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，改进了核对齐遗憾界。先前算法在计算复杂度（空间和每轮时间）为$O(\sqrt{\mathcal{A}_TT\ln{T}})$时，达到了$O((\mathcal{A}_TT\ln{T})^{\frac{1}{4}})$的遗憾界，其中$\mathcal{A}_T$称为\textit{核对齐}。我们提出了一种算法，其遗憾界和计算复杂度均优于先前结果。我们的结果依赖于核矩阵特征值的衰减率。若核矩阵特征值呈指数衰减，则算法在计算复杂度为$O(\ln^2{T})$时享有$O(\sqrt{\mathcal{A}_T})$的遗憾界。否则，算法在计算复杂度为$O(\sqrt{\mathcal{A}_TT})$时享有$O((\mathcal{A}_TT)^{\frac{1}{4}})$的遗憾界。我们将算法推广至批量学习，得到了$O(\frac{1}{T}\sqrt{\mathbb{E}[\mathcal{A}_T]})$的过剩风险界，改进了先前$O(1/\sqrt{T})$的界。