A deterministic analysis of stochastic approximation with randomized directions

We study the convergence of two stochastic approximation algorithms with randomized directions: the simultaneous perturbation stochastic approximation algorithm and the random direction Kiefer-Wolfowitz algorithm. We establish deterministic necessary and sufficient conditions on the random directions and noise sequences for both algorithms, and these conditions demonstrate the effect of the "random" directions on the "sample-path" behavior of the algorithms studied. We discuss ideas for further research in the analysis and design of these algorithms.