The integrated transportation distance between Markov kernels: theory, optimization, and applications in risk evaluation and machine learning
Creators: name: Lin, Zhengqi
Publisher: name: No Publisher Supplied
Year: 2024

description: This thesis explores the properties and applications of the Integrated Transportation Distance (ITD), a novel metric that quantifies the dissimilarity between probability (Markov) kernels on Polish spaces. The ITD is defined by integrating the Wasserstein distance between conditional probability measures (the values of the kernels) with respect to a given marginal measure. This construction enables a comprehensive comparison of composite probability distributions, considering both the measures' marginal and conditional aspects. We establish the metric properties of the ITD and relate it to various modes of convergence in the space of kernels. Furthermore, we investigate the continuity of the ITD with respect to the marginal measures. Under suitable assumptions on the kernels, such as Lipschitz continuity, we prove that the ITD exhibits Lipschitz continuity properties. These results have important implications for the stability and robustness of the ITD, guaranteeing that small perturbations in the marginal measures and conditional kernels lead to controlled changes in the distance itself. Additionally, we demonstrate that this metric serves as the foundation for an efficient approximation technique, enabling the replacement of the original system's kernel by a kernel with a discrete support of limited cardinality. To facilitate practical implementation, we present a specialized Dual Subgradient Algorithm capable of constructing these approximate kernels quickly and efficiently, without requiring computationally expensive matrix operations. This advancement offers new possibilities for the streamlined analysis and manipulation of stochastic systems represented by kernels. The findings of this research contribute to a deeper understanding of the properties and behavior of the ITD, showcasing its potential as a powerful tool in optimal transport theory and related fields. The ITD offers a promising framework for analyzing and comparing complex probability distributions in various applications. Among them, we highlight the significant contributions of the ITD in Dynamic Risk Evaluation in Markov Systems, Distributed Machine Learning, and Federated Learning. Based on the ITD, we propose a method for approximating solutions to multi-time stage forward-backward Markov systems and develop a particle-based numerical procedure with finite support sets. The ITD is applied to build a recursive approximation scheme for the forward system of a Markov process and estimate the error of the risk evaluation by the errors of individual kernel approximations. This methodology is illustrated by stopping problems and well-known risk measures. We apply the proposed approach to pricing American basket options in financial problems. Furthermore, we illustrate the potential of utilizing the ITD in Federated Learning (FL). It is a Distributed Machine Learning (ML) framework where diverse clients or users collaborate to train a model on a central server while upholding the decentralized nature of sensitive client data. Crucial characteristics of FL include its emphasis on statistical heterogeneity and privacy preservation. We demonstrate that the ITD is perfectly suitable for approximating heterogeneous devices' data distributions, particularly in scenarios involving a large number of devices. In particular, the Dual Subgradient Algorithm is specifically modified for privacy-preserving settings, enabling sensitive data to remain on local devices. This Federated Dual Subgradient Algorithm also finds immediate applications in clustering tasks under the federated setting.