Learning Complex Stochastic Systems
学习复杂的随机系统
基本信息
- 批准号:2246815
- 负责人:
- 金额:$ 20万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2023
- 资助国家:美国
- 起止时间:2023-08-15 至 2026-07-31
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
Differential equations are often used to model temporal evolutions of a variety of systems. However, most realistic systems including those arising from biology, environmental science, engineering, physics, medicine and financial markets exhibit randomness in their behavior. Accurate analysis of such systems thus needs differential equations that can incorporate this randomness. Stochastic differential equations are powerful tools for this purpose. Understanding behaviors of these systems requires not just building mathematical models but integrating them with available data. This in turn requires various types of learning algorithms. It is important to judge the effectiveness of these algorithms by rigorous mathematical analysis, which is the primary objective of this project. The dynamics of these stochastic systems are however intricate with convoluted correlation structures, and there is a critical lack of mathematical results in the literature investigating learning methods for such complex data. The work done by the investigator will fill some of this gap by deriving mathematical results that will not only be able to answer if the algorithms become more accurate with data observed over longer periods of time but will be able to provide valuable insight on how to fine-tune the key parameters for optimal efficiency. Building such data-driven stochastic models backed by rigorous mathematics enhances our understanding of complex systems across multiple domains and empowers informed decision-making in the presence of randomness. The project will involve undergraduate and graduate students and will teach them valuable skills through a combination of theoretical knowledge, practical application, and hands-on experience with coding. It will enable them to excel in the digital age and adapt to the demands of an increasingly data-driven and technologically advanced world. The results of the project will be disseminated through publications in well-known scientific journals and presentations at domestic and international conferences.The project will study important learning problems for a broad class of stochastic differential equations (SDEs). These problems lie on the interface of stochastic analysis and statistical learning theory, and there is a paucity of theoretical results in probability, statistics and machine learning literature addressing them. The project is divided into three interconnected parts, each of which plays an important role in the other. Part I will address important problems on parametric inference including point estimation and testing of hypotheses. It will derive asymptotic results including law of large numbers, central limit theorems and large deviation principles for estimators of a finite dimensional parameter of a broad class of SDEs. Unlike some existing works in this direction which assume data to be in the form of a continuous trajectory, the investigator's work will consider the realistic case of availability of only discrete data points. Since asymptotic analysis requires the time horizon to go to infinity, the effect of time-gap (or discretization step) between the observations on the accuracy of these estimators over long time is not clear, and it is known that naive discretization of estimators based on a continuous trajectory of an SDE can lead to erroneous inference. The project will introduce appropriate scaling frameworks to quantify this effect and analyze the errors in different scaling regimes. Next, these results will be utilized to design tests for composite hypotheses-testing problems so that the probability of type I error decays rapidly and which are asymptotically uniformly powerful within a class of tests having similar level of type I error. Part II of the project concerns itself with the important topic of decision-making. Decision-making involves (constrained) minimization of suitable cost functions depending on model parameters. Since these latter quantities are unknown, data-driven versions of such minimization problems are necessary in practice. In particular, it is necessary to construct suitable estimators of the cost functions so that decisions based on their minimization are close to the true decisions. The investigator will study a novel approach based on large deviation analysis and results of Part I which aims to guarantee that under appropriate conditions this can be achieved with a very high probability. Part III is devoted to nonparametric learning of SDEs. The last part falls in the realm of infinite-dimensional learning theory where the goal is to learn the entire driving functions of the SDE-based models as opposed to estimating finite-dimensional parameters. A rigorous computational framework combining Bayesian techniques with the theory of Reproducing Kernel Hilbert Space will be developed toward this end, and the theoretical properties of the resulting learning algorithms will also be studied.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.
微分方程经常被用来模拟各种系统的时间演化。然而,大多数现实的系统,包括那些从生物学,环境科学,工程学,物理学,医学和金融市场表现出随机性的行为。因此,对这种系统的精确分析需要能够包含这种随机性的微分方程。随机微分方程是实现这一目的的有力工具。理解这些系统的行为不仅需要建立数学模型,还需要将它们与可用数据相结合。这反过来又需要各种类型的学习算法。重要的是通过严格的数学分析来判断这些算法的有效性,这是本项目的主要目标。然而,这些随机系统的动力学是复杂的,具有复杂的相关结构,并且在研究这种复杂数据的学习方法的文献中严重缺乏数学结果。研究人员所做的工作将通过得出数学结果来填补这一空白,这些数学结果不仅能够回答算法是否会在较长时间内观察到更准确的数据,而且还能够就如何微调关键参数以实现最佳效率提供有价值的见解。 建立这种由严格数学支持的数据驱动的随机模型,可以增强我们对多个领域复杂系统的理解,并在存在随机性的情况下做出明智的决策。 该项目将涉及本科生和研究生,并将通过理论知识,实际应用和编码实践经验的结合来教授他们宝贵的技能。它将使他们能够在数字时代脱颖而出,并适应日益数据驱动和技术先进的世界的需求。 该项目的成果将通过在知名科学期刊上发表文章以及在国内和国际会议上发表演讲的方式进行传播,该项目将研究一大类随机微分方程的重要学习问题。这些问题在于随机分析和统计学习理论的接口,并且在概率,统计和机器学习文献中缺乏解决它们的理论结果。该项目分为三个相互关联的部分,每个部分都在另一个部分中发挥重要作用。第一部分将讨论参数推断的重要问题,包括点估计和假设检验。它将导出渐近结果,包括大数定律,中心极限定理和大偏差原理的估计的有限维参数的广泛的一类SDES。与这方面的一些现有工作不同,这些工作假设数据是以连续轨迹的形式出现的,调查人员的工作将考虑只有离散数据点可用的现实情况。由于渐近分析要求时间范围趋于无穷大,因此观测值之间的时间间隔(或离散化步骤)对这些估计量在长时间内的准确性的影响尚不清楚,并且已知基于连续轨迹的估计量的朴素离散化可能导致错误的推断。该项目将引入适当的缩放框架来量化这种影响,并分析不同缩放制度中的误差。接下来,这些结果将被用来设计测试复合假设测试问题,使I型错误的概率迅速衰减,并在一类具有类似水平的I型错误的测试是渐近一致强大的。该项目的第二部分涉及决策这一重要议题。决策涉及(约束)最小化的适当的成本函数取决于模型参数。 由于后者的数量是未知的,数据驱动的版本,这样的最小化问题是必要的,在实践中。特别是,有必要构建合适的估计的成本函数,使决策的基础上,他们的最小化接近真实的决策。研究者将根据大偏差分析和第一部分的结果研究一种新方法,旨在保证在适当条件下以非常高的概率实现这一目标。第三部分致力于非参数学习的SDES。最后一部分福尔斯属于无限维学习理论的领域,其目标是学习基于PDE的模型的整个驱动函数,而不是估计有限维参数。为此,将开发贝叶斯技术与再生核希尔伯特空间理论相结合的严格计算框架,并研究由此产生的学习算法的理论特性。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。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Arnab Ganguly其他文献
A Variational Approach to Path Estimation and Parameter Inference of Hidden Diffusion Processes
隐扩散过程路径估计和参数推断的变分方法
- DOI:
- 发表时间:
2015 - 期刊:
- 影响因子:6
- 作者:
Tobias Sutter;Arnab Ganguly;H. Koeppl - 通讯作者:
H. Koeppl
A Linear Space Data Structure for Range LCP Queries
用于范围LCP查询的线性空间数据结构
- DOI:
- 发表时间:
2018 - 期刊:
- 影响因子:0.8
- 作者:
Arnab Ganguly;Manish Patil;Rahul Shah;Sharma V. Thankachan - 通讯作者:
Sharma V. Thankachan
Succinct Non-overlapping Indexing
- DOI:
10.1007/s00453-019-00605-5 - 发表时间:
2019-07-30 - 期刊:
- 影响因子:0.700
- 作者:
Arnab Ganguly;Rahul Shah;Sharma V. Thankachan - 通讯作者:
Sharma V. Thankachan
Context-Aware Design of Cyber-Physical Human Systems (CPHS)
信息物理人类系统的情境感知设计 (CPHS)
- DOI:
- 发表时间:
2020 - 期刊:
- 影响因子:0
- 作者:
S. Mukhopadhyay;Qun Liu;Edward Collier;Yimin Zhu;Ravindra Gudishala;Chanachok Chokwitthaya;Robert DiBiano;Alimire Nabijiang;Sanaz Saeidi;Subhajit Sidhanta;Arnab Ganguly - 通讯作者:
Arnab Ganguly
Categorical Range Reporting with Frequencies
带频率的分类范围报告
- DOI:
- 发表时间:
2019 - 期刊:
- 影响因子:0
- 作者:
Arnab Ganguly;J. Munro;Yakov Nekrich;Rahul Shah;Sharma V. Thankachan - 通讯作者:
Sharma V. Thankachan
Arnab Ganguly的其他文献
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{{ truncateString('Arnab Ganguly', 18)}}的其他基金
Complex Stochastic Systems and the Effect of Discretization
复杂随机系统和离散化的影响
- 批准号:
1855788 - 财政年份:2019
- 资助金额:
$ 20万 - 项目类别:
Standard Grant
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