### Poster

## Online learning with kernel losses

### Niladri Chatterji · Aldo Pacchiano · Peter Bartlett

##### Pacific Ballroom #185

Keywords: [ Bandits ] [ Computational Learning Theory ] [ Online Learning ] [ Statistical Learning Theory ]

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Abstract
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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 \emph{non-matching} 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 eigen-values 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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