In this paper, we improve the kernel alignment regret bound for online kernel learning in the regime of the Hinge loss function. Previous algorithm achieves a regret of $O((\mathcal{A}_TT\ln{T})^{\frac{1}{4}})$ at a computational complexity (space and per-round time) of $O(\sqrt{\mathcal{A}_TT\ln{T}})$, where $\mathcal{A}_T$ is called \textit{kernel alignment}. We propose an algorithm whose regret bound and computational complexity are better than previous results. Our results depend on the decay rate of eigenvalues of the kernel matrix. If the eigenvalues of the kernel matrix decay exponentially, then our algorithm enjoys a regret of $O(\sqrt{\mathcal{A}_T})$ at a computational complexity of $O(\ln^2{T})$. Otherwise, our algorithm enjoys a regret of $O((\mathcal{A}_TT)^{\frac{1}{4}})$ at a computational complexity of $O(\sqrt{\mathcal{A}_TT})$. We extend our algorithm to batch learning and obtain a $O(\frac{1}{T}\sqrt{\mathbb{E}[\mathcal{A}_T]})$ excess risk bound which improves the previous $O(1/\sqrt{T})$ bound.

翻译：本文针对Hinge损失函数下的在线核学习，提出了改进的核对齐遗憾界。现有算法在计算复杂度（空间和每轮时间）为$O(\sqrt{\mathcal{A}_TT\ln{T}})$时，实现了$O((\mathcal{A}_TT\ln{T})^{\frac{1}{4}})$的遗憾值，其中$\mathcal{A}_T$称为\textit{核对齐}。我们提出的算法在遗憾界和计算复杂度上均优于现有结果。我们的结果依赖于核矩阵特征值的衰减速率。若核矩阵特征值呈指数衰减，则该算法在计算复杂度为$O(\ln^2{T})$时，遗憾值为$O(\sqrt{\mathcal{A}_T})$。否则，算法在计算复杂度为$O(\sqrt{\mathcal{A}_TT})$时，遗憾值为$O((\mathcal{A}_TT)^{\frac{1}{4}})$。我们将算法扩展到批量学习场景，并得到$O(\frac{1}{T}\sqrt{\mathbb{E}[\mathcal{A}_T]})$的超额风险界，优于之前的$O(1/\sqrt{T})$界。