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.
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