Formal Comparison of Simultaneous Perturbation Stochastic Approximation and Random Direction Stochastic Approximation ∗

Stochastic approximation (SA) algorithms can be used in system optimization problems when only noisy measurements of a system are available. This paper formally compares the performance of two popular SA algorithms in a multivariate Kiefer-Wolfowitz setting of simultaneous-perturbation SA (SPSA) and the random-directions SA (RDSA). This paper provides sufficient conditions to demonstrate which algorithm has the smaller asymptotic mean squared error (MSE) and numerically presents comparison of SPSA and RDSA in a test function and a model-free control system. The theory and supporting numerics indicate that SPSA has better efficiency (lower MSE) across a broad range of problem settings.