Structured linear contextual bandits: A sharp and geometric smoothed analysis

Bandit learning algorithms typically involve the balance of exploration and exploitation. However, in many practical applications, worst-case scenarios needing systematic exploration are seldom encountered. In this work, we consider a smoothed setting for structured linear contextual bandits where the adversarial contexts are perturbed by Gaussian noise and the unknown parameter has structure, e.g., sparsity, group sparsity, low rank, etc. We propose simple greedy algorithms for both the single- A nd multi-parameter (i.e., different parameter for each context) settings and provide a unified regret analysis for ? with any assumed structure. The regret bounds are expressed in terms of geometric quantities such as Gaussian widths associated with the structure of ?. We also obtain sharper regret bounds compared to earlier work for the unstructured setting as a consequence of our improved analysis. We show there is implicit exploration in the smoothed setting where a simple greedy algorithm works.