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

Slavomír Hanzely; Dmitry Kamzolov; Dmitry Pasechnyuk; Alexander Gasnikov; Peter Richtárik; Martin Takáč

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

Slavomír Hanzely, Dmitry Kamzolov, Dmitry Pasechnyuk, Alexander Gasnikov, Peter Richtárik, Martin Takáč

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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.

Original language	English (US)
Title of host publication	Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022
Editors	S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, A. Oh
Publisher	Neural information processing systems foundation
ISBN (Electronic)	9781713871088
State	Published - 2022
Event	36th Conference on Neural Information Processing Systems, NeurIPS 2022 - New Orleans, United States Duration: Nov 28 2022 → Dec 9 2022

Publication series

Name	Advances in Neural Information Processing Systems
Volume	35
ISSN (Print)	1049-5258

Conference

Conference	36th Conference on Neural Information Processing Systems, NeurIPS 2022
Country/Territory	United States
City	New Orleans
Period	11/28/22 → 12/9/22

Bibliographical note

Funding Information:
The work of D. Pasechnyuk and A. Gasnikov was supported by a grant for research centers in the field of artificial intelligence, provided by the Analytical Center for the Government of the RF in accordance with the subsidy agreement (agreement identifier 000000D730321P5Q0002) and the agreement with the Ivannikov Institute for System Programming of the RAS dated November 2, 2021 No. 70-2021-00142.

Publisher Copyright:
© 2022 Neural information processing systems foundation. All rights reserved.

ASJC Scopus subject areas

Computer Networks and Communications
Information Systems
Signal Processing

Cite this

Hanzely, S., Kamzolov, D., Pasechnyuk, D., Gasnikov, A., Richtárik, P., & Takáč, M. (2022). A Damped Newton Method Achieves Global O(1/K²) and Local Quadratic Convergence Rate. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, & A. Oh (Eds.), Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022 (Advances in Neural Information Processing Systems; Vol. 35). Neural information processing systems foundation.

Hanzely, Slavomír ; Kamzolov, Dmitry ; Pasechnyuk, Dmitry et al. / A Damped Newton Method Achieves Global O(1/K²) and Local Quadratic Convergence Rate. Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. editor / S. Koyejo ; S. Mohamed ; A. Agarwal ; D. Belgrave ; K. Cho ; A. Oh. Neural information processing systems foundation, 2022. (Advances in Neural Information Processing Systems).

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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.",

author = "Slavom{\'i}r Hanzely and Dmitry Kamzolov and Dmitry Pasechnyuk and Alexander Gasnikov and Peter Richt{\'a}rik and Martin Tak{\'a}{\v c}",

note = "Funding Information: The work of D. Pasechnyuk and A. Gasnikov was supported by a grant for research centers in the field of artificial intelligence, provided by the Analytical Center for the Government of the RF in accordance with the subsidy agreement (agreement identifier 000000D730321P5Q0002) and the agreement with the Ivannikov Institute for System Programming of the RAS dated November 2, 2021 No. 70-2021-00142. Publisher Copyright: {\textcopyright} 2022 Neural information processing systems foundation. All rights reserved.; 36th Conference on Neural Information Processing Systems, NeurIPS 2022 ; Conference date: 28-11-2022 Through 09-12-2022",

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Hanzely, S, Kamzolov, D, Pasechnyuk, D, Gasnikov, A, Richtárik, P & Takáč, M 2022, A Damped Newton Method Achieves Global O(1/K²) and Local Quadratic Convergence Rate. in S Koyejo, S Mohamed, A Agarwal, D Belgrave, K Cho & A Oh (eds), Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. Advances in Neural Information Processing Systems, vol. 35, Neural information processing systems foundation, 36th Conference on Neural Information Processing Systems, NeurIPS 2022, New Orleans, United States, 11/28/22.

A Damped Newton Method Achieves Global O(1/K²) and Local Quadratic Convergence Rate. / Hanzely, Slavomír; Kamzolov, Dmitry; Pasechnyuk, Dmitry et al.
Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. ed. / S. Koyejo; S. Mohamed; A. Agarwal; D. Belgrave; K. Cho; A. Oh. Neural information processing systems foundation, 2022. (Advances in Neural Information Processing Systems; Vol. 35).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Takáč, Martin

N1 - Funding Information: The work of D. Pasechnyuk and A. Gasnikov was supported by a grant for research centers in the field of artificial intelligence, provided by the Analytical Center for the Government of the RF in accordance with the subsidy agreement (agreement identifier 000000D730321P5Q0002) and the agreement with the Ivannikov Institute for System Programming of the RAS dated November 2, 2021 No. 70-2021-00142. Publisher Copyright: © 2022 Neural information processing systems foundation. All rights reserved.

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N2 - 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.

AB - 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.

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Hanzely S, Kamzolov D, Pasechnyuk D, Gasnikov A, Richtárik P, Takáč M. A Damped Newton Method Achieves Global O(1/K²) and Local Quadratic Convergence Rate. In Koyejo S, Mohamed S, Agarwal A, Belgrave D, Cho K, Oh A, editors, Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. Neural information processing systems foundation. 2022. (Advances in Neural Information Processing Systems).