Anderson Acceleration For Perturbed Newton Methods

TL;DR

This paper develops a convergence theory for Anderson acceleration applied to perturbed Newton methods, including Levenberg-Marquardt, showing improved local convergence rates and adaptive safeguarding in solving nonlinear equations.

Contribution

It introduces a novel convergence analysis for Anderson acceleration with safeguarding on perturbed Newton methods, extending to Levenberg-Marquardt and demonstrating adaptive step tuning.

Findings

Anderson acceleration improves convergence rates for perturbed Newton methods.

Safeguarding detects superlinear convergence and adjusts steps accordingly.

The methods outperform standard approaches on benchmark problems.

Abstract

We present a convergence theory for Anderson acceleration (AA) applied to perturbed Newton methods (pNMs) for computing roots of nonlinear problems. Two important special cases are the classical Newton method and the Levenberg-Marquardt method. We prove that if a problem is 2-regular, then Anderson accelerated pNMs coupled with a safeguarding scheme, known as $\gamma$-safeguarding, converge locally linearly in a starlike domain of convergence, but with an improved rate of convergence compared to standard perturbed Newton methods. Since Levenberg-Marquardt methods are a special case of pNMs, we obtain a novel acceleration and local convergence result for Anderson accelerated Levenberg-Marquardt. We further show that $\gamma$-safeguarding can detect if the underlying perturbed Newton method is converging superlinearly, and respond by tuning the Anderson step down. We demonstrate the methods on several benchmark problems in the literature.