Passive stochastic approximation with averaging along trajectories
The paper deals with the problem of estimating a root \(x^*\) of the regression equation \(f(x)=0\) on the basis of measurements \[ y_n=f(x_n)+ \xi_n,\qquad n=1,2,\dots \] (\(f(\cdot):\mathbb{R}^1\to \mathbb{R}^1\); \(\{\xi_n\}\) is a random sequence corresponding to a noise). To solve this problem the authors propose a two-step recursive algorithm defined by the relations \[ \begin{aligned} z_{n+1}&= P_Q\Biggl\{ z_n-\gamma_n h_n^{-1} \varphi(y_n) K\biggl( {{x_n-z_n}\over{h_n}} \biggr)\Biggr\},\\ \overline{z}_{n+1}&= \overline{z}_n- (n+1)^{-1} (\overline{z}_n-z_n),\end{aligned} \] where \(P_Q\) is the operator of the projection onto the corresponding set \(Q\); \(\varphi(\cdot): \mathbb{R}^1\to\mathbb{R}^1\) is a monotonic, nondecreasing function; \(K(\cdot)\) is a kernel function. The assumptions under which \(\overline{z}_n= {1\over n}\sum^{n-1}_{t=0} z_t\to x^*\) are introduced in the paper and, moreover, the convergence rate is also investigated there. The paper follows some previous papers on a similar topic.
scientific article; zbMATH DE number 885290