@Article{JCM-41-6,
author = {Wang, Jiani and Xiao, Wang and Zhang, Liwei},
title = {A Stochastic Newton Method for Nonlinear Equations},
journal = {Journal of Computational Mathematics},
year = {2023},
volume = {41},
number = {6},
pages = {1192--1221},
abstract = {<p style="text-align: justify;">In this paper, we study a stochastic Newton method for nonlinear equations, whose
exact function information is difficult to obtain while only stochastic approximations are
available. At each iteration of the proposed algorithm, an inexact Newton step is first
computed based on stochastic zeroth- and first-order oracles. To encourage the possible
reduction of the optimality error, we then take the unit step size if it is acceptable by an
inexact Armijo line search condition. Otherwise, a small step size will be taken to help
induce desired good properties. Then we investigate convergence properties of the proposed
algorithm and obtain the almost sure global convergence under certain conditions. We also
explore the computational complexities to find an approximate solution in terms of calls to
stochastic zeroth- and first-order oracles, when the proposed algorithm returns a randomly
chosen output. Furthermore, we analyze the local convergence properties of the algorithm
and establish the local convergence rate in high probability. At last we present preliminary
numerical tests and the results demonstrate the promising performances of the proposed
algorithm.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.2112-m2021-0072},
url = {https://global-sci.com/article/84161/a-stochastic-newton-method-for-nonlinear-equations}
}