We investigate constrained online convex optimization, in which decisions must belong to a fixed and typically complicated domain, and are required to approximately satisfy additional time-varying constraints over the long term. In this setting, the commonly used projection operations are often computationally expensive or even intractable. To avoid the time-consuming operation, several projection-free methods have been proposed with an $\mathcal{O}(T^{3/4} \sqrt{\log T})$ regret bound and an $\mathcal{O}(T^{7/8})$ cumulative constraint violation (CCV) bound for general convex losses. In this paper, we improve this result and further establish \textit{novel} regret and CCV bounds when loss functions are strongly convex. The primary idea is to first construct a composite surrogate loss, involving the original loss and constraint functions, by utilizing the Lyapunov-based technique. Then, we propose a parameter-free variant of the classical projection-free method, namely online Frank-Wolfe (OFW), and run this new extension over the online-generated surrogate loss. Theoretically, for general convex losses, we achieve an $\mathcal{O}(T^{3/4})$ regret bound and an $\mathcal{O}(T^{3/4} \log T)$ CCV bound, both of which are order-wise tighter than existing results. For strongly convex losses, we establish new guarantees of an $\mathcal{O}(T^{2/3})$ regret bound and an $\mathcal{O}(T^{5/6})$ CCV bound. Moreover, we also extend our methods to a more challenging setting with bandit feedback, obtaining similar theoretical findings. Empirically, experiments on real-world datasets have demonstrated the effectiveness of our methods.

翻译：本文研究约束在线凸优化问题，其中决策必须属于一个固定且通常复杂的域，并需长期近似满足附加的时变约束。在此设定下，常用的投影操作通常计算成本高昂甚至难以处理。为规避这一耗时操作，已有若干无投影方法被提出，针对一般凸损失函数获得了$\mathcal{O}(T^{3/4} \sqrt{\log T})$的遗憾界和$\mathcal{O}(T^{7/8})$的累积约束违反（CCV）界。本文改进了这一结果，并在损失函数为强凸时进一步建立了\textit{新颖的}遗憾界与CCV界。核心思想是首先利用基于李雅普诺夫的技术，通过结合原始损失函数与约束函数构建一个复合代理损失。随后，我们提出经典无投影方法——在线Frank-Wolfe（OFW）算法的一个无参数变体，并在在线生成的代理损失上运行这一新扩展。理论上，对于一般凸损失，我们实现了$\mathcal{O}(T^{3/4})$的遗憾界和$\mathcal{O}(T^{3/4} \log T)$的CCV界，两者在阶数上均优于现有结果。对于强凸损失，我们建立了$\mathcal{O}(T^{2/3})$的遗憾界和$\mathcal{O}(T^{5/6})$的CCV界的新保证。此外，我们还将方法扩展到更具挑战性的带反馈设定，获得了类似的理论结论。实证方面，在真实数据集上的实验验证了我们方法的有效性。