Online learning with kernel losses

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 eigen-decay of the kernel we provide a sharp characterization of the regret for this algorithm. When we have polynomial eigen-decay ($\mu_j \le \mathcal{O}(j^{-\beta})$), we find that the regret is bounded by $\mathcal{R}_n \le \mathcal{O}(n^{\beta/2(\beta-1)})$. While under the assumption of exponential eigen-decay ($\mu_j \le \mathcal{O}(e^{-\beta j })$) we get an even tighter bound on the regret $\mathcal{R}_n \le \tilde{\mathcal{O}}(n^{1/2})$. When the eigen-decay is polynomial we also show a