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

DOI10.1287/MOOR.2021.1176arXiv1911.03539OpenAlexW2985420217MaRDI QIDQ6118316

Author name not available (Why is that?)

Publication date: 23 February 2024

Published in: Mathematics of Operations Research (Search for Journal in Brave)

Abstract: 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 choosing a prior -- that is, a pair of signal and noise distributions ranging over independent Wasserstein balls -- with the goal to minimize and maximize the expected squared estimation error, respectively. We show that if the Wasserstein balls are centered at normal distributions, then the zero-sum game admits a Nash equilibrium, where the players' optimal strategies are given by an {em affine} estimator and a {em normal} prior, respectively. We further prove that this Nash equilibrium can be computed by solving a tractable convex program. Finally, we develop a Frank-Wolfe algorithm that can solve this convex program orders of magnitude faster than state-of-the-art general purpose solvers. We show that this algorithm enjoys a linear convergence rate and that its direction-finding subproblems can be solved in quasi-closed form.

Full work available at URL: https://arxiv.org/abs/1911.03539

zbMATH Keywords

Wasserstein distance minimum mean square error estimation distributionally robust optimization affine estimator

Mathematics Subject Classification ID

Mathematical programming (90Cxx)