The linear contextual bandit literature is mostly focused on the design of efficient learning algorithms for a given representation. However, a contextual bandit problem may admit multiple linear representations, each one with different characteristics that directly impact the regret of the learning algorithm. In particular, recent works showed that there exist "good" representations for which constant problem-dependent regret can be achieved. In this paper, we first provide a systematic analysis of the different definitions of "good" representations proposed in the literature. We then propose a novel selection algorithm able to adapt to the best representation in a set of $M$ candidates. We show that the regret is indeed never worse than the regret obtained by running LinUCB on the best representation (up to a $\ln M$ factor). As a result, our algorithm achieves constant regret whenever a "good" representation is available in the set. Furthermore, we show that the algorithm may still achieve constant regret by implicitly constructing a "good" representation, even when none of the initial representations is "good". Finally, we empirically validate our theoretical findings in a number of standard contextual bandit problems.

翻译：线性背景土匪文献主要侧重于设计针对特定代表的高效学习算法。 但是,背景土匪问题可能承认多个线性表述,每个具有不同特征的表述都直接影响到学习算法的遗憾。 特别是,最近的工作表明,存在“ 良好” 的表述,可以持续地因问题而感到遗憾。 在本文中,我们首先对文献中提议的“良好”表述的不同定义进行系统分析,然后我们提出一种新的选择算法,能够适应一组美元候选人中的最佳表述。我们表明,在最佳代表法上运行LinUCB所获得的遗憾(最高达1美元要素 ), 确实不会比运行LinUCB获得的遗憾更糟糕。 结果,当“良好”表达法在集中出现时,我们的算法总是会后悔。 此外,我们表明,即使最初的表述没有“良好”表述,我们还是可以通过隐含地构建一个“良好”的表达法来实现持续遗憾, 即使最初的表述没有“良好” 。 最后,我们从经验上证实我们在一些标准背景部落问题的理论结论。