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Bridging Bayesian and Minimax Mean Square Error Estimation via Wasserstein Distributionally Robust Optimization
- Source :
- Mathemtics of Operations Research, Mathematics of Operations Research, 48(1)
- Publication Year :
- 2023
-
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. Funding: This research was supported by the Swiss National Science Foundation [Grants BSCGI0_ 157733 and 51NF40_180545], an Early Postdoc.Mobility Fellowship [Grant P2ELP2_195149], and the European Research Council [Grant TRUST-949796].
- Subjects :
- FOS: Computer and information sciences
Computer Science - Machine Learning
distributionally robust optimization
Wasserstein metric
General Mathematics
Machine Learning (stat.ML)
Mathematics - Statistics Theory
02 engineering and technology
Statistics Theory (math.ST)
Management Science and Operations Research
01 natural sciences
Machine Learning (cs.LG)
Statistics - Machine Learning
0202 electrical engineering, electronic engineering, information engineering
FOS: Mathematics
Wasserstein distance
0101 mathematics
Mathematics - Optimization and Control
010102 general mathematics
Statistics
020206 networking & telecommunications
minimum mean square error estimation
Computer Science Applications
Convex optimization
Optimization and Control (math.OC)
affine estimator
Subjects
Details
- Language :
- English
- ISSN :
- 0364765X
- Volume :
- 48
- Issue :
- 1
- Database :
- OpenAIRE
- Journal :
- Mathematics of Operations Research
- Accession number :
- edsair.doi.dedup.....272fa409704c8a3763848cc9394e4430