SecondOrder Kernel Online Convex Optimization with Adaptive Sketching
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Proceedings of the 34th International Conference on Machine Learning, PMLR 70:645653, 2017.
Abstract
Kernel online convex optimization (KOCO) is a framework combining the expressiveness of nonparametric kernel models with the regret guarantees of online learning. Firstorder KOCO methods such as functional gradient descent require only $O(t)$ time and space per iteration, and, when the only information on the losses is their convexity, achieve a minimax optimal $O(\sqrt{T})$ regret. Nonetheless, many common losses in kernel problems, such as squared loss, logistic loss, and squared hinge loss posses stronger curvature that can be exploited. In this case, secondorder KOCO methods achieve $O(\log(\mathrm{Det}(K)))$ regret, which we show scales as $O(deff \log T)$, where $deff$ is the effective dimension of the problem and is usually much smaller than $O(\sqrt{T})$. The main drawback of secondorder methods is their much higher $O(t^2)$ space and time complexity. In this paper, we introduce kernel online Newton step (KONS), a new secondorder KOCO method that also achieves $O(deff\log T)$ regret. To address the computational complexity of secondorder methods, we introduce a new matrix sketching algorithm for the kernel matrix~$K$, and show that for a chosen parameter $\gamma \leq 1$ our SketchedKONS reduces the space and time complexity by a factor of $\gamma^2$ to $O(t^2\gamma^2)$ space and time per iteration, while incurring only $1/\gamma$ times more regret.
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