Cubic Regularization with Momentum for Nonconvex Optimization

Zhe Wang, Yi Zhou, Yingbin Liang, Guanghui Lan
Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, PMLR 115:313-322, 2020.

Abstract

Momentum is a popular technique to accelerate the convergence in practical training, and its impact on convergence guarantee has been well-studied for first-order algorithms. However, such a successful acceleration technique has not yet been proposed for second-order algorithms in nonconvex optimization. In this paper, we apply the momentum scheme to cubic regularized (CR) Newton’s method and explore the potential for acceleration. Our numerical experiments on various nonconvex optimization problems demonstrate that the momentum scheme can substantially facilitate the convergence of cubic regularization, and perform even better than the Nesterov’s acceleration scheme for CR. Theoretically, we prove that CR under momentum achieves the best possible convergence rate to a second-order stationary point for nonconvex optimization. Moreover, we study the proposed algorithm for solving problems satisfying an error bound condition and establish a local quadratic convergence rate. Then, particularly for finite-sum problems, we show that the proposed algorithm can allow computational inexactness that reduces the overall sample complexity without degrading the convergence rate.

Cite this Paper

BibTeX
@InProceedings{pmlr-v115-wang20c, title = {Cubic Regularization with Momentum for Nonconvex Optimization}, author = {Wang, Zhe and Zhou, Yi and Liang, Yingbin and Lan, Guanghui}, booktitle = {Proceedings of The 35th Uncertainty in Artificial Intelligence Conference}, pages = {313--322}, year = {2020}, editor = {Adams, Ryan P. and Gogate, Vibhav}, volume = {115}, series = {Proceedings of Machine Learning Research}, month = {22--25 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v115/wang20c/wang20c.pdf}, url = {https://proceedings.mlr.press/v115/wang20c.html}, abstract = {Momentum is a popular technique to accelerate the convergence in practical training, and its impact on convergence guarantee has been well-studied for first-order algorithms. However, such a successful acceleration technique has not yet been proposed for second-order algorithms in nonconvex optimization. In this paper, we apply the momentum scheme to cubic regularized (CR) Newton’s method and explore the potential for acceleration. Our numerical experiments on various nonconvex optimization problems demonstrate that the momentum scheme can substantially facilitate the convergence of cubic regularization, and perform even better than the Nesterov’s acceleration scheme for CR. Theoretically, we prove that CR under momentum achieves the best possible convergence rate to a second-order stationary point for nonconvex optimization. Moreover, we study the proposed algorithm for solving problems satisfying an error bound condition and establish a local quadratic convergence rate. Then, particularly for finite-sum problems, we show that the proposed algorithm can allow computational inexactness that reduces the overall sample complexity without degrading the convergence rate.} }
Endnote
%0 Conference Paper %T Cubic Regularization with Momentum for Nonconvex Optimization %A Zhe Wang %A Yi Zhou %A Yingbin Liang %A Guanghui Lan %B Proceedings of The 35th Uncertainty in Artificial Intelligence Conference %C Proceedings of Machine Learning Research %D 2020 %E Ryan P. Adams %E Vibhav Gogate %F pmlr-v115-wang20c %I PMLR %P 313--322 %U https://proceedings.mlr.press/v115/wang20c.html %V 115 %X Momentum is a popular technique to accelerate the convergence in practical training, and its impact on convergence guarantee has been well-studied for first-order algorithms. However, such a successful acceleration technique has not yet been proposed for second-order algorithms in nonconvex optimization. In this paper, we apply the momentum scheme to cubic regularized (CR) Newton’s method and explore the potential for acceleration. Our numerical experiments on various nonconvex optimization problems demonstrate that the momentum scheme can substantially facilitate the convergence of cubic regularization, and perform even better than the Nesterov’s acceleration scheme for CR. Theoretically, we prove that CR under momentum achieves the best possible convergence rate to a second-order stationary point for nonconvex optimization. Moreover, we study the proposed algorithm for solving problems satisfying an error bound condition and establish a local quadratic convergence rate. Then, particularly for finite-sum problems, we show that the proposed algorithm can allow computational inexactness that reduces the overall sample complexity without degrading the convergence rate.
APA
Wang, Z., Zhou, Y., Liang, Y. & Lan, G.. (2020). Cubic Regularization with Momentum for Nonconvex Optimization. Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, in Proceedings of Machine Learning Research 115:313-322 Available from https://proceedings.mlr.press/v115/wang20c.html.

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