Variance-Reduced and Projection-Free Stochastic Optimization
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.
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.}
}