Exploiting Problem Geometry in Safe Linear Bandits

The safe linear bandit problem is a version of the classic linear bandit problem where the learner's actions must satisfy an uncertain linear constraint at all rounds. Due its applicability to many real-world settings, this problem has received considerable attention in recent years. We find that by exploiting the geometry of the specific problem setting, we can achieve improved regret guarantees for both well-separated problem instances and action sets that are finite star convex sets. Additionally, we propose a novel algorithm for this setting that chooses problem parameters adaptively and enjoys at least as good regret guarantees as existing algorithms. Lastly, we introduce a generalization of the safe linear bandit setting where the constraints are convex and adapt our algorithms and analyses to this setting by leveraging a novel convex-analysis based approach. Simulation results show improved performance over existing algorithms for a variety of randomly sampled settings.