An efficient high-probability algorithm for Linear Bandits

For the linear bandit problem, we extend the analysis of algorithm CombEXP from Combes et al. [2015] to the high-probability case against adaptive adversaries, allowing actions to come from an arbitrary polytope. We prove a high-probability regret of O(T2/3) for time horizon T. While this bound is weaker than the optimal O( √ T) bound achieved by GeometricHedge in Bartlett et al. [2008], CombEXP is computationally efficient, requiring only an efficient linear optimization oracle over the convex hull of the actions.