A Damped Newton Method Achieves Global O(1/K²) and Local Quadratic Convergence Rate

Authors

Slavomír Hanzely, Mohamed Bin Zayed University of Artificial Intelligence
Dmitry Kamzolov, Mohamed Bin Zayed University of Artificial IntelligenceFollow
Dmitry Pasechnyuk, Mohamed Bin Zayed University of Artificial IntelligenceFollow
Alexander Gasnikov, Moscow Institute of Physics and Technology
Peter Richtárik, King Abdullah University of Science and Technology
Martin Takáč, Mohamed Bin Zayed University of Artificial IntelligenceFollow

Document Type

Conference Proceeding

Publication Title

Advances in Neural Information Processing Systems

Abstract

In this paper, we present the first stepsize schedule for Newton method resulting in fast global and local convergence guarantees. In particular, a) we prove an O (1/k2) global rate, which matches the state-of-the-art global rate of cubically regularized Newton method of Polyak and Nesterov (2006) and of regularized Newton method of Mishchenko (2021) and Doikov and Nesterov (2021), b) we prove a local quadratic rate, which matches the best-known local rate of second-order methods, and c) our stepsize formula is simple, explicit, and does not require solving any subproblem. Our convergence proofs hold under affine-invariance assumptions closely related to the notion of self-concordance. Finally, our method has competitive performance when compared to existing baselines, which share the same fast global convergence guarantees.

Publication Date

12-2022

Keywords

Damped Newton's methods, Global conver-gence, Local Convergence, Local quadratic convergence, Newton's methods, Quadratic convergence rates, Quadratic rates, Regularized Newton method, State of the art, Step size

Comments

Open Access version available on NeurIPS Proceedings

Recommended Citation

S. Hanzely et al., "A Damped Newton Method Achieves Global O(1/K²) and Local Quadratic Convergence Rate," Advances in Neural Information Processing Systems, vol. 35, Dec 2022.