In this paper, we explore online convex optimization (OCO) and introduce a new analysis that provides fast rates by exploiting the curvature of feasible sets. In online linear optimization, it is known that if the average gradient of loss functions is larger than a certain value, the curvature of feasible sets can be exploited by the follow-the-leader (FTL) algorithm to achieve a logarithmic regret. This paper reveals that algorithms adaptive to the curvature of loss functions can also leverage the curvature of feasible sets. We first prove that if an optimal decision is on the boundary of a feasible set and the gradient of an underlying loss function is non-zero, then the algorithm achieves a regret upper bound of $O(\rho \log T)$ in stochastic environments. Here, $\rho > 0$ is the radius of the smallest sphere that includes the optimal decision and encloses the feasible set. Our approach, unlike existing ones, can work directly with convex loss functions, exploiting the curvature of loss functions simultaneously, and can achieve the logarithmic regret only with a local property of feasible sets. Additionally, it achieves an $O(\sqrt{T})$ regret even in adversarial environments where FTL suffers an $\Omega(T)$ regret, and attains an $O(\rho \log T + \sqrt{C \rho \log T})$ regret bound in corrupted stochastic environments with corruption level $C$. Furthermore, by extending our analysis, we establish a regret upper bound of $O\Big(T^{\frac{q-2}{2(q-1)}} (\log T)^{\frac{q}{2(q-1)}}\Big)$ for $q$-uniformly convex feasible sets, where uniformly convex sets include strongly convex sets and $\ell_p$-balls for $p \in [1,\infty)$. This bound bridges the gap between the $O(\log T)$ regret bound for strongly convex sets ($q=2$) and the $O(\sqrt{T})$ regret bound for non-curved sets ($q\to\infty$).

翻译：本文研究在线凸优化（OCO），通过利用可行集的曲率提出一种新的分析方法以获得快速收敛速率。在线线性优化中，已知当损失函数平均梯度大于某个阈值时，可通过"跟随领导者"（FTL）算法利用可行集的曲率实现对数级遗憾。本文揭示，对损失函数曲率自适应的算法同样能利用可行集的曲率。我们首先证明，若最优决策位于可行集边界且潜在损失函数梯度非零，则随机环境下算法可实现$O(\rho \log T)$的遗憾上界，其中$\rho>0$为包含最优决策且外接可行集的最小球半径。与现有方法不同，本方法可直接处理凸损失函数并同时利用损失函数曲率，仅需可行集局部性质即可实现对数遗憾。此外，在FTL会遭受$\Omega(T)$遗憾的对抗环境中，本方法仍能达到$O(\sqrt{T})$遗憾；在腐蚀水平为$C$的腐蚀随机环境中，可取得$O(\rho \log T + \sqrt{C \rho \log T})$的遗憾界。通过进一步扩展分析，我们建立了$q$-均匀凸可行集（均匀凸集包含强凸集及$p \in [1,\infty)$的$\ell_p$球）的遗憾上界$O\Big(T^{\frac{q-2}{2(q-1)}} (\log T)^{\frac{q}{2(q-1)}}\Big)$。该界弥合了强凸集（$q=2$）的$O(\log T)$遗憾界与非弯曲集（$q\to\infty$）的$O(\sqrt{T})$遗憾界之间的理论鸿沟。