Batching Adaptive Variance Reduction
针对自适应蒙特卡洛方差缩减中初始化阶段耗时不确定的问题，提出一种批处理程序，并给出学习率公式以最小化理论方差上界，适用于有限计算预算下的模拟。
Adaptive Monte Carlo variance reduction is an effective framework for running a Monte Carlo simulation along with a parameter search algorithm for variance reduction, whereas an initialization step is required for preparing problem parameters in some instances. In spite of the effectiveness of adaptive variance reduction in various fields of application, the length of the preliminary phase has often been left unspecified for the user to determine on a case-by-case basis, much like in typical sequential frameworks. This uncertain element may possibly be even fatal in realistic finite-budget situations, since the pilot run may take most of the budget, or possibly use up all of it. To unnecessitate such an ad hoc initialization step, we develop a batching procedure in adaptive variance reduction, and provide an implementable formula of the learning rate in the parameter search which minimizes an upper bound of the theoretical variance of the empirical batch mean. We analyze decay rates of the minimized upper bound towards the minimal estimator variance with respect to the predetermined computing budget, and provide convergence results as the computing budget increases progressively when the batch size is fixed. Numerical examples are provided to support theoretical findings and illustrate the effectiveness of the proposed batching procedure.