An Efficient Finite Difference Approximation via a Double Sample-Recycling Approach
CoRR(2024)
Abstract
Estimating stochastic gradients is pivotal in fields like service systems
within operations research. The classical method for this estimation is the
finite difference approximation, which entails generating samples at perturbed
inputs. Nonetheless, practical challenges persist in determining the
perturbation and obtaining an optimal finite difference estimator in the sense
of possessing the smallest mean squared error (MSE). To tackle this problem, we
propose a double sample-recycling approach in this paper. Firstly, pilot
samples are recycled to estimate the optimal perturbation. Secondly, recycling
these pilot samples again and generating new samples at the estimated
perturbation, lead to an efficient finite difference estimator. We analyze its
bias, variance and MSE. Our analyses demonstrate a reduction in asymptotic
variance, and in some cases, a decrease in asymptotic bias, compared to the
optimal finite difference estimator. Therefore, our proposed estimator
consistently coincides with, or even outperforms the optimal finite difference
estimator. In numerical experiments, we apply the estimator in several
examples, and numerical results demonstrate its robustness, as well as
coincidence with the theory presented, especially in the case of small sample
sizes.
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