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Thompson sampling is a heuristic algorithm for the multi-armed bandit problem which has a long tradition in machine learning. The algorithm has a Bayesian spirit in the sense that it selects arms based on posterior samples of reward probabilities of each arm. By forging a connection between combinatorial binary bandits and spike-and-slab variable selection, we propose a stochastic optimization approach to subset selection called Thompson Variable Selection (TVS). TVS is a framework for interpretable machine learning which does not rely on the underlying model to be linear. TVS brings together Bayesian reinforcement and machine learning in order to extend the reach of Bayesian subset selection to non-parametric models and large datasets with very many predictors and/or very many observations. Depending on the choice of a reward, TVS can be deployed in offline as well as online setups with streaming data batches. Tailoring multiplay bandits to variable selection, we provide regret bounds without necessarily assuming that the arm mean rewards be unrelated. We show a very strong empirical performance on both simulated and real data. Unlike deterministic optimization methods for spike-and-slab variable selection, the stochastic nature makes TVS less prone to local convergence and thereby more robust.