Structured Projection-free Online Convex Optimization with Multi-point Bandit Feedback

We consider structured online convex optimization (OCO) with bandit feedback, where either the loss function is smooth or the constraint set is strongly convex. Projection-free methods are among the most popular and computationally efficient algorithms for solving this problem, mainly due to their ability to handle convex constraints appearing in machine learning for which computing projections is often impractical in high-dimensional settings. Despite the improved regret bound results for the full-information setting where the gradients of the functions are readily available, it remains unclear whether simple projection-free zero-order algorithms become more efficient for structured OCO problems in the case when multiple function values can be sampled at each time instance. In this paper, we develop some simple projection-free algorithms and prove that they indeed achieve the same improved regret bounds as the full-information case under various additional problem structures. This implies that leveraging the structural properties of the problem compensates for the lack of access to the gradients. Experiments on the online matrix completion reveal several attractive advantages of the proposed algorithms, including their simplicity, easy implementation, and effectiveness, as they outperform other competing algorithms.