Riemannian Stochastic Recursive Gradient Algorithm

Hiroyuki Kasai, Hiroyuki Sato, Bamdev Mishra
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2516-2524, 2018.

Abstract

Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions on a Riemannian manifold. The present paper proposes a Riemannian stochastic recursive gradient algorithm (R-SRG), which does not require the inverse of retraction between two distant iterates on the manifold. Convergence analyses of R-SRG are performed on both retraction-convex and non-convex functions under computationally efficient retraction and vector transport operations. The key challenge is analysis of the influence of vector transport along the retraction curve. Numerical evaluations reveal that R-SRG competes well with state-of-the-art Riemannian batch and stochastic gradient algorithms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v80-kasai18a, title = {{R}iemannian Stochastic Recursive Gradient Algorithm}, author = {Kasai, Hiroyuki and Sato, Hiroyuki and Mishra, Bamdev}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {2516--2524}, 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/kasai18a/kasai18a.pdf}, url = {https://proceedings.mlr.press/v80/kasai18a.html}, abstract = {Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions on a Riemannian manifold. The present paper proposes a Riemannian stochastic recursive gradient algorithm (R-SRG), which does not require the inverse of retraction between two distant iterates on the manifold. Convergence analyses of R-SRG are performed on both retraction-convex and non-convex functions under computationally efficient retraction and vector transport operations. The key challenge is analysis of the influence of vector transport along the retraction curve. Numerical evaluations reveal that R-SRG competes well with state-of-the-art Riemannian batch and stochastic gradient algorithms.} }
Endnote
%0 Conference Paper %T Riemannian Stochastic Recursive Gradient Algorithm %A Hiroyuki Kasai %A Hiroyuki Sato %A Bamdev Mishra %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-kasai18a %I PMLR %P 2516--2524 %U https://proceedings.mlr.press/v80/kasai18a.html %V 80 %X Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions on a Riemannian manifold. The present paper proposes a Riemannian stochastic recursive gradient algorithm (R-SRG), which does not require the inverse of retraction between two distant iterates on the manifold. Convergence analyses of R-SRG are performed on both retraction-convex and non-convex functions under computationally efficient retraction and vector transport operations. The key challenge is analysis of the influence of vector transport along the retraction curve. Numerical evaluations reveal that R-SRG competes well with state-of-the-art Riemannian batch and stochastic gradient algorithms.
APA
Kasai, H., Sato, H. & Mishra, B.. (2018). Riemannian Stochastic Recursive Gradient Algorithm. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:2516-2524 Available from https://proceedings.mlr.press/v80/kasai18a.html.

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