A Derivative-Free Gauss-Newton Method

TL;DR

The paper introduces DFO-GN, a derivative-free Gauss-Newton method for nonlinear least-squares problems that simplifies existing models and offers comparable accuracy with faster runtime and better scalability.

Contribution

It presents a simplified derivative-free Gauss-Newton algorithm using linear models, improving runtime and scalability over previous methods.

Findings

DFO-GN achieves similar objective evaluation performance as prior methods.

DFO-GN has substantially faster runtime.

DFO-GN demonstrates improved scalability.

Abstract

We present DFO-GN, a derivative-free version of the Gauss-Newton method for solving nonlinear least-squares problems. As is common in derivative-free optimization, DFO-GN uses interpolation of function values to build a model of the objective, which is then used within a trust-region framework to give a globally-convergent algorithm requiring $O(\epsilon^{-2})$ iterations to reach approximate first-order criticality within tolerance $\epsilon$. This algorithm is a simplification of the method from [H. Zhang, A. R. Conn, and K. Scheinberg, A Derivative-Free Algorithm for Least-Squares Minimization, SIAM J. Optim., 20 (2010), pp. 3555-3576], where we replace quadratic models for each residual with linear models. We demonstrate that DFO-GN performs comparably to the method of Zhang et al. in terms of objective evaluations, as well as having a substantially faster runtime and improved scalability.