An Approach to Robust Performance Analysis Using Optimal Transport

本研究的目标是研究一套全面的工具，通过建立一个框架来系统地评估建模错误的影响，从而实现强大的性能分析和决策。本研究的总体思路如下。随机模型几乎无处不在，其中许多模型都很方便，因为它们可以很容易地校准和/或因为性能分析或优化可以很容易地以封闭形式或算法完成。但我们都认识到，在模型的保真度（即它们复制现实的能力）和可追溯性之间存在权衡。这个项目研究了一种系统的方法，可以用来解释这种权衡的影响。PI研究了广泛的称为随机网络的模型，这些模型用于描述几乎任何存在资源争夺的概率系统。这些系统用于物流、运输、通信和系统风险等领域。PI计划将开发的方法应用于研究与大流量利用和此类系统中罕见事件的随机网络相关的鲁棒性问题。该项目研究了一套全面的工具，这些工具可以量化各种随机系统的性能分析和控制中的模型误差。研究者的策略结合了数学的各个领域，包括凸优化，概率论和蒙特卡罗方法。研究者将利用一般对偶结果，这些结果用于从基线概率模型（通常为可追溯性而选择）获得在一定公差范围内的所有概率模型中最坏情况期望的显式表达式。描述模型邻域的度量基于最优输运理论。这些结果适用于随机过程水平（对于在一般波兰空间上取值的随机元素），因此它们可用于近似复杂随机系统的样本路径期望。程序中的一个关键因素是，给定事件的最坏情况概率可以根据基线（可处理）模型下修改（显式）事件的概率显式表示。研究者将研究与随机网络的罕见事件分析和大流量近似相关的广泛问题，这些问题广泛应用于通信网络、呼叫中心、制造系统和化学反应网络等应用领域。该奖项反映了美国国家科学基金会的法定使命，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。

The goal of this research is to investigate a comprehensive set of tools to enable robust performance analysis and decision making by building a framework which systematically evaluates the impact of modeling errors. The general philosophy of this research is as follows. Stochastic models are used virtually everywhere and many of these models are convenient because they can be easily calibrated and/or because performance analysis or optimization can be easily done in closed form or algorithmically. But we all recognize that there are trade-offs between the model's fidelity (i.e. their ability to replicate reality) and its tractability. This project investigates a systematic approach which can be used to account for the impact of this trade-off. The PI studies a wide range of models called stochastic networks, which are used to describe virtually any probabilistic system in which there is resource contention. These systems are used in logistics, transportation, communications and systemic risk, among others. The PI plans to apply the developed approach to study robustness questions related to stochastic networks in heavy-traffic utilization and rare events in such systems. This project investigates a comprehensive set of tools which enables the quantification of model errors in the performance analysis and control of a wide range of stochastic systems. The investigator's strategy combines various areas of mathematics, including convex optimization, probability theory, and Monte Carlo methods. The investigator will exploit general duality results which are used to obtain explicit expressions for worst-case expectations among all probability models within a certain tolerance from a baseline probabilistic model (typically chosen for tractability). The metric describing the neighborhood of models is based on optimal transport theory. These results are applicable at the stochastic-process level (for random elements taken values on general Polish spaces), so they can be used to approximate sample-path expectations of complex stochastic systems. A key element in the program is that the worst-case probability of a given event can be expressed explicitly in terms of the probability of a modified (explicit) event under the baseline (tractable) model. The investigator will study a wide range of questions related to rare-event analysis and heavy-traffic approximations of stochastic networks, which are widely used in application areas such as communication networks, call centers, manufacturing systems, and chemical reaction networks, among others.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.