Leveraging Good Representations in Linear Contextual Bandits
This work addresses the challenge of representation selection in linear contextual bandits, which is incremental as it builds on prior definitions of good representations to propose an adaptive algorithm.
The paper tackles the problem of selecting the best linear representation in contextual bandits to minimize regret, showing that their algorithm achieves constant regret when a good representation is available and can implicitly construct one otherwise, with regret bounded by that of the best representation up to a logarithmic factor.
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.