Abstract
A difficulty in using Simultaneous Perturbation Stochastics Approximation (SPSA) is its performance sensitivity to the step sizes chosen at the initial stage of the iteration. If the step size is too large, the solution estimate may fail to converge. The proposed adaptive stepping method automatically reduces the initial step size of the SPSA so that reduction of the objective function value occurs more reliably. Ten mathematical functions each with three different noise levels were used to empirically show the effectiveness of the proposed idea. A parameter estimation example of a nonlinear dynamical system is also included.
Highlights
Simultaneous Perturbation Stochastic Approximation (SPSA) (Spall 1992) is an optimization algorithm that uses only objective function measurements in the search of solutions
The number of iteration needed for convergence to the optimum is said to be more or less the same with Finite-Difference Stochastic Approximation (FDSA) (Kiefer and Wolfowitz 1952), which in essence is an approximate steepest-descent method that uses finite-differencing to approximate the partial derivatives along each of the D parameters
This paper provides a solution to determine the appropriate values of a by introducing an adaptive scheme as discussed in “Adaptive initial step sizes” section
Summary
Simultaneous Perturbation Stochastic Approximation (SPSA) (Spall 1992) is an optimization algorithm that uses only objective function measurements in the search of solutions. This paper provides a solution to determine the appropriate values of a by introducing an adaptive scheme as discussed in “Adaptive initial step sizes” section It does not require any additional objective function evaluations per iteration nor extra problem dependent parameters to set up. The SPSA with the proposed initial step size reduction, on the other hand, effectively mitigates this divergence problem producing smaller objective values in general as the (a priori) initial step size is increased This is because if the two function evaluations in the iteration are not smaller than the starting point value f (θ0), the algorithm will reduce the step size (by halving a) and restart at θb, which is the point that gave the smallest output in the history of iterations.
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