A Distributionally Robust Optimization Approach for Multivariate Linear Regression under the Wasserstein Metric

We present a Distributionally Robust Optimization (DRO) approach for Multivariate Linear Regression (MLR), where multiple correlated response variables are to be regressed against a common set of predictors. We develop a regularized MLR formulation that is robust to large perturbations in the data, where the regularizer is the dual norm of the regression coefficient matrix in the sense of a newly defined matrix norm. We establish bounds on the prediction bias of the solution, offering insights on the role of the regularizer in controlling the prediction error. Experimental results show that, compared to a number of popular MLR methods, our approach leads to a lower out-of-sample Mean Squared Error (MSE) in various scenarios.