IEEE Transactions on Information Theory
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Decision making formulated as finding a strategy that maximizes a utility function depends critically on knowing the problem parameters precisely. The obtained strategy can be highly sub-optimal and/or infeasible when parameters are subject to uncertainty, a typical situation in practice. Robust optimization, and more generally robust decision making, addresses this issue by treating uncertain parameters as an arbitrary element of a pre-defined set and solving solutions based on a worst-case analysis. In this thesis we contribute to two closely related fields of robust decision making.First, we address two limitations of robust decision making. Namely, a lack of theoretical justification and conservatism in sequential decision making. Specifically, we provide an axiomatic justification of robust optimization based on the MaxMin Expected Utility framework from decision theory. Furthermore, we propose three less conservative decision criteria for sequential decision making tasks, which include: (1) In uncertain Markov decision processes we propose an alternative formulation of the parameter uncertainty – the nested-set structured parameter uncertainty – and find the strategy that achieves maxmin expected utility to mitigate the conservatism of the standard robust Markov decision processes. (2) We investigate uncertain Markov decision processes where each strategy is evaluated comparatively by its gap to the optimum value. Two formulations, namely minimax regret and mean-variance tradeoff of the regret, were proposed and their computational cost studied. (3) We propose a novel Kalman filter design based on trading-off the likely performance and the robustness under parameter uncertainty. Second, we apply robust decision making into machine learning both theoretically and algorithmically. Specifically, on the theoretical front, we show that the concept of robustness is essential to “successful” learning. In particular, we prove that both SVM and Lasso are special cases of robust optimization, and such robustness interpretation implies consistency and sparsity naturally. We further establish a more general duality between robustness and generalizability – the former is a necessary and sufficient condition to the latter for an arbitrary learning algorithm – thus providing an answer to the fundamental question of what makes a learning algorithm work. On the algorithmic front, we propose novel robust learning algorithms that include (1) a robust classifier with controlled conservatism by extending robust SVM to a soft notion of robustness known as comprehensive robustness; (2) a High-dimensional Robust Principal Component Analysis (HR-PCA) algorithm for reducing dimensionality in the case that outlying observation exists and the dimensionality is comparable to the number of observations.