A FAST Method for Nested Estimation

Abstract

Nested estimation involves estimating an expectation of a function of a conditional expectation and has many important applications in operations research and machine learning. Nested simulation is a classic approach to this estimation, and the convergence rate of the mean squared error (MSE) of nested simulation estimators is only of order [Formula: see text], where Γ is the simulation budget. To accelerate the convergence, in this paper, we establish a jackkniFe-bAsed neSted simulaTion (FAST) method for nested estimation, and a unified theoretical analysis for general functions in the nested estimation shows that the MSE of the proposed method converges at the faster rate of [Formula: see text] or even [Formula: see text]. We also provide an efficient algorithm that ensures the estimator’s MSE decays at its optimal rate in practice. In numerical experiments, we apply the proposed estimator in portfolio risk measurement and Bayesian experimental design in operations research and machine learning areas, respectively, and numerical results are consistent with the theory presented. History: Accepted by Bruno Tuffin, Area Editor for Simulation. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72031006, 72101260, and 72394375]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0118 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0118 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .