Online learning with kernel losses
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:971980, 2019.
Abstract
We present a generalization of the adversarial linear bandits framework, where the underlying losses are kernel functions (with an associated reproducing kernel Hilbert space) rather than linear functions. We study a version of the exponential weights algorithm and bound its regret in this setting. Under conditions on the eigendecay of the kernel we provide a sharp characterization of the regret for this algorithm. When we have polynomial eigendecay ($\mu_j \le \mathcal{O}(j^{\beta})$), we find that the regret is bounded by $\mathcal{R}_n \le \mathcal{O}(n^{\beta/2(\beta1)})$. While under the assumption of exponential eigendecay ($\mu_j \le \mathcal{O}(e^{\beta j })$) we get an even tighter bound on the regret $\mathcal{R}_n \le \tilde{\mathcal{O}}(n^{1/2})$. When the eigendecay is polynomial we also show a nonmatching minimax lower bound on the regret of $\mathcal{R}_n \ge \Omega(n^{(\beta+1)/2\beta})$ and a lower bound of $\mathcal{R}_n \ge \Omega(n^{1/2})$ when the decay in the eigenvalues is exponentially fast. We also study the full information setting when the underlying losses are kernel functions and present an adapted exponential weights algorithm and a conditional gradient descent algorithm.
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