Variance-Reduced and Projection-Free Stochastic Optimization

Elad Hazan, Haipeng Luo
Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:1263-1271, 2016.

Abstract

The Frank-Wolfe optimization algorithm has recently regained popularity for machine learning applications due to its projection-free property and its ability to handle structured constraints. However, in the stochastic learning setting, it is still relatively understudied compared to the gradient descent counterpart. In this work, leveraging a recent variance reduction technique, we propose two stochastic Frank-Wolfe variants which substantially improve previous results in terms of the number of stochastic gradient evaluations needed to achieve 1-εaccuracy. For example, we improve from O(\frac1ε) to O(\ln\frac1ε) if the objective function is smooth and strongly convex, and from O(\frac1ε^2) to O(\frac1ε^1.5) if the objective function is smooth and Lipschitz. The theoretical improvement is also observed in experiments on real-world datasets for a multiclass classification application.

Cite this Paper

BibTeX
@InProceedings{pmlr-v48-hazana16, title = {Variance-Reduced and Projection-Free Stochastic Optimization}, author = {Hazan, Elad and Luo, Haipeng}, booktitle = {Proceedings of The 33rd International Conference on Machine Learning}, pages = {1263--1271}, year = {2016}, editor = {Balcan, Maria Florina and Weinberger, Kilian Q.}, volume = {48}, series = {Proceedings of Machine Learning Research}, address = {New York, New York, USA}, month = {20--22 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v48/hazana16.pdf}, url = {https://proceedings.mlr.press/v48/hazana16.html}, abstract = {The Frank-Wolfe optimization algorithm has recently regained popularity for machine learning applications due to its projection-free property and its ability to handle structured constraints. However, in the stochastic learning setting, it is still relatively understudied compared to the gradient descent counterpart. In this work, leveraging a recent variance reduction technique, we propose two stochastic Frank-Wolfe variants which substantially improve previous results in terms of the number of stochastic gradient evaluations needed to achieve 1-εaccuracy. For example, we improve from O(\frac1ε) to O(\ln\frac1ε) if the objective function is smooth and strongly convex, and from O(\frac1ε^2) to O(\frac1ε^1.5) if the objective function is smooth and Lipschitz. The theoretical improvement is also observed in experiments on real-world datasets for a multiclass classification application.} }
Endnote
%0 Conference Paper %T Variance-Reduced and Projection-Free Stochastic Optimization %A Elad Hazan %A Haipeng Luo %B Proceedings of The 33rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2016 %E Maria Florina Balcan %E Kilian Q. Weinberger %F pmlr-v48-hazana16 %I PMLR %P 1263--1271 %U https://proceedings.mlr.press/v48/hazana16.html %V 48 %X The Frank-Wolfe optimization algorithm has recently regained popularity for machine learning applications due to its projection-free property and its ability to handle structured constraints. However, in the stochastic learning setting, it is still relatively understudied compared to the gradient descent counterpart. In this work, leveraging a recent variance reduction technique, we propose two stochastic Frank-Wolfe variants which substantially improve previous results in terms of the number of stochastic gradient evaluations needed to achieve 1-εaccuracy. For example, we improve from O(\frac1ε) to O(\ln\frac1ε) if the objective function is smooth and strongly convex, and from O(\frac1ε^2) to O(\frac1ε^1.5) if the objective function is smooth and Lipschitz. The theoretical improvement is also observed in experiments on real-world datasets for a multiclass classification application.
RIS
TY - CPAPER TI - Variance-Reduced and Projection-Free Stochastic Optimization AU - Elad Hazan AU - Haipeng Luo BT - Proceedings of The 33rd International Conference on Machine Learning DA - 2016/06/11 ED - Maria Florina Balcan ED - Kilian Q. Weinberger ID - pmlr-v48-hazana16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 48 SP - 1263 EP - 1271 L1 - http://proceedings.mlr.press/v48/hazana16.pdf UR - https://proceedings.mlr.press/v48/hazana16.html AB - The Frank-Wolfe optimization algorithm has recently regained popularity for machine learning applications due to its projection-free property and its ability to handle structured constraints. However, in the stochastic learning setting, it is still relatively understudied compared to the gradient descent counterpart. In this work, leveraging a recent variance reduction technique, we propose two stochastic Frank-Wolfe variants which substantially improve previous results in terms of the number of stochastic gradient evaluations needed to achieve 1-εaccuracy. For example, we improve from O(\frac1ε) to O(\ln\frac1ε) if the objective function is smooth and strongly convex, and from O(\frac1ε^2) to O(\frac1ε^1.5) if the objective function is smooth and Lipschitz. The theoretical improvement is also observed in experiments on real-world datasets for a multiclass classification application. ER -
APA
Hazan, E. & Luo, H.. (2016). Variance-Reduced and Projection-Free Stochastic Optimization. Proceedings of The 33rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 48:1263-1271 Available from https://proceedings.mlr.press/v48/hazana16.html.

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