Stability Analysis of Newton-MR Under Hessian Perturbations

Recently, stability of Newton-CG under Hessian perturbations, i.e., inexact curvature information, have been extensively studied. Such stability analysis has subsequently been leveraged in designing variants of Newton-CG in which, to reduce the computational costs involving the Hessian matrix, the curvature is suitably approximated. Here, we do that for Newton-MR, which extends Newton-CG in the same manner that MINRES extends CG. Unlike the stability analysis of Newton-CG, which relies on spectrum preserving perturbations in the sense of L\"{o}wner partial order, our work here draws from matrix perturbation theory to estimate the distance between the underlying exact and perturbed sub-spaces. Numerical experiments demonstrate great degree of stability for Newton-MR, amounting to a highly efficient algorithm in large-scale problems.

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