  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.
