Minimax asymptotics
Mika Meitz, Alexander Shapiro
TL;DR
This work addresses the problem of understanding the asymptotic behavior of estimators in parametric minimax problems, focusing on the limiting distributions of the optimal value and the optimal solutions. It develops a sensitivity-analysis framework for parameterized optimization problems and applies a functional delta method together with a quadratic-approximation approach to derive asymptotics. The key finding is that the limiting distributions are typically non-Gaussian and depend on the multiplicity of optical points and Lagrange multipliers, with Gaussian limits arising only in simple singleton or convex-concave cases, providing explicit representations in terms of gradients and Hessians at the optimum. These results enable statistical inference for minimax estimators and connect to broader minimax contexts such as empirical likelihood and distributionally robust optimization, offering practical guidance for constructing confidence sets and hypothesis tests in minimax settings.
Abstract
In this paper, we consider asymptotics of the optimal value and the optimal solutions of parametric minimax estimation problems. Specifically, we consider estimators of the optimal value and the optimal solutions in a sample minimax problem that approximates the true population problem and study the limiting distributions of these estimators as the sample size tends to infinity. The main technical tool we employ in our analysis is the theory of sensitivity analysis of parameterized mathematical optimization problems. Our results go well beyond the existing literature and show that these limiting distributions are highly non-Gaussian in general and normal in simple specific cases. These results open up the way for the development of statistical inference methods in parametric minimax problems.