Projection-free Online Learning over Strongly Convex Sets
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866929395770327040 |
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| author | Wan, Yuanyu Zhang, Lijun |
| author_facet | Wan, Yuanyu Zhang, Lijun |
| contents | To efficiently solve online problems with complicated constraints, projection-free algorithms including online frank-wolfe (OFW) and its variants have received significant interest recently. However, in the general case, existing efficient projection-free algorithms only achieved the regret bound of $O(T^{3/4})$, which is worse than the regret of projection-based algorithms, where $T$ is the number of decision rounds. In this paper, we study the special case of online learning over strongly convex sets, for which we first prove that OFW can enjoy a better regret bound of $O(T^{2/3})$ for general convex losses. The key idea is to refine the decaying step-size in the original OFW by a simple line search rule. Furthermore, for strongly convex losses, we propose a strongly convex variant of OFW by redefining the surrogate loss function in OFW. We show that it achieves a regret bound of $O(T^{2/3})$ over general convex sets and a better regret bound of $O(\sqrt{T})$ over strongly convex sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_08177 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Projection-free Online Learning over Strongly Convex Sets Wan, Yuanyu Zhang, Lijun Machine Learning Optimization and Control To efficiently solve online problems with complicated constraints, projection-free algorithms including online frank-wolfe (OFW) and its variants have received significant interest recently. However, in the general case, existing efficient projection-free algorithms only achieved the regret bound of $O(T^{3/4})$, which is worse than the regret of projection-based algorithms, where $T$ is the number of decision rounds. In this paper, we study the special case of online learning over strongly convex sets, for which we first prove that OFW can enjoy a better regret bound of $O(T^{2/3})$ for general convex losses. The key idea is to refine the decaying step-size in the original OFW by a simple line search rule. Furthermore, for strongly convex losses, we propose a strongly convex variant of OFW by redefining the surrogate loss function in OFW. We show that it achieves a regret bound of $O(T^{2/3})$ over general convex sets and a better regret bound of $O(\sqrt{T})$ over strongly convex sets. |
| title | Projection-free Online Learning over Strongly Convex Sets |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2010.08177 |