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

Viet Anh Nguyen (viet-anh.nguyen***at***stanford.edu)
Soroosh Shafieezadeh-Abadeh (soroosh.shafiee***at***epfl.ch)
Daniel Kuhn (daniel.kuhn***at***epfl.ch)
Peyman Mohajerin Esfahani (P.MohajerinEsfahani***at***tudelft.nl)

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 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: Wasserstein distance, mean square error, affine estimator

Category 1: Stochastic Programming

Category 2: Infinite Dimensional Optimization

Category 3: Applications -- Science and Engineering (Statistics )

Citation:

Download: [PDF]

Entry Submitted: 11/08/2019
Entry Accepted: 11/11/2019
Entry Last Modified: 11/12/2019

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