An Efficient Finite-Difference Approximation

Estimating stochastic gradients is pivotal in fields like service systems in 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 with 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 and generating new samples at the estimated perturbation lead to an efficient finite-difference estimator. In numerical experiments, we apply the estimator in two examples, and numerical results demonstrate its robustness, as well as coincidence with the theory presented, especially in the case of small sample sizes.