TY - GEN

T1 - Riemannian trust regions with finite-difference hessian approximations are globally convergent

AU - Boumal, Nicolas

PY - 2015/1/1

Y1 - 2015/1/1

N2 - The Riemannian trust-region algorithm (RTR) is designed to optimize differentiable cost functions on Riemannian manifolds. It proceeds by iteratively optimizing local models of the cost function. When these models are exact up to second order, RTR boasts a quadratic convergence rate to critical points. In practice, building such models requires computing the Riemannian Hessian, which may be challenging. A simple idea to alleviate this difficulty is to approximate the Hessian using finite differences of the gradient. Unfortunately, this is a nonlinear approximation, which breaks the known convergence results for RTR. We propose RTR-FD: a modification of RTR which retains global convergence when the Hessian is approximated using finite differences. Importantly, RTR-FD reduces gracefully to RTR if a linear approximation is used. This algorithm is available in the Manopt toolbox.

AB - The Riemannian trust-region algorithm (RTR) is designed to optimize differentiable cost functions on Riemannian manifolds. It proceeds by iteratively optimizing local models of the cost function. When these models are exact up to second order, RTR boasts a quadratic convergence rate to critical points. In practice, building such models requires computing the Riemannian Hessian, which may be challenging. A simple idea to alleviate this difficulty is to approximate the Hessian using finite differences of the gradient. Unfortunately, this is a nonlinear approximation, which breaks the known convergence results for RTR. We propose RTR-FD: a modification of RTR which retains global convergence when the Hessian is approximated using finite differences. Importantly, RTR-FD reduces gracefully to RTR if a linear approximation is used. This algorithm is available in the Manopt toolbox.

KW - Convergence

KW - Manopt

KW - Optimization on manifolds

KW - RTR-FD

UR - http://www.scopus.com/inward/record.url?scp=84950342083&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84950342083&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-25040-3_50

DO - 10.1007/978-3-319-25040-3_50

M3 - Conference contribution

AN - SCOPUS:84950342083

SN - 9783319250397

SN - 9783319250397

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 467

EP - 475

BT - Geometric Science of Information - 2nd International Conference, GSI 2015, Proceedings

A2 - Nielsen, Frank

A2 - Nielsen, Frank

A2 - Nielsen, Frank

A2 - Barbaresco, Frederic

A2 - Barbaresco, Frederic

A2 - Nielsen, Frank

PB - Springer Verlag

T2 - 2nd International Conference on Geometric Science of Information, GSI 2015

Y2 - 28 October 2015 through 30 October 2015

ER -