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

Viet Anh Nguyen (), Soroosh Shafieezadeh-Abadeh (), Daniel Kuhn () and Peyman Mohajerin Esfahani ()
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Viet Anh Nguyen: Department of Management Science and Engineering, Stanford University, Stanford, California 94305
Soroosh Shafieezadeh-Abadeh: Tepper School of Business, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Daniel Kuhn: Risk Analytics and Optimization Chair, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Peyman Mohajerin Esfahani: Delft Center for Systems and Control, Delft University of Technology, 2628 CD Delft, Netherlands

Mathematics of Operations Research, 2023, vol. 48, issue 1, 1-37

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, by which the players’ optimal strategies are given by an affine estimator and a 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.

Keywords: Primary: 90-10; 90B99; distributionally robust optimization; minimum mean square error estimation; Wasserstein distance; affine estimator (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:48:y:2023:i:1:p:1-37

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