# Variance-Reduced and Projection-Free Stochastic Optimization

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Elad Hazan,
Haipeng Luo
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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.

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