Stochastic Variance-Reduced Cubic Regularized Newton Methods

Dongruo Zhou, Pan Xu, Quanquan Gu
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:5990-5999, 2018.

Abstract

We propose a stochastic variance-reduced cubic regularized Newton method (SVRC) for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an $(\epsilon,\sqrt{\epsilon})$-approximate local minimum within $\tilde{O}(n^{4/5}/\epsilon^{3/2})$ second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.

Cite this Paper

BibTeX
@InProceedings{pmlr-v80-zhou18d, title = {Stochastic Variance-Reduced Cubic Regularized {N}ewton Methods}, author = {Zhou, Dongruo and Xu, Pan and Gu, Quanquan}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {5990--5999}, year = {2018}, editor = {Dy, Jennifer and Krause, Andreas}, volume = {80}, series = {Proceedings of Machine Learning Research}, month = {10--15 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v80/zhou18d/zhou18d.pdf}, url = {https://proceedings.mlr.press/v80/zhou18d.html}, abstract = {We propose a stochastic variance-reduced cubic regularized Newton method (SVRC) for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an $(\epsilon,\sqrt{\epsilon})$-approximate local minimum within $\tilde{O}(n^{4/5}/\epsilon^{3/2})$ second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.} }
Endnote
%0 Conference Paper %T Stochastic Variance-Reduced Cubic Regularized Newton Methods %A Dongruo Zhou %A Pan Xu %A Quanquan Gu %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-zhou18d %I PMLR %P 5990--5999 %U https://proceedings.mlr.press/v80/zhou18d.html %V 80 %X We propose a stochastic variance-reduced cubic regularized Newton method (SVRC) for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an $(\epsilon,\sqrt{\epsilon})$-approximate local minimum within $\tilde{O}(n^{4/5}/\epsilon^{3/2})$ second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.
APA
Zhou, D., Xu, P. & Gu, Q.. (2018). Stochastic Variance-Reduced Cubic Regularized Newton Methods. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:5990-5999 Available from https://proceedings.mlr.press/v80/zhou18d.html.

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