Adaptive system optimization using (simultaneous) random directions stochastic approximation

We present the first adaptive random directions Newton algorithm under i.i.d., symmetric, uniformly distributed perturbations for a general problem of optimization under noisy observations. We also present a simple gradient search scheme under the aforementioned perturbation random variates. Our Newton algorithm requires generating N perturbation variates and three simulations at each iteration unlike the well studied simultaneous perturbation Newton search algorithm of Spall [2000] that requires 2N iterates and four simulations. We prove the convergence of our algorithms to a local minimum and also present rate of convergence results. Our asymptotic mean square errors (AMSE) analysis indicates that our gradient algorithm requires 60% less number of simulations to achieve a given accuracy as compared to a similar algorithm in Kushner and Clark [1978], Chin [1997] that incorporates Gaussian perturbations. Moreover, our adaptive Newton search algorithm results in an AMSE that is on par and sometimes even better than the Newton algorithm of Spall [2000]. Our experiments are seen to validate the theoretical observations. In particular, our experiments show that our Newton algorithm 2RDSA requires only 75% of the total number of loss function measurements as required by the Newton algorithm of Spall [2000] while providing the same accuracy levels as the latter.