Proceedings of the 2011 American Control Conference

Abstract

The SPSA (simultaneous perturbation stochastic approximation) method for function minimization developed in [15] is analyzed for optimization problems without measurement noise. We prove the striking result that under appropriate technical conditions the estimator sequence converges to the optimum with geometric rate with probability 1. Numerical experiments support the conjecture that the top Lyapunov-exponent of defined in terms of the SPSA method is smaller than the Lyapunov-exponent of its deterministic counterpart. \Ve conclude that randomization improves convergence rate while dramatically reducing the number of function evaluations. SPSA for noise-free optimization was briefly considered in [7]. It was shown there that under suitable technical conditions the rate of convergence for the Lq-norms of the estimations error is O(k-l/2), for any q > 1. In fact, in the noise-free case the SPSA procedure can be analyzed using results for Robbins-Monroe-type procedures. In particular, the asymptotic covariance of kl/2(O k-0") can be determined using classical results of [11]. It is easy to see that, due to the multiplicative effect of the noise, this asymptotic covariance is equal to zero. Hence a convergence rate faster than O(k-1/2) is expected. In fact, using the analysis of [7] in an inductive argument and exploiting the multiplicative nature of the noise it can be shown that the convergence rate is O(k-") for any finite m.