Improved Kernel Alignment Regret Bound for Online Kernel Learning
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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((𝒜_TTlnT)^1/4) at a computational complexity (space and per-round time) of O(√(𝒜_TTlnT)), where 𝒜_T is called 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(√(𝒜_T)) at a computational complexity of O(ln^2T). Otherwise, our algorithm enjoys a regret of O((𝒜_TT)^1/4) at a computational complexity of O(√(𝒜_TT)). We extend our algorithm to batch learning and obtain a O(1/T√(𝔼[𝒜_T])) excess risk bound which improves the previous O(1/√(T)) bound.
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