Online learning with kernel losses
This work addresses online learning challenges for scenarios with non-linear losses, offering theoretical guarantees that could benefit machine learning practitioners in sequential decision-making, though it appears incremental as it extends existing frameworks.
The paper tackles the problem of online learning with kernel losses by generalizing adversarial linear bandits to kernel functions, and it provides regret bounds for an exponential weights algorithm, achieving rates such as O(n^{β/(2(β-1))}) for polynomial eigendecay and O(n^{1/2}log(n)^{1/2}) for exponential eigendecay.
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 $μ_j \le \mathcal{O}(j^{-β})$, we find that the regret is bounded by $\mathcal{R}_n \le \mathcal{O}(n^{β/(2(β-1))})$; while under the assumption of exponential eigendecay $μ_j \le \mathcal{O}(e^{-βj })$, we get an even tighter bound on the regret $\mathcal{R}_n \le \mathcal{O}(n^{1/2}\log(n)^{1/2})$. 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.