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In this paper, we address the stochastic contextual linear bandit problem,where a decision maker is provided a context (a random set of actions drawnfrom a distribution). The expected reward of each action is specified by theinner product of the action and an unknown parameter. The goal is to design analgorithm that learns to play as close as possible to the unknown optimalpolicy after a number of action plays. This problem is considered morechallenging than the linear bandit problem, which can be viewed as a contextualbandit problem with a \emph{fixed} context. Surprisingly, in this paper, weshow that the stochastic contextual problem can be solved as if it is a linearbandit problem. In particular, we establish a novel reduction framework thatconverts every stochastic contextual linear bandit instance to a linear banditinstance, when the context distribution is known. When the context distributionis unknown, we establish an algorithm that reduces the stochastic contextualinstance to a sequence of linear bandit instances with small misspecificationsand achieves nearly the same worst-case regret bound as the algorithm thatsolves the misspecified linear bandit instances.  As a consequence, our results imply a $O(d\sqrt{T\log T})$ high-probabilityregret bound for contextual linear bandits, making progress in resolving anopen problem in (Li et al., 2019), (Li et al., 2021).  Our reduction framework opens up a new way to approach stochastic contextuallinear bandit problems, and enables improved regret bounds in a number ofinstances including the batch setting, contextual bandits withmisspecifications, contextual bandits with sparse unknown parameters, andcontextual bandits with adversarial corruption.

Contexts can be Cheap: Solving Stochastic Contextual Bandits with Linear Bandit Algorithms