A modified second‐order SPSA optimization algorithm for finite samples

Abstract

AbstractWe propose a modification to the simultaneous perturbation stochastic approximation (SPSA) methods based on the comparisons made between the first‐ and second‐order SPSA (1SPSA and 2SPSA) algorithms from the perspective of loss function Hessian. At finite iterations, the accuracy of the algorithm depends on the matrix conditioning of the loss function Hessian. The error of 2SPSA algorithm for a loss function with an ill‐conditioned Hessian is greater than the one with a well‐conditioned Hessian. On the other hand, the 1SPSA algorithm is less sensitive to the matrix conditioning of loss function Hessians. The modified 2SPSA (M2SPSA) eliminates the error amplification caused by the inversion of an ill‐conditioned Hessian. This leads to significant improvements in its algorithm efficiency in problems with an ill‐conditioned Hessian matrix. Asymptotically, the efficiency analysis shows that M2SPSA is also superior to 2SPSA in a large parameter domain. It is shown that the ratio of the mean square errors for M2SPSA to 2SPSA is always less than one except for a perfectly conditioned Hessian or for an asymptotically optimal setting of the gain sequence. Copyright © 2002 John Wiley & Sons, Ltd.