Distributionally Robust Structural Learning

Decision-making under uncertainty is common in various areas of study. Structural learning is a decision problem that involves seeking the optimal structure typically from an exponential number of structures. The task is usually performed on a finite set of samples observed from uncertain environments, which may be subject to unexpected contamination thus unreliable. The combinatorial nature and uncertainty pose challenges to relevant algorithms, particularly in the large-scale setting. We suggest that a successful structural learning method should have low time complexity, high sample efficiency, estimator consistency and robustness at the same time. In this dissertation, we propose a statistical learning framework that fulfills these requirements to tackle several structural learning problems based on techniques in the emerging fields of distributionally robust optimization (DRO). Our models hedge against a set of distributions consistent with data in terms of certain a priori assumptions. The set constitutes our uncertainty about the underlying data-generating mechanism and can be constructed in a flexible way. We establish desirable theoretical guarantees and put forward practical algorithms for specific learning problems with judiciously chosen uncertainty sets.