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Bandit learning algorithms typically involve the balance of exploration andexploitation. However, in many practical applications, worst-case scenariosneeding systematic exploration are seldom encountered. In this work, weconsider a smoothed setting for structured linear contextual bandits where theadversarial contexts are perturbed by Gaussian noise and the unknown parameter$\theta^*$ has structure, e.g., sparsity, group sparsity, low rank, etc. Wepropose simple greedy algorithms for both the single- and multi-parameter(i.e., different parameter for each context) settings and provide a unifiedregret analysis for $\theta^*$ with any assumed structure. The regret boundsare expressed in terms of geometric quantities such as Gaussian widthsassociated with the structure of $\theta^*$. We also obtain sharper regretbounds compared to earlier work for the unstructured $\theta^*$ setting as aconsequence of our improved analysis. We show there is implicit exploration inthe smoothed setting where a simple greedy algorithm works.

Structured Linear Contextual Bandits: A Sharp and Geometric Smoothed Analysis