A Stochastic PCA and SVD Algorithm with an Exponential Convergence Rate

Ohad Shamir
Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:144-152, 2015.

Abstract

We describe and analyze a simple algorithm for principal component analysis and singular value decomposition, VR-PCA, which uses computationally cheap stochastic iterations, yet converges exponentially fast to the optimal solution. In contrast, existing algorithms suffer either from slow convergence, or computationally intensive iterations whose runtime scales with the data size. The algorithm builds on a recent variance-reduced stochastic gradient technique, which was previously analyzed for strongly convex optimization, whereas here we apply it to an inherently non-convex problem, using a very different analysis.

Cite this Paper

BibTeX
@InProceedings{pmlr-v37-shamir15, title = {A Stochastic PCA and SVD Algorithm with an Exponential Convergence Rate}, author = {Shamir, Ohad}, booktitle = {Proceedings of the 32nd International Conference on Machine Learning}, pages = {144--152}, year = {2015}, editor = {Bach, Francis and Blei, David}, volume = {37}, series = {Proceedings of Machine Learning Research}, address = {Lille, France}, month = {07--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v37/shamir15.pdf}, url = {https://proceedings.mlr.press/v37/shamir15.html}, abstract = {We describe and analyze a simple algorithm for principal component analysis and singular value decomposition, VR-PCA, which uses computationally cheap stochastic iterations, yet converges exponentially fast to the optimal solution. In contrast, existing algorithms suffer either from slow convergence, or computationally intensive iterations whose runtime scales with the data size. The algorithm builds on a recent variance-reduced stochastic gradient technique, which was previously analyzed for strongly convex optimization, whereas here we apply it to an inherently non-convex problem, using a very different analysis.} }
Endnote
%0 Conference Paper %T A Stochastic PCA and SVD Algorithm with an Exponential Convergence Rate %A Ohad Shamir %B Proceedings of the 32nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2015 %E Francis Bach %E David Blei %F pmlr-v37-shamir15 %I PMLR %P 144--152 %U https://proceedings.mlr.press/v37/shamir15.html %V 37 %X We describe and analyze a simple algorithm for principal component analysis and singular value decomposition, VR-PCA, which uses computationally cheap stochastic iterations, yet converges exponentially fast to the optimal solution. In contrast, existing algorithms suffer either from slow convergence, or computationally intensive iterations whose runtime scales with the data size. The algorithm builds on a recent variance-reduced stochastic gradient technique, which was previously analyzed for strongly convex optimization, whereas here we apply it to an inherently non-convex problem, using a very different analysis.
RIS
TY - CPAPER TI - A Stochastic PCA and SVD Algorithm with an Exponential Convergence Rate AU - Ohad Shamir BT - Proceedings of the 32nd International Conference on Machine Learning DA - 2015/06/01 ED - Francis Bach ED - David Blei ID - pmlr-v37-shamir15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 37 SP - 144 EP - 152 L1 - http://proceedings.mlr.press/v37/shamir15.pdf UR - https://proceedings.mlr.press/v37/shamir15.html AB - We describe and analyze a simple algorithm for principal component analysis and singular value decomposition, VR-PCA, which uses computationally cheap stochastic iterations, yet converges exponentially fast to the optimal solution. In contrast, existing algorithms suffer either from slow convergence, or computationally intensive iterations whose runtime scales with the data size. The algorithm builds on a recent variance-reduced stochastic gradient technique, which was previously analyzed for strongly convex optimization, whereas here we apply it to an inherently non-convex problem, using a very different analysis. ER -
APA
Shamir, O.. (2015). A Stochastic PCA and SVD Algorithm with an Exponential Convergence Rate. Proceedings of the 32nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 37:144-152 Available from https://proceedings.mlr.press/v37/shamir15.html.

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