In this paper, we introduce a new projection-free algorithm for Online Convex Optimization (OCO) with a state-of-the-art regret guarantee among separation-based algorithms. Existing projection-free methods based on the classical Frank-Wolfe algorithm achieve a suboptimal regret bound of $O(T^{3/4})$, while more recent separation-based approaches guarantee a regret bound of $O(\kappa \sqrt{T})$, where $\kappa$ denotes the asphericity of the feasible set, defined as the ratio of the radii of the containing and contained balls. However, for ill-conditioned sets, $\kappa$ can be arbitrarily large, potentially leading to poor performance. Our algorithm achieves a regret bound of $\widetilde{O}(\sqrt{dT} + \kappa d)$, while requiring only $\widetilde{O}(1)$ calls to a separation oracle per round. Crucially, the main term in the bound, $\widetilde{O}(\sqrt{d T})$, is independent of $\kappa$, addressing the limitations of previous methods. Additionally, as a by-product of our analysis, we recover the $O(\kappa \sqrt{T})$ regret bound of existing OCO algorithms with a more straightforward analysis and improve the regret bound for projection-free online exp-concave optimization. Finally, for constrained stochastic convex optimization, we achieve a state-of-the-art convergence rate of $\widetilde{O}(\sigma/\sqrt{T} + \kappa d/T)$, where $\sigma$ represents the noise in the stochastic gradients, while requiring only $\widetilde{O}(1)$ calls to a separation oracle per iteration.

翻译：本文提出了一种新的无投影算法，用于基于分离预言机的在线凸优化（OCO），其在同类分离基算法中实现了最先进的遗憾界。基于经典Frank-Wolfe算法的现有无投影方法仅能达到次优的$O(T^{3/4})$遗憾界，而更近期的分离基方法可保证$O(\kappa \sqrt{T})$的遗憾界，其中$\kappa$表示可行集的非球性，定义为包含球与被包含球半径之比。然而，对于病态集合，$\kappa$可能任意大，从而导致性能不佳。我们的算法实现了$\widetilde{O}(\sqrt{dT} + \kappa d)$的遗憾界，且每轮仅需$\widetilde{O}(1)$次分离预言机调用。关键的是，界中的主要项$\widetilde{O}(\sqrt{d T})$独立于$\kappa$，从而解决了先前方法的局限性。此外，作为我们分析的副产品，我们以更简洁的分析恢复了现有OCO算法的$O(\kappa \sqrt{T})$遗憾界，并改进了无投影在线指数凹优化的遗憾界。最后，对于约束随机凸优化，我们实现了$\widetilde{O}(\sigma/\sqrt{T} + \kappa d/T)$的最先进收敛率，其中$\sigma$表示随机梯度中的噪声，且每次迭代仅需$\widetilde{O}(1)$次分离预言机调用。