# On the Global Linear Convergence of Frank-Wolfe Optimization Variants

@inproceedings{LacosteJulien2015OnTG, title={On the Global Linear Convergence of Frank-Wolfe Optimization Variants}, author={Simon Lacoste-Julien and Martin Jaggi}, booktitle={NIPS}, year={2015} }

The Frank-Wolfe (FW) optimization algorithm has lately re-gained popularity thanks in particular to its ability to nicely handle the structured constraints appearing in machine learning applications. However, its convergence rate is known to be slow (sublinear) when the solution lies at the boundary. A simple less-known fix is to add the possibility to take 'away steps' during optimization, an operation that importantly does not require a feasibility oracle. In this paper, we highlight and…

## 309 Citations

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A unified convergence analysis for FW algorithm and its variants under the setting of nonconvex but smooth objective with a convex, compact constraint set and a novel observation on the so-called Frank-Wolfe gap is presented.

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Variants of Away-steps and Pairwise FW that lift both restrictions simultaneously and inherit all the favorable convergence properties of the exact line-search version, including linear convergence for strongly convex functions over polytopes are proposed.

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- Computer Science, MathematicsNeurIPS
- 2020

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- Computer ScienceICML
- 2018

Two novel randomized variants of the Frank-Wolfe (FW) or conditional gradient algorithm are analyzed, one of which achieves a sublinear convergence rate as in the deterministic counterpart, and the other reaches linear (i.e., exponential) convergence rate making it the first provably convergent randomized variant of Away-step FW.

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- Computer ScienceICML
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One Sample Stochastic Frank-Wolfe

- Computer ScienceAISTATS
- 2020

This paper proposes the first one-sample stochastic Frank-Wolfe algorithm, called 1-SFW, that avoids the need to carefully tune the batch size, step size, learning rate, and other complicated hyper parameters, and achieves the optimal convergence rate of $\mathcal{O}(1/\epsilon^2)$.

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- Computer Science
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A general setting for which Frank-Wolfe algorithms with open loop step-size rules converges non-asymptotically faster than with line search or short-step is characterized, several accelerated convergence results for FW are derived, and potential gaps are highlighted in current understanding of the FW method in general.

Frank-Wolfe Method is Automatically Adaptive to Error Bound Condition

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