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 Wasserstein ambiguity set to recover an unknown signal from noisy observation. The proposed can be viewed as zero-sum game between statistician choosing estimator—that is, measurable function of the observation—and fictitious adversary prior—that pair and noise distributions ranging over independent balls—with goal minimize maximize expected squared error, respectively. show that, if balls are centered at normal distributions, then admits Nash equilibrium, by which players’ optimal strategies given affine estimator prior, further prove that this equilibrium computed solving tractable convex program. Finally, we develop Frank–Wolfe algorithm solve program orders magnitude faster than state-of-the-art general-purpose solvers. enjoys linear convergence rate its direction-finding subproblems solved in quasi-closed form. Funding: This research was supported Swiss National Science Foundation [Grants BSCGI0_ 157733 51NF40_180545], Early Postdoc.Mobility Fellowship [Grant P2ELP2_195149], European Research Council TRUST-949796].