Sobol' sensitivity index estimators for stochastic models are functions of nested Monte Carlo estimators, which are estimators built from two nested Monte Carlo loops. The outer loop explores the input space and, for each of the explorations, the inner loop repeats model runs to estimate conditional expectations. Although the optimal allocation between explorations and repetitions of one's computational budget is well-known for nested Monte Carlo estimators, it is less clear how to deal with functions of nested Monte Carlo estimators, especially when those functions have unbounded Hessian matrices, as it is the case for Sobol' index estimators. To address this problem, a regularization method is introduced to bound the mean squared error of functions of nested Monte Carlo estimators. Based on a heuristic, an allocation strategy that seeks to minimize a bias-variance trade-off is proposed. The method is applied to Sobol' index estimators for stochastic models. A practical algorithm that adapts to the level of intrinsic randomness in the models is given and illustrated on numerical experiments.

翻译：对于随机模型的索博尔敏感性指数估计器，其本质是嵌套蒙特卡洛估计器的函数，这些估计器由两层嵌套的蒙特卡洛循环构建。外层循环探索输入空间，对于每次探索，内层循环重复运行模型以估计条件期望。尽管嵌套蒙特卡洛估计器在计算预算的探索与重复之间的最优分配已得到充分研究，但对于嵌套蒙特卡洛估计器的函数（特别是当这些函数具有无界海森矩阵时，如索博尔指数估计器）的处理方式尚不明确。为解决此问题，本文引入一种正则化方法以约束嵌套蒙特卡洛估计器函数的均方误差。基于启发式策略，提出一种旨在最小化偏差-方差权衡的分配方案。该方法被应用于随机模型的索博尔指数估计器，并给出一个能够自适应模型内在随机性水平的实用算法，通过数值实验加以验证。