In continuum-armed bandit problems where the underlying function resides in a reproducing kernel Hilbert space (RKHS), namely, the kernelised bandit problems, an important open problem remains of how well learning algorithms can adapt if the regularity of the associated kernel function is unknown. In this work, we study adaptivity to the regularity of translation-invariant kernels, which is characterized by the decay rate of the Fourier transformation of the kernel, in the bandit setting. We derive an adaptivity lower bound, proving that it is impossible to simultaneously achieve optimal cumulative regret in a pair of RKHSs with different regularities. To verify the tightness of this lower bound, we show that an existing bandit model selection algorithm applied with minimax non-adaptive kernelised bandit algorithms matches the lower bound in dependence of $T$, the total number of steps, except for log factors. By filling in the regret bounds for adaptivity between RKHSs, we connect the statistical difficulty for adaptivity in continuum-armed bandits in three fundamental types of function spaces: RKHS, Sobolev space, and H\"older space.

翻译：在连续臂赌博机问题中，当潜在函数属于再生核希尔伯特空间（RKHS）时，即所谓的基于核的赌博机问题，一个重要的未解难题在于：若相关核函数的正则性未知，学习算法如何实现有效适应。本研究探讨了赌博机背景下平移不变核正则性的适应性问题，该类核的正则性由核的傅里叶变换衰减速率刻画。我们推导出一个适应性下界，证明无法在一对具有不同正则性的RKHS中同时达到最优累积遗憾。为验证该下界的紧致性，我们表明：将现有赌博机模型选择算法与最小最大非自适应核赌博机算法相结合时，其在总步数$T$上的依赖性与下界相匹配（除对数因子外）。通过填补RKHS之间适应性的遗憾界限，我们揭示了连续臂赌博机在三种基础函数空间（RKHS、索伯列夫空间和赫尔德空间）中适应性的统计难度关联。