Bridging Bayesian and Minimax Mean Square Error Estimation via Wasserstein Distributionally Robust Optimization

We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator—that is, a measurable function of the observation—and a fictitious adversary...

Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

We introduce a distributionally robust maximum likelihood estimation model with a Wasserstein ambiguity set to infer the inverse covariance matrix of a p-dimensional Gaussian random vector from n independent samples. The proposed model minimizes the worst case (maximum) of Stein’s loss across all normal reference distributions within a...

From data to decisions: Distributionally robust optimization is optimal

We study stochastic programs where the decision maker cannot observe the distribution of the exogenous uncertainties but has access to a finite set of independent samples from this distribution. In this setting, the goal is to find a procedure that transforms the data to an estimate of the expected cost function under the unknown data...

From static to dynamic anomaly detection with application to power system cyber security

Developing advanced diagnosis tools to detect cyber attacks is the key to security of power systems. It has been shown that multivariate data injection attacks can bypass bad data detection schemes typically built on static behavior of the systems, which misleads operators to disruptive decisions. In this article, we depart from the existing...

Regularization via mass transportation

The goal of regression and classification methods in supervised learning is to minimize the empirical risk, that is, the expectation of some loss function quantifying the prediction error under the empirical distribution. When facing scarce training data, overfitting is typically mitigated by adding regularization terms to the objective that...