Reliable Decision-Making Under Uncertainty Through The Lens of Statistics and Optimization
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Wang, Jie
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Abstract
In this thesis, we develop computationally efficient algorithms with statistical guarantees for problems of decision-making under uncertainty, particularly in the presence of large-scale, noisy, and high-dimensional data. In Chapter 2, we propose a kernelized projected Wasserstein distance for high-dimensional hypothesis testing, which finds the nonlinear mapping that maximizes the discrepancy between projected distributions. In Chapter 3, we provide an in-depth analysis of the computational and statistical guarantees of the kernelized projected Wasserstein distance. In Chapter 4, we study the variable selection problem in two-sample testing, aiming to select the most informative variables to determine whether two datasets follow the same distribution. In Chapter 5, we present a novel framework for distributionally robust stochastic optimization (DRO), which seeks an optimal decision that minimizes expected loss under the worst-case distribution within a specified set. This worst-case distribution is modeled using a variant of the Wasserstein distance based on entropic regularization. In Chapter 6, we incorporate Phi-divergence regularization into the infinity-type Wasserstein DRO, which is a formulation particularly useful for adversarial machine learning tasks. Chapter 7 concludes with an overview of promising future research directions.
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2025-04-15
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Dissertation