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# Improved Kernel Alignment Regret Bound for Online Kernel Learning

Dec 2022

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

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