In this paper, we examine the Sample Average Approximation (SAA) procedure within a framework where the Monte Carlo estimator of the expectation is biased. We also introduce Multilevel Monte Carlo (MLMC) in the SAA setup to enhance the computational efficiency of solving optimization problems. In this context, we conduct a thorough analysis, exploiting Cram\'er's large deviation theory, to establish uniform convergence, quantify the convergence rate, and determine the sample complexity for both standard Monte Carlo and MLMC paradigms. Additionally, we perform a root-mean-squared error analysis utilizing tools from empirical process theory to derive sample complexity without relying on the finite moment condition typically required for uniform convergence results. Finally, we validate our findings and demonstrate the advantages of the MLMC estimator through numerical examples, estimating Conditional Value-at-Risk (CVaR) in the Geometric Brownian Motion and nested expectation framework.

翻译：本文在蒙特卡洛期望估计存在偏差的框架下，研究了样本平均逼近（SAA）方法。我们进一步在SAA框架中引入多水平蒙特卡洛（MLMC）方法，以提升求解优化问题的计算效率。在此背景下，我们利用克拉默大偏差理论进行了深入分析，为标准蒙特卡洛和MLMC范式建立了一致收敛性，量化了收敛速度，并确定了样本复杂度。此外，我们借助经验过程理论的工具进行了均方根误差分析，从而推导出不依赖于一致收敛结果通常所需的有限矩条件的样本复杂度。最后，我们通过数值算例验证了研究结果，并在几何布朗运动和嵌套期望框架下估计条件风险价值（CVaR），展示了MLMC估计器的优势。