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We consider the stochastic contextual bandit problem under the highdimensional linear model. We focus on the case where the action space is finiteand random, with each action associated with a randomly generated contextualcovariate. This setting finds essential applications such as personalizedrecommendation, online advertisement, and personalized medicine. However, it isvery challenging as we need to balance exploration and exploitation. We proposedoubly growing epochs and estimating the parameter using the best subsetselection method, which is easy to implement in practice. This approachachieves $ \tilde{\mathcal{O}}(s\sqrt{T})$ regret with high probability, whichis nearly independent in the ``ambient'' regression model dimension $d$. Wefurther attain a sharper $\tilde{\mathcal{O}}(\sqrt{sT})$ regret by using the\textsc{SupLinUCB} framework and match the minimax lower bound oflow-dimensional linear stochastic bandit problems. Finally, we conductextensive numerical experiments to demonstrate the applicability and robustnessof our algorithms empirically.

Nearly Dimension-Independent Sparse Linear Bandit over Small Action Spaces via Best Subset Selection