量仍可能表现出标准经验均值的重尾类型的大偏差行为。在这个方向上的贡献将显着扩展 Donsker-Varadhan 的大偏差理论（这是概率的基础）。相反，具有重尾平稳测度的马尔可夫链通常不具有谱间隙，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 将利用计算统计和机器学习 (ML) 随机算法中出现的重尾平稳度量，研究并建立马尔可夫链收敛到平衡的系统理论处理。 该项目将涉及学生和博士后同事，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。科学成果将在出现此类目标的计算统计和机器学习的许多子领域中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智力价值和更广泛的评估进行评估，被认为值得支持。影响审查标准。

项目成果

Wasserstein Distributionally Robust Linear-Quadratic Estimation under Martingale Constraints

鞅约束下的 Wasserstein 分布鲁棒线性二次估计

• 发表时间: 2023-08
• 期刊: Proceedings of Machine Learning Research
• 作者: Lotidis, Kyriakos;Bambos, Nicholas;Li, Jiajin
• 通讯作者: Li, Jiajin

Statistical Limit Theorems in Distributionally Robust Optimization

分布鲁棒优化中的统计极限定理

• 发表时间: 2023-03
• 期刊: arXivorg
• 作者: Blanchet, Jose;Shapiro, Alexander
• 通讯作者: Shapiro, Alexander

Unbiased Optimal Stopping via the MUSE

通过 MUSE 进行无偏最优停止

• DOI: 10.1016/j.spa.2022.12.007
• 发表时间: 2022-12
• 期刊: Stochastic Processes and their Applications
• 影响因子: 1.4
• 作者: Zhou, Zhengqing;Wang, Guanyang;Blanchet, Jose H.;Glynn, Peter W.
• 通讯作者: Glynn, Peter W.

Tikhonov Regularization is Optimal Transport Robust under Martingale Constraints.

吉洪诺夫正则化是鞅约束下的最优传输鲁棒性。

• 发表时间: 2022-07
• 期刊: Advances in neural information processing systems
• 作者: Li, Jiajin;Lin, Sirui;Blanchet, Jose H.;Nguyen, Viet
• 通讯作者: Nguyen, Viet

Distributionally Robust Q-Learning

分布式鲁棒 Q-Learning

• 发表时间: 2022-07
• 期刊: Proceedings of Machine Learning Research
• 作者: Liu, Zijian;Bai, Qinxun;Blanchet, Jose H.;Dong, Perry;Xu, Wei;Zhou, Zhengqing;Zhou, Zhengyuan
• 通讯作者: Zhou, Zhengyuan