A one-measurement form of simultaneous perturbation stochastic approximation
The paper deals with the optimization problem to find \(\min L(\theta)\), where \(L(\cdot)\) is a real-valued function of a \(p\)-dimensional vector of parameters. It is assumed that the solution \(\theta^*\) can be found as root of the equation \[ g(\theta)= {{\partial L(\theta)}\over{\partial\theta}}=0 \] and, simultanously, it is assumed that only measurements of \(L(\theta)\) (typically with additive noise) are available. No direct measurements of \(g(\theta)\) (with or without noise) are available. Solution algorithms based on the gradient approximations obtained by two measurements of the loss function (regardless of the problem dimension) have been already introduced. This paper presents a form of such algorithm based on the gradient approximation obtained by one measurement of the loss function only. The new algorithm as well as the former algorithms based on two measurements (presented by the author) are analyzed in the paper. Moreover, their comparison is presented. A simple numerical example completes the paper.
scientific article; zbMATH DE number 987796