A Comparison of the Finite Difference and Simultaneous Perturbation Gradient Estimation Methods with Noisy Function Evaluations

Abstract

Gradient information is useful in many applications such as optimization and sensitivity analysis, but is often inaccessible, providing a need for gradient estimation methods. This paper presents a comparison between the finite difference (FD) and simultaneous perturbation (SP) methods for gradient estimation. In practical experiments, function evaluations correspond to incurred costs, so the number of function evaluations used to form an estimate must be taken into account. Our theoretical results, supported by our numerical experiments, show that under certain circumstances the SP estimate has a smaller mean squared error (MSE) given a fixed number of function evaluations, and that the benefit gained from the SP method becomes more pronounced as the observation environment becomes noisier. We also discuss the performance of both methods in the noise-free case. We summarize guidelines for practitioners to determine which method is preferred, depending on the dimension of the function, noise magnitude, underlying gradient magnitude, and number of function evaluations available.