Optimal random perturbations for stochastic approximation using a simultaneous perturbation gradient approximation

• Published in: IEEE transactions on automatic control Vol. 43; no. 10; pp. 1480 - 1484
• Main Authors: Sadegh, P., Spall, J.C.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.1998; Institute of Electrical and Electronics Engineers
• Subjects: Applied sciences; Automatic control; Computer science; control theory; systems; Constraint optimization; Control systems; Control theory. Systems; Discrete time systems; Exact sciences and technology; Linear matrix inequalities; Mathematics; Operational research and scientific management; Operational research. Management science; Optimal control; Optimization. Search problems; Riccati equations; State feedback; Stochastic processes; Stochastic approximation; Algorithm; Optimal control; Optimization
• ISSN: 0018-9286
• DOI: 10.1109/9.720513

The simultaneous perturbation stochastic approximation (SPSA) algorithm has attracted considerable attention for challenging optimization problems where it is difficult or impossible to obtain a direct gradient of the objective (say, loss) function. The approach is based on a highly efficient simultaneous perturbation approximation to the gradient based on loss function measurements. SPSA is based on picking a simultaneous perturbation (random) vector in a Monte Carlo fashion as part of generating the approximation to the gradient. This paper derives the optimal distribution for the Monte Carlo process. The objective is to minimize the mean square error of the estimate. The authors also consider maximization of the likelihood that the estimate be confined within a bounded symmetric region of the true parameter. The optimal distribution for the components of the simultaneous perturbation vector is found to be a symmetric Bernoulli in both cases. The authors end the paper with a numerical study related to the area of experiment design.