This paper develops projection-free algorithms for online convex optimization with stochastic constraints. We design an online primal-dual projection-free framework that can take any projection-free algorithms developed for online convex optimization with no long-term constraint. With this general template, we deduce sublinear regret and constraint violation bounds for various settings. Moreover, for the case where the loss and constraint functions are smooth, we develop a primal-dual conditional gradient method that achieves $O(\sqrt{T})$ regret and $O(T^{3/4})$ constraint violations. Furthermore, for the setting where the loss and constraint functions are stochastic and strong duality holds for the associated offline stochastic optimization problem, we prove that the constraint violation can be reduced to have the same asymptotic growth as the regret.

翻译：本文针对具有随机约束的在线凸优化问题，提出了无投影算法。我们设计了一个在线原始-对偶无投影框架，该框架可兼容任何针对无长期约束的在线凸优化所开发的无投影算法。基于这一通用模板，我们在多种场景下推导出了次线性遗憾与约束违规界。此外，当损失函数与约束函数光滑时，我们提出了一种原始-对偶条件梯度方法，实现了$O(\sqrt{T})$的遗憾和$O(T^{3/4})$的约束违规。进一步地，对于损失函数与约束函数为随机且对应离线随机优化问题满足强对偶性的设定，我们证明了约束违规可被降至与遗憾具有相同渐近增长阶。