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

• 成果类型: Article
• 署名作者: Viet Anh Nguyen; Shafieezadeh-Abadeh, Soroosh; Kuhn, Daniel; Esfahani, Peyman Mohajerin
• 署名单位: Stanford University; Carnegie Mellon University; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Delft University of Technology
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1176
• 发表日期: 2023
• 页码: 1-37
• 关键词: conditional gradient algorithms; linear-systems; noise; MODEL; CONVERGENCE; distance; bounds; rates

摘要

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.