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The trade-off between regret and computational cost is a fundamental problemfor online kernel regression, and previous algorithms worked on the trade-offcan not keep optimal regret bounds at a sublinear computational complexity. Inthis paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which cankeep nearly optimal regret bounds at a sublinear computational complexity, andgive sufficient conditions under which our algorithms work. Both algorithmsdynamically maintain a group of nearly orthogonal basis used to approximate thekernel mapping, and keep nearly optimal regret bounds by controlling theapproximate error. The number of basis depends on the approximate error and thedecay rate of eigenvalues of the kernel matrix. If the eigenvalues decayexponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of$O(\sqrt{L(f)})$ and $O(\mathrm{d}_{\mathrm{eff}}(\mu)\ln{T})$ at acomputational complexity in $O(\ln^2{T})$. If the eigenvalues decaypolynomially with degree $p\geq 1$, then our algorithms keep the same regretbounds at a computational complexity in $o(T)$ in the case of $p>4$ and $p\geq10$, respectively. $L(f)$ is the cumulative losses of $f$ and$\mathrm{d}_{\mathrm{eff}}(\mu)$ is the effective dimension of the problem. Thetwo regret bounds are nearly optimal and are not comparable.

Nearly Optimal Algorithms with Sublinear Computational Complexity for Online Kernel Regression