Bridging Bayesian and Minimax Mean Square Error Estimation via Wasserstein Distributionally Robust Optimization
This work addresses robust estimation for statisticians and engineers by bridging Bayesian and minimax approaches, offering a computationally efficient solution, though it is incremental as it builds on existing distributionally robust optimization methods.
The paper tackles the problem of estimating an unknown signal from noisy observations by introducing a distributionally robust minimax mean square error estimation model with a Wasserstein ambiguity set, showing that under normal distributions, it admits a Nash equilibrium with an affine estimator and normal prior that can be computed via a tractable convex program solved orders of magnitude faster by a Frank-Wolfe algorithm with linear convergence.
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.