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We consider the problem of learning the exact skeleton of general discrete Bayesian networks from potentially corrupted data. Building on distributionally robust optimization and a regression approach, we propose to optimize the most adverse risk over a family of distributions within bounded Wasserstein distance or KL divergence to the empirical distribution. The worst-case risk accounts for the effect of outliers. The proposed approach applies for general categorical random variables without assuming faithfulness, an ordinal relationship or a specific form of conditional distribution. We present efficient algorithms and show the proposed methods are closely related to the standard regularized regression approach. Under mild assumptions, we derive non-asymptotic guarantees for successful structure learning with logarithmic sample complexities for bounded-degree graphs. Numerical study on synthetic and real datasets validates the effectiveness of our method. 

Structured prediction of tree-shaped objects is heavily studied under the name of syntactic dependency parsing. Current practice based on maximum likelihood or margin is either agnostic to or inconsistent with the evaluation loss. Risk minimization alleviates the discrepancy between training and test objectives but typically induces a non-convex problem. These approaches adopt explicit regularization to combat overfitting without probabilistic interpretation. We propose a momentbased distributionally robust optimization approach for tree structured prediction, where the worst-case expected loss over a set of distributions within bounded moment divergence from the empirical distribution is minimized. We develop efficient algorithms for arborescences and other variants of trees. We derive Fisher consistency, convergence rates and generalization bounds for our proposed method. We evaluate its empirical effectiveness on dependency parsing benchmarks. 

We consider the problem of learning the underlying structure of a general discrete pairwise Markov network. Existing approaches that rely on empirical risk minimization may perform poorly in settings with noisy or scarce data. To overcome these limitations, we propose a computationally efficient and robust learning method for this problem with near-optimal sample complexities. Our approach builds upon distributionally robust optimization (DRO) and maximum conditional log-likelihood. The proposed DRO estimator minimizes the worst-case risk over an ambiguity set of adversarial distributions within bounded transport cost or f-divergence of the empirical data distribution. We show that the primal minimax learning problem can be efficiently solved by leveraging sufficient statistics and greedy maximization in the ostensibly intractable dual formulation. Based on DRO’s approximation to Lipschitz and variance regularization, we derive near-optimal sample complexities matching existing results. Extensive empirical evidence with different corruption models corroborates the effectiveness of the proposed methods.