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

TL;DR

This paper develops a Wasserstein distributionally robust estimation framework that finds optimal affine estimators under normal distribution ambiguity sets, with efficient algorithms for practical computation.

Contribution

It introduces a novel game-theoretic model linking Bayesian and minimax estimation via Wasserstein ambiguity sets, and provides a tractable convex program with a fast algorithm for solution.

Findings

Nash equilibrium characterized by affine estimator and normal prior.

Convex program for equilibrium computation is tractable.

Frank-Wolfe algorithm achieves linear convergence and is computationally efficient.

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.