  We consider stochastic approximations which arise from such applications as
data communications and image processing. We demonstrate why constraints are
needed in a stochastic approximation and how a constrained approximation can be
incorporated into a preconditioning technique to derive the pre-conditioned
stochastic gradient method (PSGM). We perform convergence analysis to show that
the PSGM converges to the theoretical best approximation under some simple
assumptions on the preconditioner and on the independence of samples drawn from
a stochastic process. Simulation results are presented to demonstrate the
effectiveness of the constrained and precondi-tioned stochastic gradient
method.
