Matrix Conditioning and Adaptive Simultaneous Perturbation Stochastic Approximation Method

This paper proposes a modification to the simultaneous per tu rba t ion stochastic approximation (SPSA) methods based on the comparisons made between the first o rder and the second order SPSA (1SPSA and 2SPSA) algori thms f rom the perspective of loss function Hessian. At finite iterations, the convergence rate depends on the matr ix conditioning of the loss function Hessian. It is shown that 2SPSA converges more slowly for a loss function with an ill-conditioned Hessian than the one with a well-conditioned Hessian. On the other hand, the convergence rate of 1SPSA is less sensitive to the matr ix conditioning of loss function Hessians. The modified 2SPSA (M2SPSA) eliminates the e r ro r amplification caused by the inversion of an ill-conditioned Hessian at finite iterations which leads to significant improvements in its convergence rate in problems with an ill-conditioned Hessian matrix. Asymptotically, the efficiency analysis shows that M2SPSA is also superior to 2SPSA in terms of their convergence rate coefficients. It is shown that for the same asymptotic convergence rate, the ratio of the mean square er rors for M2SPSA to 2SPSA is always less than one except for a perfectly conditioned Hessian.