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We develop Distributionally Robust Optimization (DRO) formulations forMultivariate Linear Regression (MLR) and Multiclass Logistic Regression (MLG)when both the covariates and responses/labels may be contaminated by outliers.The DRO framework uses a probabilistic ambiguity set defined as a ball ofdistributions that are close to the empirical distribution of the training setin the sense of the Wasserstein metric. We relax the DRO formulation into aregularized learning problem whose regularizer is a norm of the coefficientmatrix. We establish out-of-sample performance guarantees for the solutions toour model, offering insights on the role of the regularizer in controlling theprediction error. Experimental results show that our approach improves thepredictive error by 7% -- 37% for MLR, and a metric of robustness by 100% forMLG.

Robustified Multivariate Regression and Classification Using Distributionally Robust Optimization under the Wasserstein Metric