Online learning with kernel losses

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 eigendecay (μj ≤ O(j−β)), we find that the regret is bounded by Rn ≤ O(nβ/2β−1). While under the assumption of exponential eigen-decay (μj ≤ O(e−βj)) we get an even tighter bound on the regretRn ≤ Õ(n). 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.