This work studies and develop projection-free algorithms for online learning with linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, this work (i) provides an improved (optimized) variant of an online Frank-Wolfe algorithm along with its conceptually simple potential-based proof, and (ii) shows how to leverage semidefinite programming to jointly design and analyze online Frank-Wolfe-type algorithms numerically in a variety of settings-that include the design of the variant (i). Based on the semidefinite technique, we conclude with strong numerical evidence suggesting that no pure online Frank-Wolfe algorithm within our model class can have a regret guarantee better than O(T^3/4) (T is the time horizon) without additional assumptions, that the current algorithms do not have optimal constants, that the algorithm benefits from similar anytime properties O(t^3/4) not requiring to know T in advance, and that multiple linear optimization rounds do not generally help to obtain better regret bounds.

翻译：本文研究并开发了用于在线学习的无投影算法，该算法利用线性优化预言（即Frank-Wolfe方法）来处理约束集。具体而言，本文(i) 提出了一种改进（优化）的在线Frank-Wolfe算法变体，并附有概念上简单的势函数证明；(ii) 展示了如何利用半定规划在多种设置中联合设计并数值分析在线Frank-Wolfe类算法——包括变体(i)的设计。基于半定规划技术，我们得到了强有力的数值证据，表明在没有额外假设的前提下，本文模型类中的纯在线Frank-Wolfe算法无法获得优于O(T^{3/4})（T为时间范围）的遗憾界；现有算法未达到最优常数；该算法具有无需预先知晓T的类似任意时间性质O(t^{3/4})；且多次线性优化迭代通常无助于获得更优的遗憾界。