In this paper, we improve the regret bound for online kernel selection under bandit feedback. Previous algorithm enjoys a $O((\Vert f\Vert^2_{\mathcal{H}_i}+1)K^{\frac{1}{3}}T^{\frac{2}{3}})$ expected bound for Lipschitz loss functions. We prove two types of regret bounds improving the previous bound. For smooth loss functions, we propose an algorithm with a $O(U^{\frac{2}{3}}K^{-\frac{1}{3}}(\sum^K_{i=1}L_T(f^\ast_i))^{\frac{2}{3}})$ expected bound where $L_T(f^\ast_i)$ is the cumulative losses of optimal hypothesis in $\mathbb{H}_{i}=\{f\in\mathcal{H}_i:\Vert f\Vert_{\mathcal{H}_i}\leq U\}$. The data-dependent bound keeps the previous worst-case bound and is smaller if most of candidate kernels match well with the data. For Lipschitz loss functions, we propose an algorithm with a $O(U\sqrt{KT}\ln^{\frac{2}{3}}{T})$ expected bound asymptotically improving the previous bound. We apply the two algorithms to online kernel selection with time constraint and prove new regret bounds matching or improving the previous $O(\sqrt{T\ln{K}} +\Vert f\Vert^2_{\mathcal{H}_i}\max\{\sqrt{T},\frac{T}{\sqrt{\mathcal{R}}}\})$ expected bound where $\mathcal{R}$ is the time budget. Finally, we empirically verify our algorithms on online regression and classification tasks.

翻译：本文改进了基于Bandit反馈的在线核选择的遗憾界。先前算法对于Lipschitz损失函数具有$O((\Vert f\Vert^2_{\mathcal{H}_i}+1)K^{\frac{1}{3}}T^{\frac{2}{3}})$的期望界。我们证明了两种类型的遗憾界改进了先前结果。对于光滑损失函数，我们提出了一种具有$O(U^{\frac{2}{3}}K^{-\frac{1}{3}}(\sum^K_{i=1}L_T(f^\ast_i))^{\frac{2}{3}})$期望界的算法，其中$L_T(f^\ast_i)$是$\mathbb{H}_{i}=\{f\in\mathcal{H}_i:\Vert f\Vert_{\mathcal{H}_i}\leq U\}$中最优假设的累积损失。该数据依赖界保持了先前最坏情况界，且当大多数候选核与数据匹配良好时更小。对于Lipschitz损失函数，我们提出了一种具有$O(U\sqrt{KT}\ln^{\frac{2}{3}}{T})$期望界的算法，渐近改进了先前界。将两种算法应用于具有时间约束的在线核选择，并证明了新的遗憾界，匹配或改进了先前$O(\sqrt{T\ln{K}} +\Vert f\Vert^2_{\mathcal{H}_i}\max\{\sqrt{T},\frac{T}{\sqrt{\mathcal{R}}}\})$的期望界，其中$\mathcal{R}$是时间预算。最后，我们通过在线回归和分类任务实验验证了算法有效性。