Optimality in problems and optimization algorithms under indeterminacy (Q1083375)

Considered is the minimization of a function \(J(x)\) on \({\mathbb{R}}^ r\) assuming that only estimates \(y(n,x)\) of the gradient \(\nabla J(x)\) can be obtained having one of the forms (a) \(y(n,x)=\nabla J(x)+\xi_ 1\) (additive noise) or (b) \(y(n,x)=Diag(\xi_ n)\nabla J(x)\) (multiplicative noise), where \((\xi_ n)\) is a sequence of independent, identically distributed, zero mean random r-vectors. Replacing \(y(n,x)\) by \(\tilde y(n,x)=\phi(y(n,x))\), formulas are given for the transformation \(\phi\) minimizing the asymptotic error covariance matrix of the resulting stochastic gradient procedure with the transformed gradient ỹ(n,x) and having an optimal gain matrix of the type \(\Gamma_ 0(n)=n^{-1}\Gamma_ 0\).